MATERIAL MODELING OF STRAIN RATE DEPENDENT POLYMER AND 2D TRI-AXIALLY BRAIDED COMPOSITES. A Dissertation. Presented to

Transcription

1 MATERIAL MODELING OF STRAIN RATE DEPENDENT POLYMER AND 2D TRI-AXIALLY BRAIDED COMPOSITES A Dissertation Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Jingyun Cheng May, 2006

2 MATERIAL MODELING OF STRAIN RATE DEPENDENT POLYMER AND 2D TRI-AXIALLY BRAIDED COMPOSITES Jingyun Cheng Dissertation Approved: Accepted: Advisor Dr. Wieslaw K. Binienda Department Chair Dr. Wieslaw K. Binienda Committee Member Dr. Robert K. Goldberg Dean of the College Dr. George K. Haritos Committee Member Dr. Atef F. Saleeb Dean of the Graduate School Dr. George R. Newkome Committee Member Dr. Pizhong Qiao Date Committee Member Dr. Xiaosheng Gao Committee Member Dr. Kevin L. Kreider ii

3 ABSTRACT 2D Tri-axially braided polymer matrix composites are beginning to be widely used in the engine fan case. A potential problem with composites in this application is cracking and delamination when the composite fan case is subjected to large deformation during a blade-out event. The capability to predict deformation and failure is crucial to engine fan case design. For polymer matrix based graphite composites, the nonlinear strain rate dependent response is due mainly to the matrix constituent. In the first part of this work, the state variable constitutive equations based on the Bodner viscoplasticity model have been modified to analyze the deformation of polymer matrix materials. In addition, the effects of hydrostatic stresses on the inelastic deformation have been accounted for by modifying the effective stress and effective plastic strain definitions through the use of a variation of the Drucker-Prager yield criterion. In addition, a simple damage model is proposed to simulate the unloading behavior. This model has been implemented as a user defined material model (UMATs) in the explicit transient finite element code LS-DYNA. The tensile and shear deformation of two representative polymers have been accurately simulated using the constitutive model. The verified strain rates are from 10-5 /s to 500/s. Additionally, the numerical simulations suggest that this model has the ability to reproduce the unloading behavior of the polymer approximately. The second part of this work is focused on an explicit finite element model developed for engine fan case design. A novel methodology called simplified braiding through thickness integration points has been proposed to model 2D tri-axially braided composites. In this model, unit cell of braided composites was divided into four sub-cells, and each sub-cell was modeled with layer composites using one shell element with several through the thickness integration points, which represented different fiber tow orientation. In LS-DYNA, Belytschko-Lin-Tsay shell element combined with user defined integration rules is used to model layer composites. Therefore, the fiber tow size and fiber architecture are considered inherently. The process of determining composites material constants is also discussed in details. Ballistic impact test on braided composite flat panel was simulated to verify this methodology. The LS-DYNA simulation results show that this model can capture the deformation of composites, and it agrees well with the experimental measurements through using 3D image correlation photogrammetry system. Specifically, LS-DYNA simulations can capture the failure shape and cracking direction for both 0 /±60 braided composites and 0 /±45 braided composites. Additionally, the coupon specimen simulation also shows that the tensile stress strain curve for two fiber preform architecture braided composites could be captured by this methodology. iii

4 ACKNOWLEDGEMENTS First and foremost, I would like to express my sincere gratitude to my advisor, Professor Wieslaw K. Binienda, for his guidance throughout this research and his support in my Ph.D program. Dr. Gary D. Roberts from NASA Glenn Research Center is greatly appreciated for his advice, discussion and generously providing the experimental results in my research. I would also thank Dr. Robert K. Goldberg from NASA Glenn Research Center for his guidance to polymer modeling and Dr. Kelly S. Carney from NASA Glenn Research Center for his discussion on LS-DYNA numerical simulations. Particularly, Professor Atef F. Saleeb from the Department of Civil Engineering at The University of Akron is greatly appreciated for his suggestion on this study. I would also like to thank my friends and fellow graduate students Xiahua Zheng, Daihua Zheng, Charles R. Ruggeri, Michael J. Bennett, and Justin Littell for their discussions. Any word can not express my deep gratitude to my family and Ms. Xin Zhao for their love and encouragement throughout my education. iv

5 TABLE OF CONTENTS Page LIST OF TABLES...viii LIST OF FIGURES ix CHAPTER I. BACKGROUND..1 Introduction..1 Literature Review.3 Mechanical Behavior of Resin.3 Textile Composites...5 Composite Material Model in LS-DYNA...9 Thesis Outline..11 II. STRAIN RATE DEPENDENT POLYMER MODEL.13 Polymer Constitutive Equations 14 Associated Flow and Evolution Equations 15 Loading-Unloading 17 Determination of Material Constants.18 LS-DYNA UMAT Implementation...22 Simulation of Strain Rate Dependent Polymer Shear and Tensile Deformation...23 v

6 Conclusions 28 III. SIMPLIFIED BRAIDING THROUGH THICKNESS INTEGRATION POINTS METHODOLOGY...45 Modeling Composites with Shell Elements in LS-DYNA 46 Simplified Braiding through Integration Points Methodology..48 Unit Cell.48 Braiding Method 49 Material Model of Uni-directional Composites.55 Discussion on Element Deletion 61 Conclusion.62 IV. EXPERIMENTAL RESULTS of BRAIDED COMPOSITES D Tri-axially Braided Composites Flat Panel..63 Coupon Test Specimen..65 Gelatin Projectile for Ballistic Impact...67 Experimental Setup of Ballistic Impact Test.68 Ballistic Impact Test Results.70 Coupon Specimen Test Results.76 Tensile and Compressive Behavior...76 Shear Behavior...82 Summary of Coupon Specimen Tests 83 V. BALLISTIC IMPACT SIMULATION Composite Mode 84 Material Modeling of Gelatin 84 Arbitrary Lagrangian Eulerian (ALE) Formulation vi

7 LS-DYNA Simulation Results Coupon Specimen Simulation 87 Penetration Threshold 88 Deformation Analysis 89 Failure Analysis and Discussion 93 Conclusions..108 VI. CONCLUSIONS VII. FUTURE WORK REFERENCES APPENDIX I vii

8 LIST OF TABLES Table Page 1.1 Composite material models in LS-DYNA Material properties for polymer matrix materials The geometry of unit cell (0 /±60 ) Fiber Volume Ratio of different fiber tow (0 /±60 ) Material properties of fiber and resin Ballistic impact test matrix Ultimate stress and strain, elastic modulus and Poisson s ratio for static tension and compression in the axial (0º) and transverse loading orientations Summary of shear responses Penetration threshold velocity comparison viii

9 LIST OF FIGURES Figure Page 1.1 CF6-80C Engine FEM mesh for fiber with shell and solid elements Damage parameter Experimental and computed shear stress-shear strain curves for PR520 resin at strain rates of 7x10-5 /sec (Low Rate), 1.76 /sec (Medium Rate), and 420 /sec (High Rate) Experimental and computed tensile stress-strain curves for PR520 resin at strain rates of 5x10-5 /sec (Low Rate), 1.4 /sec (Medium Rate) and 510 /sec (High Rate) Experimental and computed shear stress-shear strain curves for resin at strain rates of 9x10-5 /sec (Low Rate), 1.91 /sec (Medium Rate) and 518 /sec (High Rate) Experimental and computed tensile stress-strain curves for resin at strain rates of 5.7x10-5 /sec (Low Rate), 1.31 /sec (Medium Rate)and 365 /sec (High Rate) Effect of hydrostatic stress effect state variable on tensile stress-strain curve for PR520 resin at strain rate of 5x10-5 /sec (Low Rate) Expression of damage parameter Tensile test stress-strain curves for E862 resin, loading and unloading rate at in/min Experimental and computed loading-unloading-reloading shear stressshear strain curves for resin at strain rate of 9x10-5 /sec (Low Rate) 37 ix

10 2.10 Experimental and computed loading-unloading-reloading shear stressshear strain curves for resin at strain rate of 1.91 /sec (Medium Rate) Experimental and computed loading-unloading-reloading shear stressshear strain curves for resin at strain rate of 518 /sec (High Rate) Experimental and computed loading-unloading-reloading tensile stress-strain curves for resin at strain rate of 5.7x10-5 /sec (Low Rate) Experimental and computed loading-unloading-reloading tensile stress-strain curves for resin at strain rate of 1.31 /sec (Medium Rate) Experimental and computed loading-unloading-reloading tensile stress-strain curves for resin at strain rate of 365 /sec (High Rate) Experimental and computed loading-unloading-reloading shear stressshear strain curves for resin at strain rates of 9x10-5 /sec (Low Rate), 1.91 /sec (Medium Rate) and 518 /sec (High Rate) Experimental and computed loading-unloading-reloading tensile stress-strain curves for resin at strain rates of 5.7x10-5 /sec (Low Rate), 1.31 /sec (Medium Rate) and 365 /sec (High Rate) /±60 tri-axially braid architecture Schematic diagram of simplified braided through the thickness integration points methodology. (a) unit cell of tri-axially braided composites (fiber only), (b) mesh scheme of unit cell, (c) specification of fiber tow orientation angle at the through thickness integration points in one unit cell (1Ply) (11 and 22 are in-plane direction, 33 is through the thickness direction) SEM image of 2D tri-axially braided composites Meshing scheme of braiding through thickness integration points method Scheme of fiber bundle strength model.(a) schematic diagram of the fiber bundle specimen, (b) schematic diagram of the fiber bundle strength distribution 60 x

11 4.1 0 /±Ө 2D tri-axially braided architecture M36[0 /±60 ] composite panel post impact with mechanical specimen locations designated Specimen design and dimensions in centimeters for two longitudinal and one transverse bowtie configurations Gelatin projectile (a) before, (b) during and (c) after impact M36[0 /±60 ] composites plate after impact at 150 m/s M36[0 /±60 ] composites plate after impact at 192 m/s M36[0 /±45 ] composites plate after impact Sketch of ARAMIS measurement Out of plane deflection along a cross-section at the plate center under impact velocity of 128 m/s Strain distribution of panel LG670 under impact velocity 126m/s. (a) before fiber broken, (b) after fiber broken, (c) before fiber broken, and (d) after fiber broken Strain and center deflection of panel LG670 under impact velocity 126m/s Straight-sided tensile response for M36 resin and both 0 /±60 and 0 /±45 fiber architectures at axial and transverse directions Compressive response for M36 resin and both 0 /±60 and 0 /±45 fiber architectures at axial and transverse directions Bowtie tensile response for M36 resin and both 0 /±60 and 0 /±45 fiber architectures at axial and transverse directions Axial strain overlay of bowtie specimens loaded axially to 97 MPa and transversely to 303 MPa ALE mesh scheme of gelatin impact on composites target Results of straight-sided for both 0 /±60 and 0 /±45 fiber architecture braided composites (LS-DYNA simulations and experiments) 88 xi

12 5.3 The deformation of gelatin and composites plate at the impact velocity of 128 m/s LS-DYNA simulations and experiments Center deflection of Epon 862[0 /±60 ] braided composites plate Out of plane deflection along a cross-section at the plate center under impact velocity of 126 m/s Out of plane deflection along a cross-section at the plate center under impact velocity of 128 m/s LS-DYNA simulation of damage pattern of M36[0 /±60 ] at impact velocity 182 m/s LS-DYNA simulation of damage pattern of M36[0 /±45 ] at impact velocity 216 m/s Stress strain curve for 1 shell element Scheme of M36[0 /±60 ] direction stress history of axial (0 ) composites of element #A. Element #A located in Sub-cell A [60 /0 /-60 ], element deleting happened at 217 μs direction stress history of bias (±60 ) composites of element #A. Element #A located in Sub-cell A [60 /0 /-60 ], element deleting happened at 217 μs direction stress history of bias (±60 ) composites of element #B. Element #B located in Sub-cell B [60 /-60 ], element deleting happened at 207 μs direction stress history of axial (0 ) composites of element #C. Element #C located in Sub-cell A [60 /0 /-60 ], element deleting happened at 138 μs direction stress history of bias (±60 ) composites of element #C. Element #C located in Sub-cell A [60 /0 /-60 ], element deleting happened at 138 μs direction stress history of bias (±60 ) composites of element #D. Element #D located in Sub-cell B [60 /-60 ], element deleting happened at 115 μs xii

13 5.17 Scheme of [0 /±45 ] direction stress history of axial (0 ) composites of element #A. Element #A located in Sub-cell A [45 /0 /-45 ], element deleting happened at 90 μs direction stress history of bias (±45 ) composites of element #A. Element #A located in Sub-cell A [45 /0 /-45 ], element deleting happened at 90 μs direction stress history of bias (±45 ) composites of element #B. Element #B located in Sub-cell B [45 /-45 ], element deleting happened at 86μs 108 xiii

14 CHAPTER I BACKGROUND 1.1 Introduction High bypass ratio turbofan engines are used to power large modern commercial aircraft because of their high overall efficiency, high thrust at low flight speeds and low fuel consumption. The fan case is the largest structural component in these engines (Figure 1.1), and metal alloys are used for the case material in all commercial engines. The use of composite materials could significantly reduce the weight of the fan case. A potential problem with composites in this application is cracking and delamination when the composite fan case is subjected to large deformation during a blade-out event. Currently, 2D tri-axially braided polymer matrix composites are beginning to be widely used in the aerospace and automotive industry, especially for those structures where a high level of impact resistance is required. Compared with unidirectional composites, to form tri-axially braided composites three systems of yarns are intertwined diagonally. This fiber architecture offers an improved resistance to interlaminar cracking and delamination during impact. However, the geometrical parameters and material behavior of the composite such as the fiber architecture (braiding angle), fiber waviness, resin properties and the fiber/matrix interface affect the deformation behavior and failure mechanisms. Developing the ability to predict the mechanical properties and the impact 1

15 response of braided composites is becoming an issue of interest in the research community. Especially for the large engine fan case structure, the geometrical complexity prevents the direct finite element meshing of the fiber and matrix even using state of art computers. Meanwhile, the matrix in the composites shows high strain rate effects when undergoing the high speed impact. Therefore, it is very important to develop a material model implemented in commercial finite element code, which can be used in the engine fan case design. The above engineering background motivates this research. The fundamental objectives of this work are as follows: To develop a material model to simulate the mechanical behavior of resin during different strain rate loading, especially considering the unloading and reloading response of the material, and implement it as a user defined material model (UMAT) in LS-DYNA. To develop a methodology that can be used to simulate the impact on braided composites using the commercial finite element code LS-DYNA, especially for predicting the deformation and the failure. 2

16 Figure 1.1 CF6-80C Engine. 1.2 Literature Review Mechanical Behavior of Resin Polymers have long been known to have a rate-dependent constitutive response. Traditionally, for very small strain response, linear viscoelastic techniques have been used to model the rate dependent behavior on a phenomelogical level. In linear viscoelastic models, combinations of springs and dashpots in series and parallel may be used to capture the rate dependent behavior[1]. When the strains are large enough that the response is no longer linear, nonlinear viscoelastic models have been developed. For example, in a model developed by Cessna and Sternstein[2], nonlinear dashpots are incorporated into the constitutive model. Empirical equations are also used to capture the rate dependent response, in which the yield stress is scaled as a function of strain rate. A more sophisticated approach to polymer constitutive modeling takes a molecular approach. In this approach, it is assumed that the deformation of a polymer is due to the 3

17 motion of molecular chains over potential energy barriers. The molecular flow is due to applied stress, and the internal viscosity is assumed to decrease with applied stress. Internal stresses can also be defined, which represent the resistance to molecular flow which tends to drive the material back towards its original configuration. Another approach [3] to polymer deformation assumes that the deformation is due to the unwinding of molecular kinks. In both approaches, constitutive models have been developed in which the deformation response is considered to be a function of parameters such as activation energy, activation volume, molecular radius, molecular angle of rotation, and thermal constants. Furthermore, the deformation is assumed to be a function of state variables which represent the resistance to molecular flow caused by a variety of mechanisms. The state variable values evolve with stress, inelastic strain and inelastic strain rate. An alternative approach to the constitutive modeling of polymers is to utilize, either directly or with some modifications, viscoplastic constitutive equations which have been developed for metals. For example, Bordonaro[4]modified the Viscoplasticity Theory. Based on Overstress developed by Krempl [5]. In Bordonaro s model, the original theory was modified to attempt to account for phenomena encountered in the deformation of polymers that are not present in metals. For example, polymers behave differently from metals under conditions such as creep, relaxation, and unloading. Other authors, such as Zhang and Moore [6], utilized techniques developed to model the deformation of metals directly with no modification. However, they primarily limited their study to analyzing the uniaxial tensile response of polymers and did not consider phenomena such as unloading, creep or relaxation. 4

18 Recently, Goldberg et al [7,8] developed a model to account for the effects of hydrostatic stresses on the nonlinear, strain rate dependent deformation of polymer matrix composites. State variable constitutive equations originally developed for metals have been modified in order to model the nonlinear, strain rate dependent deformation of polymeric materials. To account for the effects of hydrostatic stresses, which are significant in polymers, the classical J2 plasticity theory definitions of effective stress and effective inelastic strain, along with the equations used to compute the components of the inelastic strain rate tensor, were appropriately modified. To verify the revised formulation, the shear and tensile deformation of two representative polymers were computed across a wide range of strain rates. Results computed using the developed constitutive equations correlate well with experimental data. The unloading phase of the material response is critical for impact analysis, such as internal energy absorption determination. Shen et al[9,10, and 11] did a series of test on epoxy and found that the unloading behavior is nonlinear, the unloading modulus can be assumed linear and its modulus is less loading modulus at constant strain rate. Dubois[12] used the Drucker Prager model instead of Von Mises model to simulate the different behavior of thermoplastics under tension and compression, a simple damage model decreasing young s modulus is implemented to simulate the unloading behavior Textile Composites 2D tri-axially braided composites is one kind of textile composites. The use of woven and braided composites is gaining greater interest in the research community. Previously, numerous efforts have been made by other researchers to develop analytical tools to 5

19 predict the elastic properties of woven and braided composites. Ishikawa and Chou [13] developed simple one-dimensional models in which classical lamination theory was combined with suitable iso-stress and iso-strain assumptions to determine the elastic properties of plain weave and satin weave composites. These simple laminate theory models were extended by researchers such as Raju and Wang [14], Naik and Shembekar [15], and Ganesh and Naik [16] in order to conduct two-dimensional analyses of plain weave composites. By their extensions, the geometry of the fiber perform could be more accurately modeled and the effects of the woven geometry on the elastic properties could be more accurately captured. Similar laminate theory type of approaches was used by Mital, Murthy and Chamis [18,19, and20]to predict the elastic properties and microstresses for plain weave polymer matrix and ceramic matrix composites. In an extension of the laminate theory approach, Yang and Chou [21]developed an analytical method to predict the elastic behavior of triaxially woven composites. In their approach, each undulated fiber tow was considered to be a collection of straight, connected laminates. Classical laminate theory was applied to obtain the elastic properties of each laminate piece, and then appropriate iso-strain and iso-stress assumptions were applied to obtain the effective properties of the undulated fiber tow. After the properties of each rotated tow were obtained, coordinate transformations were applied to compute the elastic constants in the global axis system, and iso-strain assumptions were used to obtain the material properties for the entire triaxial woven composite. In a more sophisticated approach, Tanov and Tabiei [22], based on an earlier approach presented by Jiang, Tabiei and Simitses [23], developed a methodology in which a plane weave woven composite was discretized into a series of four connected subcells. A 6

20 through-the thickness homogenization process utilizing appropriate constant stress and constant strain assumptions were employed to obtain the elastic properties of each subcell, and then iso-stress and iso-strain assumptions were applied within the plane of the composite to compute the overall effective properties. A similar two step approach was applied by Bednarcyk [24]. In this approach, a plane weave woven composite was discretized into a number of subcells. The effective properties of each column of subcells (a through-the-thickness homogenization) were determined using the Generalized Method of Cells. GMC was then applied once again within the plane of the composite to determine the effective properties of the composite. The unit cell concept has been implemented by many researchers to investigate the mechanical properties of textile composites. Detailed finite element analyses, in which a full finite element model is developed for the unit cell, have been used to compute the effective properties of woven composites by researchers such as Whitcomb and Tang [25, and 26]. Xue et al [28] and Peng [29]implemented numerical simulation of the unit cell of woven materials, the elastic constants were calculated based on homogenization method and iteration at the unit cell level and coupon level. However, it is very hard to predict the damage using unit cell concept and homogenization method. An alternative method of determining the effective properties of woven composites was developed by Shim [30] using a pin joint bar method. In this approach, the viscoelastic fiber elements are pin-joint connected at crossover joints at so called nodes. Through calculating the displacement at nodes, the deformation gradient and stress vs. strain can be deduced. Similar methods have been used by Ivanov and Taibei [31]to model woven composites. 7

21 Several efforts have been undertaken to analyze the elastic response of braided composites. Byun [32] developed a geometrical model to determine the dimensions of a tri-axially braided composite based on a few measured parameters. To predict the analytical properties, the average properties of each fiber tow were computed by applying constant stress assumptions along the length of each fiber tow. Stiffness averaging techniques along with appropriate coordinate transformations were then used to determine the effective stiffness matrix for the entire composite. Flanagan [33] conducted a series of ballistic impact tests on two- and three- dimensionally woven and braided composites using a Lexan right cylindrical projectile. The failure modes of the material, including indentation, matrix cracking, tensile and shear fiber failure and shear plugging, were categorized in terms of the velocity regime, the number and combination of material layers and types, and the debris mass. Beard [34] did crushing tests on 2D tri-axially braided composites tubes. The experimental results showed that the load - displacement curve and overall energy absorption could be significantly affected by the fiber perform architecture. Warrior [35] did high strain rate tensile and compressive test of braided composites, and found that axial properties were relatively insensitive to strain rate, but strong strain rate dependency was seen in the transverse direction, where the effects of polymer resin were more significant. Quek et al [37]did bi-axial test on braided composites and found the failure mode involve distributed matrix cracking and local loss of stability due to tow buckling. Similarly, Karayaka [38] analyzed the failure mechanism of woven composites and found that woven composites exhibit orientation dependent strength. Cox [39, and 40]used 8

22 micromechanical methods to predict the failure based on experimental works. In general, most of research on failure of textile composites were focus on experimental work. Blankerhorn[41] used shell elements and solid elements to discretize the fiber tow of textile composites in a finite element analysis (Figure 1.2), with a contact formulation to simulate the interaction of the fiber tows both within a layer and between layers. However, direct finite element meshing like that employed in this study is very time consuming due to the large number of elements required. Figure 1.2 FEM mesh for fiber with shell and solid elements Composite Material Model in LS-DYNA The available composite material models in LS-DYNA [42] is summarized as shown in Table

23 Table 1.1 Composite material models in LS-DYNA. Shell Element Solid Element Strain Rate Failure Criteria MAT 22 (composite_damage) X X Chang-Chang MAT 54 X Chang-Chang (enhanced_composite_damage) MAT 55 X Tsai-Wu (enhanced_composite_damage MAT 58 X Hashin (laminated_composite_fabric) MAT 117 (composite_matrix) X No failure, only can used to calculate the elastic response - extensional, bending and coupling stiffness MAT 118 (composite_direct) X No failure, only can used to calculate the elastic response - extensional, bending and coupling stiffness MAT 161 (composite_msc) X X Hashin. Can simulate the delamination In summary, most composites material models were implemented for shell elements, while only MAT 161 considered the strain rate effects. The favorite failure criteria are Chang-Chang, Tsai-Wu and Hashin [43]models. MAT 22 and MAT 54 provided Chang- Chang fiber and matrix failure modes due to in-plane stresses in unidirectional lamina. In this 2D failure model, the failure mode due to out of plane shear and normal stresses are neglected. MAT 58 is based on the Hashin criteria and continuum damage mechanics. It is implemented only in the shell elements. This material model does not consider the strain rate effects. However, this material model can be used in the initial design stage due to rather small amount of material properties input information. MAT 161 is a composite lamina model based on 3D stresses field. The failure model can be used to 10

24 effectively simulate fiber failure, matrix damage and delamination behavior. Basically, lamina failure model for fabric composites has been extended to model the progressive post-failure behavior by adopting the continuum damage mechanics approach, which characterizes the growth of damage by decreasing the material stiffness. In order to account for the experimentally observed nonlinear and rate dependent behavior, a general rate dependent progressive failure model has been developed. The disadvantage of this material model is that need quite large number of material property input, which is not easily accessed by standard material test. This also causes difficulties in material calibration. 1.3 Thesis Outline The thesis is organized into as follows: Chapter 2 describes the strain rate dependent constitutive model of the polymer and its implementation as a user defined material model in LS-DYNA. This material model allows consideration of hydrostatic stress effect and unloading behavior. Chapter 3 describes the simplified braiding through thickness integration point methodology. This method especially considers fiber perform architecture. Chapter 4 presents experimental results of ballistic impact on 2D tri-axially braided composites flat panel for two different fiber preform architectures using soft projectiles, 3D image correlation photogrammetry measurement technique was used to measure the displacement and the strain during impact, those tests were conducted at NASA Glenn Research Center. The static tests were conducted on coupon specimen of 2D tri-axially braided composites, and the relevant results are also discussed in this Chapter. Chapter 5 presents ballistic impact on braided composites flat panel simulations 11

25 using simplified braiding through thickness integration point methodology. LS-DYNA simulation results and its correlation with experimental observation are described. Chapter 6 presents the conclusion of this work. Finally, the future research work of material modeling of 2D triaxially braided composites are discussed in Chapter 7. 12

26 CHAPTER II STRAIN RATE DEPENDENT POLYMER MODEL To design a composite containment system, the ability to correctly predict the nonlinear, strain rate dependent deformation and failure of the composite under high strain rate loading conditions is very important. For polymer matrix based graphite composites, the nonlinear strain rate dependent response is mainly from the matrix part. In previous research on this project, Goldberg et al [7, and 8] developed a nonlinear, strain rate dependent polymer model with the hydrostatic stress effects appropriately accounted for. However, the unloading phase of material is critical for impact analysis, such as the determination of internal energy absorption. Experimental results showed that unloading stress strain curve of polymer is nonlinear, but it shows the unloading modulus is smaller than original modulus. Therefore, we need a model that accounts for this material behavior at the unloading stage. In this chapter, a revised model based on the original Goldberg model is developed, in which the unloading behavior is approximated by adding a simple damage parameter to decrease the original Young s modulus, where damage parameter increases with the increment of inelastic strain. Then this model is implemented as a user defined material model (UMAT) in LS-DYNA, numerical 13

27 simulations show this model can reproduce the experimental stress and strain curves approximately. 2.1 Polymer Constitutive Equations In this study, the Bodner state variable constitutive equations [44], which were originally developed to analyze the viscoplastic deformation of metals above one-half of the melting temperature, were modified to analyze the strain rate dependent, nonlinear deformation of the polymeric matrix material. In state variable constitutive equations, a single unified strain variable is defined to represent all inelastic strains [44]. Furthermore, in the state variable approach there is no defined yield stress. Inelastic strains are assumed to be present at all values of stress, only at a very small level in the elastic range of deformation. State variables, which evolve with stress and inelastic strain, are defined to represent the average effects of the deformation mechanisms. In terms of unloading behavior of polymer, Xia[11] did a series of test on epoxy and found that the unloading behavior is nonlinear, but the unloading modulus can be assumed linear and its modulus is lower than loading modulus at constant strain rate. Bordonaro[4] found that if the polymer was loaded and unloaded at the same strain rate magnitude, the resulting slopes of the unloading curves were independent of the strain rate. Also an unloading curve initiating from a higher stress would fall to the left of unloading curve which was initiated at a lower stress. Several assumptions and limitations have been specified in the development of the constitutive equations. Currently, temperature effects are neglected. While the deformation response of polymers varies significantly with temperature, only room temperature data have been obtained at this time, so the effects of temperature are not 14

28 considered. Moisture effects, while possibly significant in polymer matrix composites, are also not included at present but may be added in the future. Here, a simple damage model is proposed to model the unloading behavior of polymer at the different strain rate. The damage state does not change during unloading. During reloading, damage growth is typically very small in comparison to first loading as long as the previous strains are not exceeded [46, and 47 ]. These additional damage contributions along the reloading path will also be neglected in the model, since it is not intended to account for fatigue. It is important to keep in mind that the effective properties on the unloading and reloading path only depend on the current state of damage. Small strain theory is assumed to apply in the current analysis Associated Flow and Evolution Equations In original Goldberg model [7, and 8], a procedure very similar to that employed to derive the Prandtl-Reuss equations [48] in classical plasticity is utilized to derive the associated flow equation for the components of the inelastic strain rate tensor for polymeric materials in which the effects of the hydrostatic stresses are accounted for. By applying the Bodner-Partom model [44], the flow equation is shown in Equation 2.1 2n I 1 Z Sij & ε + ij = 2Do exp αδij (2.1) 2 σ e 2 J 2 where D0 and n are material constants. D 0 represents the maximum inelastic strain rate, and n controls the rate dependence of the material. Z is an isotropic state variable which 15

29 represents the resistance to molecular flow (internal stress). α is a material constant which controls the level of the hydrostatic stress effects. J 2 is the second invariant of the deviatoric stress tensor, S ij are the components of the deviatoric stress tensor and δ ij is the Kronecker delta. The effective stress, σ e is defined as follows σ = 3 f = 3J + 2 3ασ (2.2) e kk The rate of evolution of the internal stress state variable Z are defined by the equations Z & ( Z Z ) e& I e = q 1 (2.3) where q is a material constant representing the hardening rate, and Z1 is material constant representing the maximum values of Z. The initial value of Z is defined by the material constant Zo. The term I e& e in Equation 2.3 represents the effective deviatoric inelastic strain rate, which was defined in Equation 2.4. An important point to note is that in the original Bodner model [44], the inelastic work rate was used instead of the effective inelastic strain rate in the evolution equation for the internal stress state variable. e& I e e& I ij 2 = e& e& 3 I ij I ij = & ε & ε I ij I m (2.4) 16

30 where I e& e is the effective deviatoric inelastic strain rate and I ε& m is the mean inelastic strain rate, which matches the effective inelastic strain rate definition given by Pan and co-workers [49] Loading-Unloading A simple damage model is proposed to model the unloading behavior of polymer, as shown in Equation 2.5 E damaged = E = ( 1 d )E (2.5) unloading where d is damage parameters, Eunloading Rearrange Equation 2.5, d becomes is the unloading Young s modulus. d E Eunloading = (2.6) E Through measuring unloading modulus at the different stress level, a series of d vs. inelastic strain test points could be obtained as plotted in Figure 2.1. Therefore, damage function is inferred by curve fitting of the test points. 17

31 Damage Function Damage Parameter d Test Curve Fitting Inelastic Strain Figure 2.1 Damage parameter d. The damage parameter d evolves during loading as shown in Figure Determination of Material Constants The material constants that need to be determined include d, D 0, n, Z 0, Z1, α, and q. The procedure of determining material constants is summarized here. More details on the general approach can be found in Stouffer and Dame [45] and Bodner [44]. The values of D 0, n and Z1 are characterized as follows using Equation 2.7. The value of D 0 is currently assumed to be equal to a value of 10 4 times the maximum applied strain rate, which correlates with the maximum inelastic strain rate. Usually, uni-axial tension and pure shear data are easily accessed. The pure shear experimental data is firstly used to calculate n, Z 0, and Z1 due to two reasons. First, hydrostatic stress effects are not present 18

32 19 in the case of pure shear loading, which can simplify the material characterization. Second, since polymers tend to be more ductile in shear than in uniaxial tension, the shear stress-shear strain curves obtained experimentally are more likely to display a defined saturation stress. The saturation, or yield, stress is defined as the stress level where the inelastic strain rate equals the total applied strain rate. The Equation 2.1 is simplified for pure shear case τ τ τ γ = n o I Z D exp 2 & (2.7) where I γ& is the engineering shear strain rate, τ is the shear stress, and the remainder of the terms are as defined earlier. Apply natural logarithm on the both sides of Equation 2.7 twice and rearrange the expression as follows ) 3 ln( 2 ) ln( 2 2 2ln ln 0 τ γ n Z n D o = & (2.8) At corresponding values at saturation point are substituted into the Equation 2.8, which obtains the following ) 3 ln( 2 ) ln( 2 2 2ln ln 1 0 s o n Z n D τ γ = & (2.9)

33 where τ s is the saturation shear stress, γ& o is the constant applied total engineering shear strain rate in a constant strain rate shear test, and the remaining terms are as defined earlier. With a set of shear stress-strain curves obtained from constant strain rate tests. Each curve in this set is obtained at a different constant strain rate. Data pairs of the total strain rate and saturation shear stress values from each curve are taken. For each strain rate, the data values are substituted into Equation 2.7, and represent a point on a master curve. The number of points in the master curve equal the number of strain rates at which tensile tests were conducted. A least squares regression analysis is then performed on the master curve. As suggested by Equation 2.9, the slope of the best-fit line is equal to -2n. The intercept of the best-fit line is equal to 2n(ln (Z1)). To determine the value of Z 0, first Equation 2.7 is rearranged as follows Z & γ o = 2ln 2D 0 1 2n 3τ (2.10) where all of terms are as defined earlier. The value of the shear stress where the stressstrain curve becomes nonlinear for a particular constant strain rate shear test is used for the value of τ. The point where the stress-strain curve becomes nonlinear is defined as the approximate point where the curve appreciably deviates from a linear extrapolation of the initial data. The value of I γ& is set equal to the approximate inelastic shear strain rate when the stress-strain curve becomes nonlinear. The shear strain rate used in the constant 20

34 strain rate test divided by 100 was found by trial and error to approximate this value reasonably well. Using this data, Equation 2.10 is solved for Z, which is assumed to be equal to the value of Z 0. Using the data from the lowest strain rate test available has been found to give adequate values of Z 0. To determine the value of q, Equation 2.3 is integrated for the case of pure shear loading, resulting in the following relation Z q I = Z1 ( Z1 Zo ) exp γ (2.11) 3 I where γ is the inelastic shear strain. At saturation, the value of the internal stress Z is assumed to approach Z1, resulting in the exponential term approaching zero. Assuming that saturation occurs when the following condition is satisfied q I exp γ s = ξ (2.12) 3 I the equation is solved for q, where γ s is the inelastic shear strain at saturation which can be measured through unloading test at saturation point, ξ is a small number, which can be assumed to equal If the inelastic shear strain at saturation is found to vary with strain rate, the parameter q is computed at each strain rate and regression techniques are utilized to determine an expression for the variation of q. In the Goldberg s model, the inelastic shear strain at saturation point is calculated using the following: 21

35 γ I s τ s = γ s (2.13) G Where is γ s total strain at saturation point, is τ s shear stress at saturation point, and G is shear modulus. One thing need to emphasize here is that the inelastic strain at saturation is calculated based on the current shear modulus. The above calculated value is usually bigger than the measured inelastic strain, as shown in Figure 2.8. To obtain the values of α, Equation 2.2 is used in combination with stress-strain data from constant strain rate uniaxial tensile tests and constant strain rate shear tests. The primary assumption used at this point is that the effective stress at saturation under uniaxial tensile loading at a particular strain rate is equal to the effective stress at saturation under pure shear loading at the same equivalent strain rate. s ( 1 3α ) = τ s σ + 3 (2.14) where σs and τs are the tensile and shear stresses at saturation, respectively. The required constants can then be determined from these equations. The values of the material constants are assumed to be rate independent, so the results from only one strain rate need to be used to find the needed parameters. 2.3 LS-DYNA UMAT Implementation The algorithm of this model [52]is described as follows: 1. Initialize the history variable and update Young s modulus E 22

36 E damaged n = E 1 ( d ) n Where damaged En 2. Calculate the inelastic strain rate is the value of Young s modulus at n th step. & ε I ij 2D 1 Z exp 2 σ e 2n Sij 2 J + αδ = o ij 2 3. Update the inelastic strain increment I I Δε = & ε Δt ij 4. Calculate the elastic strain increment and the stress increment Δ ε e ij = Δε T ij Δε I ij Where T Δε ij is total strain increment. Δ σ damaged e ij = En Δε ij σ n+1 ij = σ n ij + Δσ ij 5. Update the state variables Z n+ 1, α n+ 1 and damage function d n Simulation of Strain Rate Dependent Polymer Shear and Tensile Deformation To demonstrate the ability of the developed constitutive equations to correctly analyze the mean stress dependent deformation response of polymers, two representative toughened epoxies, PR520 and 977-2, were chosen to analyze [53]. Longitudinal tensile tests and pure shear tests were conducted at room temperature on the materials at strain rates of about 5x10-5 /sec, 1 /sec and 400 /sec. The low and moderate strain rate tests were conducted using an Instron hydraulic testing machine. Whilst the high strain rate tests were conducted using a split Hopkinson bar. 23

37 Shear stress-shear strain curves for PR520 obtained under pure shear loading are shown in Figure 2.2 for each of the strain rates examined, while tensile stress-strain curves are shown in Figure 2.3. Similarly, shear stress-shear strain curves for are shown in Figure 2.4, and tensile stress-strain curves are shown in Figure 2.5. Both materials exhibit a strain rate dependent, nonlinear deformation response under both shear and tensile loading. At high strain rates, the sharp increase in stress at the beginning of the loading with negligible increase in strain observed for both materials in under shear loading and for PR520 under tensile loading is most likely the result of a lack of stress equilibrium at the start of loading. The oscillations seen in the tensile response of at high strain rates are most likely due to the specimen geometry leading to the stress waves being visible in the response. The tensile specimen design for high strain rate testing was changed between the times when the tensile data was obtained and when the PR520 data was obtained, which is the reason that the oscillations do not appear in the PR520 data. The failure stresses under tensile loading for appear not to vary with strain rate, and the failure stress for PR520 does not appear to increase in going from the low to the moderate strain rates. Preliminary investigations indicate that the measured failure stresses at the higher strain rates may be artificially low due to the presence of strain gages on the specimen and the geometry of the specimen. Further details of these studies will be given in a future report. The material constants for both polymers were determined using the procedures described earlier in this report and are listed in Table 2.1. The shear stress-shear strain curves computed using LS-DYNA UMAT for all three strain rates, along with the experimental results for comparison, are shown in Figure

38 for PR520 and Figure 2.4 for Overall, the computed results correlate well with the experimental values for all strain rates for both materials. Specifically, the nonlinearity and rate dependence of the experimental results are captured qualitatively, and the quantitative match between the experimental and computed results is reasonably good. The high strain rate results are somewhat under predicted for both materials (particularly for PR520) at lower strains, but this is most likely due to the fact that in the experiments the initial stresses increased significantly with a negligible increase in strain, and thus the initial modulus of the material was computed using data obtained after the strain became non-negligible. The tensile stress-strain curves computed using LS-DYNA UMAT for all three strain rates, along with the experimental results for comparison, are shown in Figure 2.3 for PR520 and Figure 2.5 for For both materials, qualitatively the nonlinearity and rate dependence of the experimental results is captured. Quantitatively, for PR520 at the medium strain rate the stresses in the nonlinear range are somewhat under predicted and for at the low strain rate the stresses are somewhat over predicted. Overall, however, quantitatively the comparison between the experimental and computed results is reasonably good. There is not currently a good explanation for the cause of the discrepancies, but the maximum total error is still less than ten percent. For the tensile results, the important point to note is that the material constants were primarily computed using the shear data, and the comparison of the tensile data to the computed results is overall still reasonably good. While not shown here, as discussed in Goldberg, Roberts and Gilat [53] if the hydrostatic stress effects are not properly accounted for and the material is characterized 25

39 based on the results from shear testing alone, the predicted stresses would be much higher than the experimental values in the nonlinear range. Furthermore, if the value of the hydrostatic stress effect state variable α is kept constant at its final value, the stresses in the early part of the loading curve would be over predicted. This result is significant in that in previous efforts in the literature account for the effects of hydrostatic stresses in the deformation response of polymers, the effects of the hydrostatic stresses have been assumed to be constant over the entire range of loading. To further explore the significance of properly accounting for the mean stress effects in the analysis, the tensile stress-strain curve for PR520 at the low strain rate of 5x10-5 /sec is once again considered. In Figure 2.6, the experimental stress-strain curve, along with the original computed curve, is presented. One additional computed result is given in the figure. First, a tensile curve computed without the mean stress effect included (α=0) is given. Second, a set of results computed with a constant α is presented. The tensile curve computed without accounting for mean stress effects significantly overpredicts the stresses as compared to the experimental results, indicating that mean stress effects are significant for polymers, and accounting for them in an analysis is crucial. Since the unloading and reloading test data is unavailable right now, the following numerical procedure is used to verify the UMAT in which damage is considered. First, the damage function for is assumed as I ( 11.8 ) d = 1 exp ε e (2.29) 26

40 I where ε e is effective inelastic strain. Figure 2.8 shows some preliminary tensile test results of EPON862 at the low strain rates. It is shown that the unloading modulus is smaller than original young s modulus, and it decreases with the increment of inelastic strain. As mentioned before, the unloading phase of material is critical for impact analysis, such as the determination of internal energy absorption. Therefore, d could be approximated through comparing the internal energy between the real test and the approximation curve using linear unloading. In Figure 2.8, the internal energy is represented by the area covered with corresponding stress and strain curve The loading-unloading-reloading shear stress-shear strain curves computed using LS- DYNA UMAT for all the three strain rates, along with the experimental results for comparison, are shown in Figure 2.9, Figure 2.10 and Figure 2.11 for In general, the computed results correlate well with the experimental values for all strain rates for at the final loading part. Additionally, the nonlinearity and rate dependence of the experimental results are captured qualitatively, and the quantitative match between the experimental and computed results is reasonably good. Specifically, unloading and reloading stress-strain curves is linear and the unloading Young s modulus is decreased, and the unloading Young s modulus decreases for unloading initiation at increasing stress range, which corresponds to the increased inelastic strain. The loading-unloading-reloading tensile stress-tensile strain curves computed using LS-DYNA UMAT for all the three strain rates, along with the experimental results for comparison, are shown in Figure 2.12, Figure 2.13 and Figure 2.14 for It is find that tensile results have similar pattern with pure shear simulation results. The 27

41 summarized results are plotted again in Figure 2.15 and One important result is that unloading and reloading stress-strain curves is linear and the unloading Young s modulus is decreased. Moreover, the unloading Young s modulus decreases for unloading initiation at increasing stress range. 2.5 Conclusions An analytical model has been developed to analyze the strain rate dependent, nonlinear deformation of polymers and polymer matrix composites in which the effects of hydrostatic stresses on the nonlinear deformation are systematically accounted for. State variable constitutive equations based on the Bodner viscoplasticity model have been modified to analyze the deformation of polymer matrix materials. The effects of hydrostatic stresses on the inelastic deformation have been accounted for by modifying the effective stress and effective plastic strain definitions through the use of a variation of the Drucker-Prager yield criterion. Specifically, a simple damage model is proposed to simulate the unloading behavior. The tensile and shear deformation of two representative polymers have been accurately simulated using the constitutive model. Additionally, numerical simulations show this model has the ability to reproduce the unloading behavior of polymer approximately. As limitations of the current implementation we can cite the following: The unloading approximation part of this method is: No micromechanics are considered in this model. 28

42 The application is limited to ductile plastics that are initially isotropic and remain isotropic throughout the deformation process. This will not be the case for most polymeric materials. More investigations including both unloading tests and numerical simulations need to be implemented to improve this model. Table 2.1 Material properties for polymer matrix materials. PR Strain Rate /sec Modulus GPa 7x x Poisson s Ratio D o 1/sec n Z o MPa Z 1 MPa q α o α x x

43 Stress (Mpa) Experiment: 7x10-5 /s Experiment: 1.76/s Experiment: 420/s LS-DYNA: 7X10-5 /s LS-DYNA: 1.76/s LS-DYNA: 420/s Strain Figure 2.2 Experimental and computed shear stress-shear strain curves for PR520 resin at strain rates of 7x10-5 /sec (Low Rate), 1.76 /sec (Medium Rate), and 420 /sec (High Rate). 30

44 Stress (Mpa) Experiment: 5x10-5 /s Experiment: 1.4/s Experiment: 510/s LS-DYNA: 5X10-5 /s LS-DYNA: 1.4/s LS-DYNA: 510/s Strain Figure 2.3 Experimental and computed tensile stress-strain curves for PR520 resin at strain rates of 5x10-5 /sec (Low Rate), 1.4 /sec (Medium Rate) and 510 /sec (High Rate). 31

45 Stress (Mpa) Experiment: 9x10-5 /s Experiment: 1.91/s Experiment: 518/s LS-DYNA: 9X10-5 /s LS-DYNA: 1.91/s LS-DYNA: 518/s Strain Figure 2.4 Experimental and computed shear stress-shear strain curves for resin at strain rates of 9x10-5 /sec (Low Rate), 1.91 /sec (Medium Rate) and 518 /sec (High Rate). 32

46 Stress (Mpa) Experiment: 5.7x10-5 /s Experiment: 1.31/s Experiment: 365/s LS-DYNA: 5.7X10-5 /s LS-DYNA: 1.31/s LS-DYNA: 365/s Strain Figure 2.5 Experimental and computed tensile stress-strain curves for resin at strain rates of 5.7x10-5 /sec (Low Rate), 1.31 /sec (Medium Rate) and 365 /sec (High Rate). 33

47 Stress (Mpa) Experiment: 5x10-5 /s LS-DYNA: 5X10-5 /s LS-DYNA: 5X10-5 /s (wo/ considering hydrostatic stress) Strain Figure 2.6 Effect of hydrostatic stress effect state variable on tensile stress-strain curve for PR520 resin at strain rate of 5x10-5 /sec (Low Rate). 34

48 Damage Parameter d Damage Parameter d Effective Inelastic Strain Figure 2.7 Expression of damage parameter d. 35

49 Stress (Mpa) Test 1 Test 2 Test 3 Test Strain (%) Figure 2.8 Tensile test stress-strain curves for E862 resin, loading and unloading rate at in/min 36

50 80 60 Stress (Mpa) Experiment: 9x10-5 /s LS-DYNA: 9X10-5 /s LS-DYNA (wo/damage): 9X10-5 /s Strain Figure 2.9 Experimental and computed loading-unloading-reloading shear stress-shear strain curves for resin at strain rate of 9x10-5 /sec (Low Rate). 37

51 Stress (Mpa) Experiment: 1.91/s LS-DYNA: 1.91/s LS-DYNA(wo/damage): 1.91/s Strain Figure 2.10 Experimental and computed loading-unloading-reloading shear stress-shear strain curves for resin at strain rate of 1.91 /sec (Medium Rate). 38

52 Stress (Mpa) Experiment: 518/s LS-DYNA: 518/s LS-DYNA(wo/damage): 518/s Strain Figure 2.11 Experimental and computed loading-unloading-reloading shear stress-shear strain curves for resin at strain rate of 518 /sec (High Rate). 39

53 Stress (Mpa) Experiment: 5.7x10-5 /s LS-DYNA: 5.7X10-5 /s LS-DYNA(wo/damage): 5.7X10-5 /s Strain Figure 2.12 Experimental and computed loading-unloading-reloading tensile stress-strain curves for resin at strain rate of 5.7x10-5 /sec (Low Rate). 40

54 Stress (Mpa) Experiment: 1.31/s LS-DYNA: 1.31/s Strain Figure 2.13 Experimental and computed loading-unloading-reloading tensile stress-strain curves for resin at strain rate of 1.31 /sec (Medium Rate). 41

55 Stress (Mpa) Experiment: 365/s LS-DYNA: 365/s LS-DYNA(wo/damage: 365/s Strain Figure 2.14 Experimental and computed loading-unloading-reloading tensile stress-strain curves for resin at strain rate of 365 /sec (High Rate). 42

56 Stress (Mpa) Experiment: 9x10-5 /s Experiment: 1.91/s Experiment: 518/s LS-DYNA: 9X10-5 /s LS-DYNA: 1.91/s LS-DYNA: 518/s Strain Figure2.15 Experimental and computed loading-unloading-reloading shear stressshear strain curves for resin at strain rates of 9x10-5 /sec (Low Rate), 1.91 /sec (Medium Rate) and 518 /sec (High Rate). 43

57 Stress (Mpa) Experiment: 5.7x10-5 /s Experiment: 1.31/s Experiment: 365/s LS-DYNA: 5.7X10-5 /s LS-DYNA: 1.31/s LS-DYNA: 365/s Strain Figure 2.16 Experimental and computed loading-unloading-reloading tensile stress-strain curves for resin at strain rates of 5.7x10-5 /sec (Low Rate), 1.31 /sec (Medium Rate) and 365 /sec (High Rate). 44

58 CHAPTER III SIMPLIFIED BRAIDING THROUGH THICKNESS INTEGRATION POINTS METHODOLOGY To form 2D tri-axially braided composites three systems of yarns are intertwined diagonally. This fiber architecture offers an improved resistance to interlaminar cracking and delamination during impact. However, the geometrical parameters and material behavior of the composite such as the fiber architecture (braiding angle), fiber waviness, and resin properties affect the deformation behavior and failure mechanisms. In order to understand how those geometrical and material parameters affect the mechanical performance of composites, especially during the high velocity impact, we developed a new methodology in this Chapter. First, a method of modeling composites with shell elements in LS-DYNA is described briefly, second, a novel Simplified Braiding through Thickness Integration Points Methodology (SBTIP) is proposed. Subsequently, a linear elastic composite model is used to simulate the material at the integration points. In what follows, the new methodology is described in detail and the procedures to determine the relevant material constants are provided. 45

59 3.1 Modeling Composites with Shell Elements in LS-DYNA In LS-DYNA [54], each integration point through the shell thickness allows the definition of the angle at that point, which represents the material orientation. Typically it is not limited to one point per ply. For example, in 2D tri-axially braided composites, 3 integration points are defined in one ply to represent 0 /+θ /-θ, where θ is the bias braiding angle. This can defined as an angle in *SECTION_SHELL card (55, and 56). In the implementation of the material model we first transform the stress and velocity strain tensor d ij into the material coordination system denoted by the subscript L. σ = q t σq (3.1.a) L ε L = q t dq (3.1.b) The orthogonal transformation matrix is given by cosθ sinθ 0 q = sinθ cosθ 0 (3.2) where θ is material orientation angle. In shell theory we assume a plane stress condition, i.e., that the normal stress to the mid-surface is zero. We can now incrementally update the stress state in the material coordinates σ = σ + Δσ (3.3) n+1 n n+1/ 2 L L L 46

60 where for an elastic material, it has the following expression: Δσ 11 Q11 Q d11 Q Q d Δσ n 2 σ 1/ L = Δσ = Q d Δt (3.4) Δσ Q55 0 d 23 Δσ Q d Δ + The terms are referred to as reduced components of the lamina and are defined as E 11 Q 11 = (3.5.a) 1 ν 12ν 21 E 22 Q 22 = (3.5.b) 1 ν 12ν 21 ν E Q 12 = (3.5.c) 1 ν 12ν 21 Q 44 = G 12 (3.5.d) Q 55 = G 23 (3.5.e) Q 66 = G 31 (3.5.f) Where is E11 Young s modulus at longitudinal direction, E22 is Young s modulus at transverse direction, ν 12 is Poisson s ratio, G12 is in-plane shear modulus, G 23 and G31 are transverse shear moduli. 47

61 After updating the stress, we transform the stress back into the local shell coordinate system. σ = qσ L q t (3.6) 3.2 Simplified Braiding through Integration Points Methodology Unit Cell 2D tri-axially braided composite s preform architecture presents a variety of size effects that are not encountered in unidirectional prepreg composites laminates. In a triaxially braided perform three yarns are interwined to form a single layer of 0 /±θ material. Each +θ yarn crosses alternatively over and under -θ yarn and vice versa. The 0 yarns were inserted between the braided yarns. A convenient way to analyze braided composites is to consider a unit cell of the material. A unit cell is defined as a smallest unit of repeated fiber architecture [58]. The size of the unit cell is dependent on a number of factors including the size of the yarns, and the angle at which they are interwined. In 2D tri-axially braided composites, the unit cell length is dependent on the mandrel diameter and the number of yarns braided. The width of the unit cell is dependent on the cell width and the braided angle. The fiber preform architecture of 0 /±60 tri-axially braided composites is shown in Figures 3.1, the unit cell is marked as a small box. The detailed fiber architecture is shown in Figure 3.2 (a). 48

62 Unit cell Width Length Figure /±60 tri-axially braid architecture Braiding Method A very popular concept for material modeling of textile composites is a unit cell micromechanical model. However, it has been found that it is very difficult to directly implement the unit cell method in 2D tri-axial composites[59]. The main difficulties came from the following two sides. First, it was very difficult to mesh the unit cell with accurate geometry using commercial FEA preprocessor tools, especially the assurance of fiber volume ratio. Second, the unit cell size of 2D tri-axially braided composites was quite large so that the micromechanical model of the unit cell combined with homogenization[60, and 61] may not capture the damage shape of 2D tri-axially braided 49

63 composites during the impact. Therefore a simplified braided through thickness integration methodology has been proposed here to model 2D tri-axially braided composites. In this method, the geometry of the braided fiber tows was simulated by varying the fiber orientation of the composite at the different integration points in the finite element mesh both through the thickness of the ply and at various locations within the plane of the composite ply. Specifically, the unit cell of the 2D tri-axially braided composite was divided into four sub-cells, with each sub-cell consisting of fiber tows with varying size, fiber orientation and ply layup based on the actual geometrical shape and location. In the model, each fiber tow was modeled as if it was a unidirectional composite. By appropriate placement of the unit cells, the whole composite structure can be modeled. In this study, the effects of strain rate and the fiber/matrix interface were not considered. The detailed fiber architecture inside the unit cell of 2D tri-axial braid composites is shown in Figure 3.2 (a). In order to clearly illuminate the unit cell, the matrix material part is not shown in Figure 3.2 (a). The unit cell was divided into four Sub cells, in which sub-cell A consists of θ /0 /-θ fiber tow and matrix materials, sub-cell B consists of θ /-θ fiber tow and matrix materials, sub-cell C consists of -θ /0 /θ fiber tow and matrix materials, and sub-cell D consists of -θ / θ fiber tow and matrix materials, where θ is the bias fiber braiding angle. The microstructure of 2D tri-axially braided composites is shown in Figure 3.3. Due to lack of microstructural geometry data of braided composites from the manufacturing company, Scanning Electron Microscopy (SEM) was used to measure the geometry of unit cell, which includes length, width, and thickness of 50

64 fiber tows of different orientation angle. In order to reduce the measurement errors, data from several locations have been chosen and averaged. The dimensional data of fiber tows measured by Scanning Electron Microscopy (SEM) is listed in Table 3.1. Table 3.1 The geometry of unit cell (0 /±60 ). Length (cm) Width (cm) Thickness (cm) Sub-cell A Sub-cell B Sub-cell C Sub-cell D

65 Subcell A Subcell B Subcell C Subcell D (a) 22 Shell #A Shell #B Shell #C Shell #D (b) -θ θ θ -θ 0 θ -θ 0 -θ θ 11 (c) Figure 3.2 Schematic diagram of simplified braided through the thickness integration points methodology. (a) unit cell of tri-axially braided composites (fiber only), (b) mesh scheme of unit cell, (c) specification of fiber tow orientation angle at the through thickness integration points in one unit cell (1Ply) (11 and 22 are in-plane direction, 33 is through the thickness direction). 52

66 Figure.3.3 SEM image of 2D tri-axially braided composites. Each sub-cell was modeled with layer composites using one shell element with several through the thickness integration points. Each different braiding angle fiber tow within this sub-cell was modeled as a uni-directional composite through defining the braiding angle at the corresponding integration point based on its stacking sequence. Four sub cells within one unit cell were modeled with four shell elements as shown in Figure 3.2, in which the braiding angle at through the thickness integration points was listed based on its location and stacking sequence. In LS-DYNA, Integration shell could be used to define layered composites through using the user defined integration rules, in which, each sub-cell was defined as a part consisted of different uni-directional composites with different orientation angle defined in its *SECTION_SHELL card. Appropriate used 53

67 defined integration rule was used to assure the bending and membrane stiffness through using *INTEGRATION_SHELL card, where weight factor, and the thickness of each layer played an important rule, and each layer inside this card was defined as a 0 degree uni-directional composites referenced by a *PART card. For example, in sub-cell A, three parts are represented as θ, 0 and -θ respectively in one ply. In sub-cell B, two parts are represented as -θ, and θ respectively in one ply. In sub-cell C, three parts are represented as -θ, 0 and θ respectively in one ply. In sub-cell D, two parts are represented as θ, and -θ respectively in one ply. Therefore, the fiber tow size, fiber architecture and fiber undulation have been modeled by layered composites. Figure 3.4 shows the mesh scheme of a global structure and its relation with the unit cell. In this particular case the braided composites has 6 braided plies. So sub-cell A and sub-cell C needed 18 integration points, sub-cell B and sub-cell D needed only 12 integration points. Therefore, different materials represented by uni-directional composites with its orientation angle and its own material properties at the relevant integration point were interconnected with each other spatially, reproducing virtual 2D tri-axially braided composites. 54

68 18 Integration Points at subcell A: θ 0 /0 0 /-θ 0 12 Integration Points at subcell B:- θ 0 /θ 0 18 Integration Points at subcell C: -θ 0 /0 0 /θ 0 12 Integration Points at subcell D: θ 0 /-θ 0 1 unit cell Figure 3.4 Meshing scheme of braiding through thickness integration points method. 3.3 Material Model of Uni-directional Composites Each part at different location of different sub-cell of 2D tri-axially braided composites has been approximated with uni-directional composites considering different fiber tow sizes and local bundle fiber volume ratio. Based on SEM measured data, the bundle fiber volume ratio was calculated and is listed in Table

69 Table 3.2 Fiber Volume Ratio of different fiber tow (0 /±60 ). 0 Fiber Volume Ratio 60 Fiber Volume Ratio -60 Fiber Volume Ratio Sub-cell A 85% 62% 62% Sub-cell B n/a 50% 50% Sub-cell C 85% 62% 62% Sub-cell D n/a 50% 50% The corresponding material properties of uni-directional composites were calculated using the following equations: E = V E + V E (3.24) x f f m m ν = V ν + V ν (3.25) x f f m m 1 E y = V E f f V + E m m (3.26) 1 E s = V G f f V + G m m (3.27) where G f E f = and (1 +ν ) 2 f G m Em = (3.28) (1 +ν ) 2 m where Ex is the Young s modulus along the fiber 11 direction, Ey is the Young s modulus in transverse 22 direction, 56

70 Es is the in plane shear modulus, and ν x is the in plane Poisson s ratio all calculated in material coordinate system 1-2. V f is the fiber volume ratio, Vm is the matrix volume ratio, E f and Em are the corresponding fiber and matrix Young s moduli, ν f and ν m are the corresponding fiber and matrix Poisson s ratios, G f and Gm are the corresponding fiber and matrix shear moduli. Here, fiber was assumed to be transversely isotopic, and matrix was assumed to be isotropic. The corresponding material properties used in this study are listed in Table 3.3. The Young s modulus, Poisson s ratio, tensile strength and failure strain of T700 fiber were obtained from the fiber producer - Toray, the value for transverse Young s modulus was taken from reference [63] based on representative carbon fiber data. The Young s modulus, Poisson s ratio and tensile strength of M36 Epoxy were from producer of the polymer - Hexply, and EPON 862, which was assumed to have similar properties as M36. Table 3.3 Material properties of fiber and resin. Toray T700s Fiber Hexply M36 Epoxy Density (g/cm 3 ) Young s modulus (Gpa) Poisson ratio Tensile Strength (Mpa) Failure Strain 2.1% n/a 57

71 Maximum stress failure criteria [64, and 65 ]was chosen for the unidirectional composite used for local laminate. There are four failure criteria: (a) Tensile fiber model (fiber rupture) σ 11 0 e = X 2 σ 11 f t 2 0 failed 1 < 0 elastic (3.29) When failed, the material is degraded through setting E a = Eb = Gab = ν ba = ν ab = 0 (b) Compressive fiber model (fiber buckling and kinking) σ < 11 0 e = X 2 σ 11 f c 2 0 failed 1 < 0 elastic (3.30) When failed, the material is degraded through E ν = ν = 0 a = ba ab (c) Tensile matrix mode (Matrix cracking under transverse tension and shearing) e τ m c σ = Yt + S 2 0 failed 1 < 0 elastic (3.31) When failed, the material is degraded through E ν = 0 G = 0 b = ab ab (d) Compressive matrix mode (Matrix cracking under transverse compression d shearing) e σ 22 τ m + 2 c Yc c c σ = 2S Yc 4S 1 + S 2 0 failed 1 < 0 elastic (3.32) 58

72 Here, When failed, the material is degraded through E ν = ν = 0 G = 0 X t is tensile strength in 11 direction, b = ba ab X c is compressive strength in 11 direction, Yt is tensile strength in 22 direction, Yc is compressive strength in 22 direction, and Sc is in plane shear strength. Experimental results and theoretical research showed that tow strength depends on the number of fibers and could be described using Weibull distribution[66]. As shown in Figure 3.5, fiber bundle strength is degraded for increasing number of fibers. The main reason is that fiber interaction insider the fiber bundle could weaken the strength. For example, the 0 /±60 braided preform had 12k flat tow fibers in the ±60 (bias) directions and 24k flat tow fibers in the 0 (axial) direction, therefore, tensile strength of the 0 tow could be assumed to be about 50% of the 60 and -60 bundles, because 0 tow contained twice as many fibers as ±60 tow. Local failure was triggered by local stress at each integration point through checking all stress failure criteria. The material was degraded at integration point if the failure criteria were met at each time step. ab 59

73 (a) Bundle Strength Number of fibers in bundle (b) Figure 3.5 Scheme of fiber bundle strength model.(a) schematic diagram of the fiber bundle specimen, (b) schematic diagram of the fiber bundle strength distribution. 60

74 3.4 Discussion on Element Deletion In LS-DYNA impact simulations, cracking induced by impact is modeled by removing the element form the mesh. This procedure is often named as element kill or element erosion method. Failure occurs when an element is subjected to a certain critical stress, strain or a user defined damage value. The advantage of this method is that it is easy to use and it does not contribute to any additional calculation cost. It also gives good results if it is used correctly with suitable element size and correct damage models. A drawback is that part of mass is lost, which could cause numerical problems. In LS- DYNA, this part of mass will be considered in the contact treatment through *CONTROL_CONTACT card. Sometimes, including this part of mass may cause numerical problems of contact. In this particular case, element is deleted in accordance with the following criteria: 1. When all integration points in one element meet its failure criteria, the element is deleted from the mesh. 2. When an effective strain damage criterion of an element is reached, the element is deleted. The damage criteria is defined as following: Other methods of element removal skill include element splitting and using cohesive elements [67 and 68]. In LS-DYNA, element splitting can be implemented by using *CONSTRAINED_TIE_BREAK and the * CONSTRAINED_TIE_NODES to Failure. Additional cohesive element combined cohesive law can also be implemented. Those methods will not cause mass losing problem, however, meshing and modeling using those methods will cause a lot tedious work and the relevant calculations are very time consuming. 61

75 3.5 Conclusion A simplified braiding through thickness integration points methodology has been proposed to model 2D tri-axial braided composites, the fiber tow size and fiber architecture have been inherently considered in this model. Additionally, the process of determining composites material constants has been discussed. In the following chapters, coupon test and ballistic impact simulation using this methodology will be discussed. 62

76 CHAPTER IV EXPERIMENTAL RESULTS of BRAIDED COMPOSITES Ballistic impact tests on 2D tri-axially braided composites flat panels using a soft gelatin projectile were preformed to identify failure modes that occur at the high strain energy density during impact loading. 3D image correlation photogrammetry technique was used to measure the deformation. The ballistic impact tests to be discussed here were conducted by members of the Ballistic Impact Laboratory at NASA Glenn Research Center (70, 71, 72. and 73). They are discussed primarily to give insight into the impact simulation which will be presented in later chapters. Coupon level mechanical tests were performed to provide the basic properties data on the composites for use. Experimental results will allow in depth comparison with LS-DYNA numerical simulation. The measurement results and experimental observation are described in this chapter D Tri-axially Braided Composites Flat Panel Three different kind of carbon/epoxy braided composites were investigated, they are braided 0 /±60 and 0 /±45 preforms reinforced with M36 resin (toughened epoxy system from HEXCEL Composites)and braided 0 /±60 preforms reinforced with Epon 862 resin (toughened epoxy system from Shell). The panels will be referred to as M36[0 /±60 ], M36[0 /±45 ] and Epon 862[0 /±60 ] in what follows. Figure 4.1 shows 63

77 the fiber architecture of braided preform. Composites were fabricated by resin film infusion or resin transfer modeling into six layers of a tri-axially braided T700 carbon fiber (from Toray) preform. The preforms were manufactured by A&P Technology. The 0 /±60 braided preform had 12k flat tow fibers in the ±60 (bias) directions and 24k flat tow fibers in the 0 (axial) direction. The 0 /±45 braided preform had 12k flat tow fibers in the ±45 (bias) directions and 36k flat tow fibers in the 0 (axial) direction. Both layups were designed to have a fiber-volume fraction of 60%. In the 0 /±60 system, 33.3% of the fibers were in the -60 direction, 33.3% were in the +60 direction, and 33.3% were in the 0 direction. Since the fiber volume in each direction was equal, both the individual plies and the entire lay-up were quasi-isotropic. In the 0 /±45 system, 48.5% of the fibers were in the ±45 direction and 51.5% were in the 0 direction. The spacing between axial tows was 0.89 cm in both layups. The spacing between the perpendicular repeating unit cells was 0.52 cm for the ±60 layup and 0.89 cm for the ±45 layup. Composite panels were formed from six plies with the 0 fibers aligned. The cured composites had a nominal thickness of about 3.2 mm, though there were systematic thickness variations associated with the braid pattern. Composites panels were fabricated in the form of 61 cm x 61 cm flat plates. The ballistic impact test matrix is shown in Table

78 0 (axial) -θ (bias) +θ (bias) Figure /±θ 2D tri-axially braided architecture. Table 4.1 Ballistic impact test matrix Fiber Architecture Resin Panel Serial Number ARAMIS Measurement Total Panel Number 0 /±60 M36 GE thru 8 No 8 0 /±45 M36 GE thru 4 No 4 0 /±60 Epon 862 LG670 &LG671 Yes Coupon Test Specimen The M36[0 /±60 ] and M36[0 /±45 ] composite panels were first used for impact testing as described in reference [Roberts ]. The panels were impacted by a soft, gelatin projectile at m/s and m/s for M36[0 /±60 ] and M36[0 /±45 ] composite, respectively. Post impact testing ultrasonic inspection suggested that the impact damage was concentrated near the impact site and in regions which were visibly damaged. 65

79 Mechanical test specimen locations were selected from undamaged regions of the panel, as illustrated in Figure 4.2. The gage width of specimens was chosen based on repeating unit cells of either the spacing between axial tows in the longitudinal direction or the tow width in the transverse direction. Straight-sided tensile specimens were 30.5 cm long with a width of 2.54 cm or 4 repeating unit cells, whichever was larger. Therefore, the transversely oriented M36[0 /±60 ] had a width of 2.54 cm and the width was 3.58 cm for M36[0 /±45 ]straight-sided tensile specimens. A bowtie specimen design was also used for tensile testing in an attempt to isolate cut fibers from the gage section. Figure 4.3 illustrates the shape and dimensions of the bowtie specimens; the gage width for all was 3 axial tow repeating unit cells. The gage width in each of the bowtie specimens was chosen to contain three axial tows. The compression specimens were 2.54 cm wide and 7.62 cm long. The Iosepescu specimens were 1.91 cm by 7.62 cm with a 1.14 gage section. Straight-sided tensile specimens were instrumented with a single axial and a single transverse strain gage. Compression specimens were instrumented with a single axial strain gage. Iosepescu specimens were instrumented with a single ±45 strain gage. The gage section of the bowtie specimens was painted with a grid of dots spaced cm apart. Additionally, 3D image correlation photogrammetry technique was used to obtain strain distribution for bowtie tensile specimens. 66

80 Figure 4.2 M36[0 /±60 ] composite panel post impact with mechanical specimen locations designated. Specimen A B C D E A ±60 longtensile ±60 trans-tensile ±45 longtensile º º º 2.68 B E D Figure 4.3 Specimen design and dimensions in centimeters for two longitudinal and one transverse bowtie configurations. (Unit:cm) C 4.3 Gelatin Projectile for Ballistic Impact Use of a soft projectile allows a large amount of kinetic energy to be transferred into strain energy in the target before penetration occurs. The soft projectiles used in the impact tests were made using a mixture of gelatin and micro balloons with a density of 67

81 960 kg/ m3. The mixture was cast in the shape of a cylinder 12.7 cm long and 7.0 cm in diameter. Figure 4.4 shows an example of a projectile before and after impact. For impact testing, a projectile was mounted in a sabot and shot into a target using a 20.3 cm diameter gas gun. The axis of the cylindrical projectile was aligned along the line of flight so that the flat end of the cylinder made first contact with the target. Impact velocity was measured digitally using a high-speed video camera with a view perpendicular to the line of flight. During a test, the projectile initially flattened into a disk approximately 25.4 cm in diameter. At low velocities the projectile could be recovered undamaged, and at high velocities the projectile was disintegrated into small pieces. At intermediate velocities the projectile rebounded to the partially damaged shape shown in Figure 4.4(b). (a) (b) (c) Figure 4.4 Gelatin projectile (a) before, (b) during and (c) after impact. 4.4 Experimental Setup of Ballistic Impact Test The data acquisition systems used for composite impact testing utilized a number of types of transducers, including high speed digital video cameras, load cells, 68

82 accelerometers, strain gages and laser displacement transducers. While it is difficult to measure forces at the location of impact, forces, strains, displacements and accelerations at other locations provide data that are critical for developing and validating numerical simulations. A particularly useful technique is available for non-contact measurement of spatial displacements and strains in composite panels using a pair of calibrated high speed digital video cameras (ARAMIS, GOM mbh, Braunschweig, Germany). This technology is so called three-dimensional image correlation photogrammetry, which requires significantly less sample preparation than moire, and has greater dynamic range and robustness than ESPI, with lower system cost. Composites panel consisted of applying a regular or random high contrast dot pattern to the surface, typically with an airbrush. Thousands of unique correlation areas known as facets (typically 15 pixels square) were defined across the entire imaging area. The center of each facet was a measurement point that can be thought of as an extensometer and strain rosette. These facet centers were tracked in each successive pair of images, with accuracy up to one hundredth of a pixel. Then, using the principles of photogrammetry, the 3D coordinates of each facet were determined for each picture set. The results were the 3D shape of the component, the 3D displacements, and the in-plane strains. Data could be presented as color plots, movies, section line diagrams, etc, and ASCII exports supported further analysis and comparison. Although only two picture sets were required to measure the change from zero to maximum load, multiple image sets provided a progressive measurement of deformations and strains. 69

83 4.5 Ballistic Impact Test Results The first set of impact tests was performed on eight M36[0 /±60 ] flat plates with impact velocities ranging from 103 m/s to 198 m/s. The measured penetration threshold was between 150 m/s and 161 m/s. Figure 4.5 shows the front of composite plate tested at 150m/s. In the figure the small whitened circular region at the center of the plate is the initial contact area. The larger, less distinctive, whitened are the region over which the gelatin projectile was spread when it was completely flattened into a disk before rebounding. Post impact check showed that the fiber broke at center of panel. Figure 4.6 shows the front and back surfaces of the composite plate tested at 192 m/s. Failure was initiated at the center of the plate where fiber tensile strain reached its maximum. Cracks then propagated along the ±60 bias fiber directions. When these cracks extended beyond the initial contact area, triangular flap fold back along 0 axial fibers. The cracks did not propagate far from the initial impact area and there was no apparent delamination between plies. Figure 4.5 M36[0 /±60 ] composites plate after impact at 150 m/s. 70

84 (a) Front (b) Back Figure 4.6 M36[0 /±60 ] composites plate after impact at 192 m/s. [*Tests were conducted by NASA Glenn Research Center personnel] The second set of impact tests was performed on four M36 [0 /±45 ] flat plates with impact velocities ranging from 215 m/s to 263 m/s. The penetration threshold was about 215 m/s. Figure 4.7 shows the front surfaces of the composite plate tested at different velocities. Failure was initiated at the center of the plate where fiber tensile strain reached its maximum. Cracks then mainly propagated along the 0 axial fiber directions. The cracks did not propagate far from the initial impact area and there was no apparent delamination between plies. 71

85 (a)215 m/s (b)237 m/s (c)251 m/s (d)263 m/s Figure 4.7 M36[0 /±45 ] composites plate after impact. [*Tests were conducted by NASA Glenn Research Center personnel] The third set of impact tests was performed on two Epon 862[0 /±60 ] flat plates with impact velocity below penetration threshold. The impact velocity of Panel LG670 is 126 m/s and the impact velocity of Panel LG671 is 128 m/s. For 3D photogrammetry measurement, the dot information would be lost if penetration happened, therefore, the main objective of using ARAMSIS system was to measure the deformation. Figure 4.8 (a) shows the sketch of ARAMIS measurement setup displaying that the field of view was located at the center of flat panels and the size was 30.5cm x 30.5 cm where the 72

86 centerline of plate is indicated as a dot line, and Figure 4.8 (b) shows the out of plane displacement in the impact. Figure 4.9 shows the measured out of plane deflection along a cross-section at the plate center under impact velocity of 128 m/s. More detailed measurement results will be discussed in combination with numerical results in chapter 5. Field of View (ARAMIS) Center (a) Specification of ARAMIS setup (b)aramis measurement results Figure 4.8 Sketch of ARAMIS measurement. [*Tests were conducted by NASA Glenn Research Center personnel] 73

87 Figure 4.9 Out of plane deflection along a cross-section at the plate center under impact velocity of 128 m/s. [*Tests were conducted by NASA Glenn Research Center personnel] Especially for flat panel LG670, the small amount of fiber was broken at the center of the back surface. Figure 4.10 shows the strain distribution at the moment of fiber broken, where ε x is strain perpendicular to the axial (0 ) direction, andε y is strain along the axial (0 ) direction. The history of the out of plane displacement and strain is shown in Figure It could be found that fiber is initially broken at the time of 150μs, and the relevant failure strain is between 0.9% and 1%. This experimental results correlate with the 74

88 material properties set in the methodology of simplified braiding through the thickness integration points discussed in the Chapter 3. (a) (b) (c) (d) Figure 4.10 Strain distribution of panel LG670 under impact velocity 126m/s. (a) ε x before fiber broken, (b) ε x after fiber broken, (c) ε y before fiber broken, and (d) ε y after fiber broken. [*Tests were conducted by NASA Glenn Research Center personnel] 75

89 Strain εx εy z displacement Out of plane displacement (mm) Time (μs) Figure 4.11 Strain and center deflection of panel LG670 under impact velocity 126m/s [*Tests were conducted by NASA Glenn Research Center personnel] 4.6 Coupon Specimen Test Results The mechanical properties of the two composite panels were evaluated through tensile, compressive, and shear testing in both axial and transverse directions by Cincinnati Testing Laboratory [71]. Three specimens were tested for each condition. All tests were conducted in constant displacement-rate control Tensile and Compressive Behavior The results of the tensile, bowtie-tensile, and compressive tests were summarized in Table 4.2. The values are averages of three test results. The M36[0 /±60 ] and 76

90 M36[0 /±45 ] tensile response curves were plotted in Figure 4.12 The M36[0 /±60 ] and M36[0 /±45 ]compressive stress-strain curves were plotted in Figure Comparison of the straight-sided tensile response and the straight-sided compressive response suggested that the moduli are similar for all the composites with 0/±60 fiber reinforcement. Figure 4.12 also shows that the strength is ordered highest to lowest as M36 along the axial fibers, M36 perpendicular to the axial fibers, The same figures of straight-sided specimen behavior indicated that the M36[0 /±45 ] specimens consistently yielded the highest strength and stiffness when tested parallel to the axial fibers and the lowest strength and stiffness when tested perpendicular to the axial fibers. The bowtie tensile tests results shown in Figure 4.14 were included in this study because the optimum specimen design was not obvious for braided architectures. The cross-sections of the straight-sided specimens in this study contained a number of cut fibers. In the bowtie specimens, the grip tapered to the reduced section at an angle that complimented the off-axis fiber orientation. This led to a gage section with nearly all the fibers continuing to the gripped region. In each case, the bowtie specimens should yield a higher strength than the straight specimens. The bowtie strengths were higher for the axially and transversely loaded M36/±45 and for the transversely loaded M36/±60. However, the longitudinally loaded M36/±60 bowtie specimens had lower strength than the straight-sided specimens. The effect of the bowtie specimen on strain response is more difficult to measure using strain gages. Additionally bowtie specimen tests were implemented at The University of Akron using 3D image correlation photogrammetry [73]. Figure 4.15 shows the strain distribution of bowtie specimen. Note that, as expected, 77

91 there is some concentration of strain near the notches in the bowtie specimens, but that the gauge area has a fairly uniform strain distribution. Table 4.2 Ultimate stress and strain, elastic modulus and Poisson s ratio for static tension and compression in the axial (0º) and transverse loading orientations. Composite Test Orientation Ult. Stress MPa Failure Strain % Modulus GPa Poisson s Ratio Tensile axial M36 / ±60 Bowtie axial M36 / ±45 Tensile axial Bowtie axial M36 / ±60 Comp. axial M36 / ±45 Comp. axial M36 / ±60 M36 / ±45 Tensile transverse Bowtie transverse Tensile transverse Bowtie transverse M36 / ±60 Comp. transverse M36 / ±45 Comp. transverse

92 Tensile Stress (Mpa) /-45 (Axial) 0+/-60 (Trans.) 0+/-60 (Axial) /-45 (Trans.) Strain Figure 4.12 Straight-sided tensile response for M36 resin and both0 /±60 and 0 /±45 fiber architectures at axial and transverse directions. [*Tests were conducted by NASA Glenn Research Center personnel] 79

93 Compressive Stress (Mpa) /-45 (Axial) 0+/-60 (Trans.) 0+/-60 (Axial) /-45 (Trans.) Strain Figure 4.13 Compressive response for M36 resin and both 0 /±60 and 0 /±45 fiber architectures at axial and transverse directions. [*Tests were conducted by NASA Glenn Research Center personnel] 80

94 /-45 (Axial) Tensile Stress (Mpa) /-60 (Trans.) 0+/-60 (Axial) /-45 (Trans.) Displacement (mm) Figure 4.14 Bowtie tensile response for M36 resin and both0 /±60 and 0 /±45 fiber architectures at axial and transverse directions. Precise strain is not available for bowtie test configuration. [*Tests were conducted by NASA Glenn Research Center personnel] 81

95 (a) (b) Figure 4.15 Axial strain overlay of bowtie specimens loaded axially to 97 MPa and transversely to 303 MPa Shear Behavior The results of the Iosepescu shear testing were summarized in Table 4.3. All of the inplane (1-2 and 2-1) shear moduli were very consistent. The ultimate strength also agreed 82

96 very well for the in-plane orientations of the isotropic M36/±60 composites. The deviation was larger for the non-isotropic 0º ±45º layup. Table 4.3 Summary of shear responses where orientation 1 is parallel to the axial fibers, orientation 2 is perpendicular to the axial fibers in the plane of the panel, and orientation 3 is perpendicular to the axial fibers through the thickness of the panel. Composite Test Orientation Ult. Stress MPa Ult. St. Dev. MPa Shear Modulus GPa Iosepescu M36 / ±60 Iosepescu M36 / ±45 Iosepescu Iosepescu Summary of Coupon Specimen Tests Tensile, compressive and shear properties were obtained for three composites and used to augment analysis of impact testing. Care must be taken in applying these results since the M36/±60 and M36/±45 coupons came from panels after impact testing. On the basis of these results, the M36/±45 configuration had the best response for properties parallel to the axial tows. M36/±60 has the more isotropic behavior, as demonstrated in the Iosepescu shear data. Also the M36/±60 had higher strengths when tested transverse to the axial tows. 83

97 CHAPTER V BALLISTIC IMPACT SIMULATION A simplified braiding through thickness integration point method has been developed to model the 2D tri-axially braided composites, particularly when it is under high velocity impact loading. In this chapter, this method has been implemented in LS-DYNA and was used to simulate ballistic impact tests on two different fiber preform architecture ([0 /±60 ] and [0 /±45 ]) braided composites, the simulation results were compared with experimental observation in two aspects: deformation and failure. Additionally, the coupon specimen tensile tests have also been simulated. Strain rate dependent material behaviors are not involved in the study. 5.1 Composite Model Braided composites were modeled using the method described in Chapter Material Modeling of Gelatin Wilbeck gave a comprehensive description of impact by a soft projectile in his work [74]. When a soft projectile impacts a target plate, the particles on the front surface of the projectile are instantaneously brought to rest relative to the target and a shock wave is formed. The pressure in the shock region is very high initially and is constant throughout 84

98 the region. As the shock propagates up the projectile, a very high pressure gradient is developing, and then a very complicated set of stress waves propagate in the projectile. Since the gelatin had a very low strength and stiffness, it can be described using the LS- DYNA Material 9 (MAT_NULL), in which the pressure-volume relation is modeled using the equation of state (EOS). This material model uses the equation of state without computing deviatoric stresses. The equation of state is defined by card *EOS_TABULATED where the pressure P is expressed as P = C( ε ) + γt ( ε E (5.1) V V ) Where ε V is the volumetric strain given by the natural logarithm of the relative volume, E is internal energy. C,γ and T are corresponding material constants. In this study, a set of proprietary material constants have been used in the numerical simulation. 5.3 Arbitrary Lagrangian Eulerian (ALE) Formulation Experimental results showed the gelatin flows during the impact process, and a conventional Lagrangian description of this kind of large deformation can induce severe mesh distortion. Therefore, an arbitrary Lagrangian Eulerian (ALE) formulation was chosen for soft projectile impact simulation. The ALE formulation is based on the arbitrary movement of a reference domain [75]. 85

99 Void Part Gelatin Material Impacting Velocity Merged nodes on material-mesh and void-mesh Composites Target Figure 5.1 ALE mesh scheme of gelatin impact on composites target. To solve the governing equations posed in an ALE reference system, LS-DYNA relies on a so-called operator split technique where each time step is split into a Lagrangian phase and an advection phase. In the Lagrangian phase, the FE model is treated as if it was purely Lagrangian. That is, the mesh is forced to follow the motion of the material s flow so that the advection phase the nodes of the mesh are repositioned to new, preferred, locations and the solution is mapped from the old configuration onto the new one. A spatially second order accurate advection algorithm, referred to as the van Leer method, was used in LS-DYNA. Both the operator split technique and the van Leer advection algorithm are described in [76]. In this application, the soft gelatin projectile was modeled with a multi-material Eulerian formulation through choosing solid element type (*SECTION_SOLID card, type 12), while the composite target was modeled with a Lagrangian formulation. Instead of contact in the pure Lagrangian formulation, a penalty-based ALE Lagrangian coupling algorithm was chosen by using the card *CONSTRAINED_LAGRANGE_IN_SOLID. The meshes used to model the projectile, 86

100 void part and the plate are shown in Figure 5.1. The validity of numerical results was examined by verifying the stability of the explicit finite element analysis through energy analysis and by comparing numerical results to the experimental data. 5.4 LS-DYNA Simulation Results Coupon Specimen Simulation Material constants, especially the moduli are mainly dependent on the fiber volume ratio of different fiber bundle modeled using simplified braided through thickness integration point methodology. Figure 5.2 shows the experimental results as well as tensile test simulation of coupon specimen, including 0 /±60 and 0 /±45 braided composites. Overall, LS-DYNA simulations correlated with experimental results, and some discrepancy has been found for 0 /±45 specimen. This is mainly due to unavailable microscopic measurement data for 0 /±45 specimen. Also, 36k instead of 24k fiber is put on 0 (axial) direction for 0 /±45 braided composites during manufacturing. The LS-DYNA simulation is achieved through changing the length of Sub-cell A and Sub-cell C, but keeping the same fiber volume ratio of different fiber tows as 0 /±60. 87

101 /-60 (Axial) /-45 (Axial) LS-DYNA Tensile Stress (Mpa) /-45 (Axial) 0+/-60 (Axial) LS-DYNA 0+/-60 (Trans.) LS-DYNA 0+/-60 (Trans.) 0+/-45 (Trans.) LS-DYNA /-45 (Trans.) Strain Figure 5.2 Results of straight-sided for both 0 /±60 and 0 /±45 fiber architecture braided composites (LS-DYNA simulations and experiments) Penetration Threshold Impact of the composite plate was simulated for velocities ranging from 91.4 m/s to 230 m/s. The penetration threshold for the 0 /±60 flat panel was determined to be between m/s and 154 m/s. Moreover, the penetration threshold for the 0 /±45 flat panel was determined to be between 215 m/s and 218 m/s. These results are summarized in Table 5.1. It shows that the numerical simulation results agree well with experimental measurements. 88

102 Table 5.1 Penetration threshold velocity comparison Penetration threshold velocity (m/s) Experimental result LSDYNA simulation Flat Panel 0 /± Flat Panel 0 /±45 About 215m/s Deformation Analysis Figure 5.3 shows the deformation behavior of both projectile and the impacted composites panels at the different stages. From the moment of impact, the projectile was gradually squashed into the panel and captured the pancake like shape, which was similar to that observed in high speed video of composite impact experiments. (a) Experiment (b) LS-DYNA (1) During the initial stage of impact 89

103 (a) Experiment (b) LS-DYNA (2) During the impact Figure 5.3 The deformation of gelatin and composites plate at the impact velocity of 128 m/s LS-DYNA simulations and experiments. Figure 5.4 shows the comparison of the out of plane displacement history of impact center taken from the numerical model with the experimental measurement by 3D image correlation photogrammetry ARAMIS system. The impact velocity was 128 m/s which was lower than penetration threshold. Comparison shows agreement between simulation and test from start till the center reach its maximum deformation and their respective maximum values were closed. Some discrepancy existed after the panel plate rebounded from the maximum out of plane displacement, when the panel was in its unloading stage. This was mainly caused by the material model of gelatin, and the inherent algorithm of ALE formulation can not prevent the energy leakage, especially when projectile rebounded from the impacted targets. 90

104 4 Center out of plane deflection (cm) Test (Velocity = 128 m/s) LS-DYNA (Velocity = 128 m/s) Time (s) Figure 5.4 Center deflection of Epon 862[0 /±60 ] braided composites plate. In addition, the displacement from panel centerline has been exported from the nodes shown in Figure 4.8 with locations corresponding to the ARAMIS measurement position. Figure 5.5 shows the panel at the impact velocity of 126 m/s and Figure 5.6 shows the panel at the impact velocity of 128 m/s. the displacement at different time has been compared. The comparisons show that the simulation results agree with the experimental measurements. It can be concluded that this method captures the main characteristics of deformation. 91

105 Out of plane deflection (cm) LS-DYNA ( t= 37 ms) Test ( t= 37 ms) LS-DYNA ( t= 110 ms) Test ( t= 110 ms) LS-DYNA ( t= 370 ms) Test ( t= 370 ms) LS-DYNA ( t= 592 ms) Test ( t= 592 ms) LS-DYNA ( t= 1370 ms) Test ( t= 1370 ms) Location (cm) Figure 5.5 Out of plane deflection along a cross-section at the plate center under impact velocity of 126 m/s. 92

106 Out of plane deflection (cm) LS-DYNA ( t= 40 ms) Test ( t= 40 ms) LS-DYNA ( t= 225 ms) Test ( t= 225 ms) LS-DYNA ( t= 484 ms) Test ( t= 484 ms) LS-DYNA ( t= 706 ms) Test ( t= 706 ms) LS-DYNA ( t= 965 ms) Test ( t= 965 ms) Location (cm) Figure 5.6 Out of plane deflection along a cross-section at the plate center under impact velocity of 128 m/s Failure Analysis and Discussion Figure 5.7 shows damage progression of M36[0 /±60 ] flat panel during impact velocity of 192 m/s. The crack initiated at the location that was close to the impact center and grew symmetrically along ±60 degree directions and then along 0 direction. The final damage shape and size matched the experimental observation shown in Figure 4.6. Figure 5.8 shows the numerical parametric study of damage progression of M36[0 /±45 ] flat panel during impact velocity of 216 m/s. The cracks initiated at the center of plate and grew along 0 degree directions and then formed the final damage shape. The damage shape, size and its propagation sequence were all similar to the experimental observations shown in Figure

107 (a) t=0μs (b) t=115μs (c) t=140μs (d) t=297μs Figure 5.7 LS-DYNA simulation of damage pattern of M36[0 /±60 ] at impact velocity 182 m/s. 94

108 (a) t=0μs (b) t=90μs (c) t=140μs (d) t=180μs Figure 5.8 LS-DYNA simulation of damage pattern of M36[0 /±45 ] at impact velocity 216 m/s. As mentioned in Chapter 3, this methodology is not intended to simulate the damage progression at the micromechanical level. It is only a numerical method combined the 95