A N ENGINEERING A P P R O A C H TO M O D E L L I N G T H E I M P A C T A N D BALLISTIC RESPONSE OF L A M I N A T E D COMPOSITE STRUCTURES by XIANGYANG QUAN B . S c , East China Institute of Technology, Nanjing, China, 1984 M . S c , East China Institute of Technology, Nanjing, China, 1987 A THESIS SUBMITTED IN P A R T I A L F U L F I L L M E N T OF THE REQUIREMENTS FOR T H E D E G R E E OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E STUDIES (Department of Civil Engineering) We accept this thesis as confirming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A November 1998 © Xiangyang Quan, 1998 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of C/ h'V n The University of British Columbia Vancouver, Canada Date DE-6 (2/88) A/hU^k* 2.7. /9?3 &€S^'- Abstract Abstract A super finite element program, SENACS, that was previously developed for transient non- linear analysis of isotropic structures, has been modified to handle the structural response of layered (laminated) composite materials. Both material and geometric non-linearities have been taken into account in the formulation. The structural analysis capabilities of the code have been demonstrated by successfully comparing the predictions with other experimental, analytical, and numerical results in the literature. Impact problems have been subdivided into two groups: nonpenetrating and penetrating. In each case, appropriate contact laws are introduced to evaluate the local impact load on the structure while the structural analysis part of S E N A C S computes the target global response. In nonpenetrating impact events, where there is only elastic indentation in the targets, the Hertzian contact law has been employed to establish the impact force as a function of the local indentation. Predictions of the nonpenetrating impact response of plain and stiffened laminated composite plate and shell structures have been found to be in good agreement with the experimental measurements previously reported in the literature. Modelling of penetrating impact problems has been guided by experimental investigations. A phenomenological analytical model for static penetration of composite materials has been developed. In this model, three major penetration mechanisms have been accounted for: hole expansion, flexural deformation of a delaminated plate (split-plate), and transverse plugging. ii Abstract The parameters of the model have been determined through material characterisation tests. The penetration model is introduced as a local contact behaviour in SENACS, and used to predict the penetrating impact response of composites to projectiles with different conical nose shapes. Ballistic impact response of two different material systems, IM7/8551-7 Carbon Fibre-Reinforced Polymer (CFRP) and S2-glass/phenolic resin Glass FibreReinforced Polymer (GFRP) laminates provides the physical background and experimental verifications for the present model. Finally, a number of numerical ballistic simulations have been carried out to investigate the influence of structural shapes and sizes, loading conditions, projectile geometry, and impact velocity on the energy absorption capability of composite materials. Table of Contents Table of Contents Abstract ii Table of Contents iv List of Tables List of Figures x xiii Nomenclature xxviii Acknowledgement xxxvii Chapter One: Introduction 1.1 Impact Events in Applications of Composite Materials 1 1 1.2 General Approaches to Impact Problems 2 1.3 Objectives and Scope of the Present Research 4 Chapter Two: Super Finite Element Formulation 7 2.1 Introduction 7 2.2 Finite Element Equations 8 8 2.2.1 Strain-Displacement Relations 2.2.2 Constitutive Relations 13 2.2.3 Virtual Work Principle 15 Table 2.3 of Solution Scheme Contents 19 2.3.1 Time Integration Method 20 2.3.2 Newton-Raphson Iteration Scheme 21 2.4 Summary Chapter Three: Nonpenetrating Impact Response 3.1 Literature Review 22 29 29 3.1.1 Analytical Work on Plates and Shells 29 3.1.2 Numerical Work on Plates and Shells 30 3.1.3 Experimental, Analytical, and Numerical Work on Stiffened Structures 32 3.2 Contact Law 33 3.3 Solution Scheme 34 3.4 Results for Laminated Plates and Shells 37 3.4.1 Small Deflection Analysis 37 3.4.2 Large Deflection Analysis 43 3.5 Results for Stiffened Plates and Shells 45 3.5.1 Verification 45 3.5.2 Numerical Experiments 46 3.6 Summary Chapter Four: Literature Review on Penetration Mechanisms and Modelling 4.1 Penetration Mechanisms 48 64 65 4.1.1 Hole Expansion 65 4.1.2 Dishing/Petalling 67 Table of Contents 4.1.3 Transverse Plugging 69 4.1.4 Delamination 71 4.2 Transition of Penetration Mechanisms 72 4.3 Penetration Modelling of Composites 73 4.3.1 Static vs. Dynamic Tests 74 4.3.2 Analytical Models for Projectiles with Spherical Head Shapes 75 4.3.3 Analytical Models for Blunt-Ended Projectiles 76 4.3.4 Analytical Models for Projectiles with Conical Head Shapes 78 4.3.5 Analytical Model for FSP (Fragment Simulating Projectiles) 80 4.3.6 Numerical Modelling 81 4.4 Summary 82 Chapter Five: Penetration Mechanisms 5.1 86 Experimental Investigations 86 5.1.1 Static Penetration Tests 86 5.1.2 Static Indentation Tests 88 5.1.3 Experimental Force-Displacement Relationships 89 5.1.4 Characterised Force-Displacement Relationships 92 5.2 Identification of Penetration Mechanisms .....95 5.2.1 Elastic Deformation 96 5.2.2 Hole Expansion 5.2.3 Flexural Deflection of a Split-Plate 103 5.2.4 Transverse Plugging 114 5.2.5 Friction 116 '. 97 vi Table 5.3 Summary Chapter Six: Static Penetration Model and Applications 6.1 Model Development of Contents 120 147 147 6.1.1 Description and Flow Chart of the Penetration Model 148 6.1.2 Sensitivity Analysis 152 6.2 Results and Discussion 155 6.2.1 CFRP Laminates 155 6.2.2 GFRP Laminates 158 6.3 Summary Chapter Seven: Ballistic Applications 160 177 7.1 Solution Scheme 177 7.2 Comparison with Experiments 181 7.2.1 CFRP Laminates 182 7.2.2 GFRP Laminates 185 7.3 Effect of Projectile Cone Angle 191 7.3.1 CFRP Laminates 191 7.3.2 GFRP Laminates 192 Effect of Impact Velocity 193 7.4 7.4.1 CFRP Laminates 194 7.4.2 GFRP Laminates 196 7.4.3 Comparison Between the Ballistic Resistance of CFRP and GFRP Laminates 199 7.5 Effect of Target Thickness 199 Table of Contents 7.6 Effect of Boundary Conditions 200 7.7 Effect of Planar Size 200 7.8 Effect of Projectile Mass 201 7.9 Summary 201 Chapter Eight: Summary, Conclusions, and Future Work 238 8.1. Summary 238 8.2. Conclusions 240 8.3. Future Work 242 References 244 Appendix A: Shape Functions for Super Shell and Beam Elements 255 A. 1 Super Shell Element 255 A.2 258 Super Beam Element Appendix B: Elastic and Plastic Constitutive Matrices 259 Appendix C: Numerical Verifications and Structural Applications of SENACS 260 C l . Static Problems 260 C. 1.1 Small Deflection, Linear-Elastic Analysis 260 C.1.2 Large Deflection, Linear-Elastic Analysis 261 C. 1.3 Small Deflection, Elasto-Plastic Analysis 262 C.2 Eigenvalue Problems C.3 Transient Problems 264 : 266 Table of Contents Appendix D: High Frequency Oscillations in Contact Force-Time Histories 283 Appendix E: Derivation of Bending Compliance of a Clamped Circular Ring 291 ix List of Tables List of Tables Table 2.1 Nodal degrees of freedom for a super shell element (Jiang et al., 1993) 24 Table 2.2 Nodal degrees of freedom for a super beam element (Jiang et al., 1993) Table 2.3 Gauss integration rule used in SENACS for a super shell element 24 Table 3.1 Properties of the laminated plate used for convergence studies 50 Table 3.2 Fundamental frequencies of the laminate defined in Table 3.1 obtained from 24 different mesh geometries 50 Table 3.3 Properties of the laminated plate used in instrumented impact tests 50 Table 3.4 Properties of the cylindrical shell used in the analysis of Christoforou and Swanson(1990) 50 Table 3.5 Properties of the laminated plate used in Matsuhashi et al. (1993) 51 Table 3.6 Material properties and lay-up of the stiffened plate shown in Figure 3.15 51 Table 3.7 Parametric studies: fundamental frequency and maximum impact force of the stiffened laminated cylindrical shells 51 Maximum impact force for a stiffened plate impacted at different sites 51 Geometric sizes and material properties of the indenters used in static penetration and indentation tests 122 Table 3.8 Table 5.1 Table 5.2 Geometric sizes and material properties of the CFRP laminated plates used in static indentation and penetration tests 122 Table 5.3 Geometric sizes and material properties of the GFRP laminated plates used in static indentation and penetration tests 122 Table 5.4 Summary of the equations used to calculate the contact force during penetration of a laminate 123 Table 7.1 Mass and length of projectiles with different cone angles in gas gun and powder gun tests on CFRP laminates. The projectile length is chosen in such a way that all the projectiles have the same shaft length (L -L ) regardless of their cone angles 204 p c List of Tables Table 7.2 Mass and length of projectiles with two different cone angles used in powder gun tests on GFRP laminates 204 Table 7.3 Partition of the predicted perforation energy for a 6.35 mm thick GFRP laminate impacted by 4.2 g projectiles with various cone angles 204 Table 7.4 Partition of the predicted perforation energy for a 12.7 mm thick GFRP laminate impacted by 13.2 g projectiles with 37° and 120° cone angles 205 Table 7.5 Partition of the predicted perforation energy for a 19.05 mm thick GFRP laminate impacted by 13.2 g projectiles with 37° and 120° cone angles 205 Table 7.6 Comparison of the predicted and measured ballistic limit V for CFRP and GFRP laminates. The projectile mass is 4.2 g for CFRP and 13.2 g for GFRP 205 Table C l Comparison of normalised stress components and transverse deflection with Table 3.7-6 in Ochoa et al. (1992) for a simply-supported square graphite/epoxy plate with a lay-up of [0/90/0] subjected to transverse sinusoidal pressure load 268 Table C.2 Comparison of normalised stress components and transverse deflection with Table 3.7-8 in Ochoa et al. (1992) for a simply-supported rectangular graphite/epoxy plate with a lay-up of [0/90/0] subjected to transverse sinusoidal pressure load 268 Table C.3 Constants of a laminated stiffened cylindrical shell panel subjected to a point load as shown in Figure C l (Goswami et al., 1994) 269 Table C.4 Static deflection of a laminated stiffened cylindrical shell panel at the loading point as shown in Figure C l 269 Table C.5 Constants of laminated square plates with lay-ups of [0] and [0/90/90/0] under uniform pressure load in large deflection analysis (Ochoa et al., 1992) 270 Table C.6 Constants of a laminated cylindrical shell panel with a lay-up of [0 /90 /0 ] subjected to a point load as shown in Figure C.4 (Lee et ah, 1993) cr 4 4 Table C.7 Material constants for [0] boron/aluminum plate subjected to uniformlydistributed pressure load (Rahman et al., 1992). Units: GPa 4 270 270 Table C.8 Constants of an orthotropic plate [0] subjected to uniformly-distributed pressure load (Owen et al, 1983) 270 Table C.9 Constants of an orthotropic cylindrical shell panel [0] subjected to uniform pressure load as shown in Figure C.10 (a) (Owen et al., 1983) 271 g 6 xi List of Tables Table C I O Properties of [0] plywood plate (Hearmon, 1959) 271 Table C l 1 Fundamental frequencies for [0] plywood plate under various boundary conditions 271 Table C 1 2 Constants of a [0] boron/epoxy plate (Whitney, 1987) 272 Table C.13 Natural frequencies of [0] and [45] boron/epoxy plates 272 Table C.14 Normalised natural frequencies for a rectangular simply-supported laminated plate with a lay-up of [0/90/0] 272 Table C.15 Constants for the isotropic cylindrical shell panel shown in Figure C.4 273 Table D. 1 Comparison of periods of high frequency oscillations in impact force time history of T800H/3900-2 CFRP plates 286 xii List of Figures List of Figures Figure 2.1 Figure 2.2 Schematic illustration of an orthogonally stiffened laminated composite cylindrical shell 25 Illustration of element connection between a super shell element and a super beam element (straight and curved) 25 Figure 2.3 Nodal configuration of super elements 26 Figure 2.4 Schematic illustration of a laminated composite plate 26 Figure 2.5 Schematic illustration of a stiffened laminated composite plate where a laminated stiffener is attached to a laminated skin (Dost et al, 1991) Flow chart of structural response calculation using SENACS 27 28 Figure 2.6 Figure 3.1 Convergence of the transient response of the laminated plate defined in Table 3.1 subjected to a 8.54 g, 3.0 m/s central impact 52 Figure 3.2 Impact force history of the laminated plate defined in Table 3.1; present analysis compared with modal analysis of Pierson and Vaziri (1996) and finite element analysis of Sun and Chen (1985) 52 Figure 3.3 Central deflection history of the laminated plate defined in Table 3.1; present analysis compared with modal analysis of Pierson and Vaziri (1996) and finite element analysis of Sun and Chen (1985) 53 Figure 3.4 Impact force history for 6.14 kg, 1.76 m/s impact on the laminate defined in Table 3.3; present model compared with drop-weight test results (Delfosse et al, 1993b) 53 Figure 3.5 Impact force history for 6.14 kg, 2.68 m/s impact on the laminate defined in Table 3.3; present model compared with drop-weight test results (Delfosse et al, 1993b) 54 Figure 3.6 Impact force history for 0.314 kg, 7.70 m/s impact on the laminate defined in Table 3.3; present model compared with gas-gun test results (Delfosse et al, 1993b) 54 Figure 3.7 Impact force history for 0.314 kg, 11.85 m/s impact on the laminate defined in Table 3.3; present model compared with gas-gun test results (Delfosse et al, 1993b) 55 List of Figures Figure 3.8 Parametric studies: Predicted impact force-displacement curves for impact events with the same incident energy but with different striking velocities on the laminate defined in Table 3.3. Also see Figure 3.9 for a description of the forcedisplacement curve with a striking velocity of 59.25 m/s 55 Figure 3.9 Predicted time histories of the projectile velocity, central target velocity, and velocity of indentation depth for the impact event shown in Figure 3.8 with a striking velocity of 59.25 m/s 56 Figure 3.10 Parametric studies: Time histories of central deflection of the target for impact events with the same incident energy but with different striking velocities on the laminate defined in Table 3.3 56 Figure 3.11 Profiles of central deflections under impacts with the same amount of energy but in different velocities on the laminate defined in Table 3.3 57 Figure 3.12 Vertical deflection history at the impact point for the Christoforou and Swanson's cylinder impact problem defined in Table 3.4. (a). Circular cylinder under an impact at the center, (b). Finite element mesh for a quarter cylinder used in present analysis, (c) Present predictions compared with Christoforou and Swanson's analytical results 58 Figure 3.13 Central deflection history for a 8.54 g, 30.0 m/s impact on a simply-supported [0/45/0/45/0] CFRP rectangular plate with the layer properties defined in Table 3.1; present results obtained from a small and large deflection analysis compared with the corresponding results of Chen and Sun (1985) 59 2S Figure 3.14 Central deflection history for a 1.53 kg, 3.0 m/s impact on a clamped-free [±45 /0 ] CFRP plate with the layer properties defined in Table 3.5; present small and large deflection analysis results compared with experimental data in Matsuhashi etal. (1993) 59 2 2 2S Figure 3.15 Super finite element mesh in nonpenetrating impact analysis of laminated composite stiffened plates (Dost et al, 1991). Material properties and lay-up are shown in Table 3.6 60 Figure 3.16 The comparison of the predicted contact force time history with the measured datafrominstrumented impact tests in Dost et al (1991). The material properties and geometric sizes are shown in Figure 3.15 and Table 3.6 60 Figure 3.17 Parametric studies: The time history of the central deflection at the impact point for stiffened cylindrical shells with various radii of curvature. The material properties and geometric sizes are shown in Figure 3.15 and Table 3.6 with arc length 6 = 0.508 m 61 xiv List of Figures Figure 3.18 Parametric studies: The time history of the impact force for stiffened cylindrical shells with various radii of curvature. The material properties and geometric sizes are shown in Figure 3.15 and Table 3.6 with arc length b = 0.508 m 61 Figure 3.19 Parametric studies: The time history of predicted non-linear central deflection for stiffened cylindrical shells with various radii of curvature. The material properties and geometric sizes are shown in Figure 3.15 and Table 3.6 with arc length^ = 0.508 m 62 Figure 3.20 Parametric studies: Comparison of deflection at different impact sites. The material properties and geometric sizes are shown in Figure 3.15 and Table 3.6 62 Figure 3.21 Parametric studies: Comparison of the impact force at different impact sites. The material properties and geometric sizes are shown in Figure 3.15 and Table 3.6 63 Figure 4.1 Schematic illustration of transition of penetration mechanisms in static penetration tests of thin metallic plates by conical indenters with different cone angles (Johnson et al, 1973) 83 Figure 4.2 Micrographs of cross-sections of IM7/8551-7 CFRP laminates in (a) static penetration tests and (b) in dynamic penetration tests (Delfosse, 1994b and Sanders, 1997) Figure 5.1 84 Schematic illustration of static penetration tests. The specimen sits on a rigid support with a circular or rectangular opening. R denotes the radius of a circular opening, while a and b denote the length and width of a rectangular opening. The indenter diameter is 2R = 7.62 mm 124 t p Figure 5.2 Schematic illustration of static indentation tests 124 Figure 5.3 Penetration force versus indenter displacement in static penetration tests on the CFRP laminates defined in Table 5.2 penetrated by indenters with included cone angles of 37°, 60°, 120°, and 180° (Delfosse, 1994b) 125 Figure 5.4 Micrographic illustration of the initial penetration stage in a 6.15 mm thick CFRP laminate penetrated by an indenter with 37° cone angle (Sanders, 1997). The material around the indenter is crushed and is pushed aside by the indenter 125 Figure 5.5 Micrographic illustration of failure in a 6.15 mm thick CFRP specimen penetrated by an indenter with a cone angle of 37° (Sanders, 1997). The material below the indenter is broken and bent away from the indenter 126 List of Figures Figure 5.6 Cross-sectional micrograph of a 6.15 mm thick CFRP laminate penetrated by the indenter with an included cone angle of 120°. The plug initiation site can be seen very clearly 126 Figure 5.7 Micrographic illustration of delamination initiation in a 6.15 mm thick CFRP specimen when penetrated by an indenter with 37° cone angle (Sanders, 1997). The delaminations are located below the indenter and matrix cracks nearby....127 Figure 5.8 Micrographic illustration of delamination development in a 6.15 mm thick CFRP specimen penetrated by an indenter with 37° cone angle (Sanders, 1997). The delaminations are well developed by the time the indenter shaft reaches the frontal surface of the specimen as shown in the figure 127 Figure 5.9 Experimental force versus indenter displacement curves for 6.35 mm, 12.70 mm, and 19.05 mm thick GFRP laminates penetrated by an indenter with 37° included cone angle (Sanders, 1997) 128 Figure 5.10 Experimental force versus indenter displacement curves for 6.35 mm, 12.70 mm, and 19.05 mm thick GFRP laminates penetrated by an indenter with 120° included cone angle (Sanders, 1997) 128 Figure 5.11 Cross-sectional micrographs of the penetrated and the perforated specimens and corresponding sections with dye penetrant for a 19.05 mm thick GFRP laminate impacted by a projectile with 37° cone angle at different striking energies (Sanders, 1997). The micro-delamination sites are shown as the black areas in the sections with dye penetrant 129 Figure 5.12 Micrographic illustration of macro-delamination for a 12.7 mm thick GFRP specimen impacted by an indenter with 120° cone angle (Sanders, 1997). The development of macro-delamination is clearly visible 130 Figure 5.13 Top and side view of a 6.35 mm thick GFRP specimen penetrated by a 120° indenter (Sanders, 1997). The indenter simply pushes the laminate through the opening of the support plate and does not actually penetrate it 131 Figure 5.14 Schematic illustration of the force-displacement curves of CFRP and GFRP laminates when penetrated by an indenter with a general cone angle. The penetration mechanisms follow the initial hole expansion mechanism {OA or OA ) are hole expansion and flexural deformation of a delaminated plate (splitplate) in curve 1 (A B ) and transverse plugging in curve 2 (A B ). Friction force initiates and gradually dominates the penetration process after B in curve 1 and B in curve 2 132 x 2 l 1 2 2 x 2 Figure 5.15 Local deformation zones: zone I, hole expansion; zone II, flexural deformation of the delaminated plate (split-plate) 133 xvi List of Figures Figure 5.16 Schematic illustration of transverse shear plugging initiation 133 Figure 5.17 Schematic illustration of hole expansion mechanism. A is the projected area of a projectile or indenter, a is the hole expansion pressure of the laminate, and (a. is the coefficient of friction of the laminate material 134 p 0 Figure 5.18 Schematic illustration of indentation pressure a in static penetration tests. The relationship between cr and a in Figure 5.17 is shown in Equation (5.2) 134 m m 0 Figure 5.19 Schematic illustration of bluntness b on the tip of a conical projectile or indenter 0 135 Figure 5.20 Calculation of mean indentation pressure based on measured force-displacement curve for the CFRP laminate. The conical indenter has an included cone angle of 37°. The specimen thickness is 6.15 mm 135 Figure 5.21 Calculation of mean indentation pressure based on the measured forcedisplacement curve for a CFRP laminate. The conical indenter has an included cone angle of 120°. The specimen thickness is 6.15 mm. The calculation of indentation pressure ends when the indenter shaft reaches the frontal surface of the laminate, i. e., 2.2 mm of indenter displacement 136 Figure 5.22 Micrograph examination of cross-section of the CFRP specimen in static indentation tests by a 120° indenter. "Sinking-in" (a<A) is clearly observed. The measured relationship between or and A is or« 0.7A 136 Figure 5.23 Calculation of mean indentation pressure based on the measured forcedisplacement curve for a S2-glass laminate where the conical indenter has an included cone angle of 37° and a diameter of 7.62 mm. The specimen thickness is 19.0 mm 137 Figure 5.24 Cross-sectional microgragh showing "piling-up" in the 19.05 mm thick GFRP specimen when penetrated by an indenter with an included cone angle of 37° and a diameter of 7.62 mm, a&\3A 137 Figure 5.25 Calculation of mean indentation pressure based on the measured forcedisplacement curve for a S2-glass laminate penetrated by a conical indenter with an included cone angle of 120° and a diameter of 7.62 mm. The specimen thickness is 19.05 mm 138 Figure 5.26 Comparison between the predicted and the measured contact force-indenter displacement curve in static indentation tests on a 6.15 mm thick CFRP specimen by an indenter with an included cone angle of 60° 138 Figure 5.27 Relationship between the mean indentation pressure and the included cone angle of an indenter for CFRP and GFRP laminates 139 xvii List of Figures Figure 5.28 Schematic illustration of hole expansion mechanism after delamination is initiated in the laminate. A is the projected area undergoing hole expansion. D is a reduction factor accounting for lateral deformation on the distal surface of the laminate 139 p Figure 5.29 Free body diagrams used in calculation of shear stresses for delamination criterion 140 Figure 5.30 The experimental force-displacement curve for a 6.15 mm thick CFRP laminate statically penetrated by a 37° indenter. The support plate has a small-geometry opening 140 Figure 5.31 Schematic illustration of delamination patterns in a T300H/F593 CFRP laminate with a stacking sequence of [45/0/-45/90] under static loading by a flat indenter (Delfosse et al, 1994b) 141 ns Figure 5.32 Schematic illustration of the geometry and loading condition of a typical split in the multi-split plate 141 Figure 5.33 Schematic illustration for calculation of the inner and outer radii of each split in the split-plate 142 Figure 5.34 Schematic illustration of growth of the outer and inner radii of delaminations. 142 Figure 5.35 Schematic illustration of flexural deformation of a split-plate in section 5.2.3.3.2 143 Figure 5.36 Nomenclature for the split-plate deflection after the projectile tip penetrates a certain layer of the laminate. OQ is the penetration depth just when delaminations have initiated 144 Figure 5.37 Estimation of the transverse shear strength and friction force between the plug and adjacent material for CFRP laminates 145 Figure 5.38 Calculation of penetration force at different positions of the projectile in the target when the projectile cone length is smaller than the local target thickness for (1) long projectiles (L - L>h+w ) and (2) short projectiles (L L<h+w ) 146 p s p s Figure 5.39 Calculation of penetration force at different positions of the projectile in the target when the projectile cone length is greater than the local target thickness for (1) long projectiles (L - L>h+w ) and (2) short projectiles (L - L<h+w ) 147 p Figure 6.1 s Flow chart of the penetration, model p s 161 xviii List Figure 6.2 Figures Illustration of the penetration model as a spring system. F denotes the penetration force, k , k , k denote the spring stiffnesses due to hole expansion, flexural deformation of a split-plate, and transverse plugging, respectively 162 h Figure 6.3 of s p Schematic illustration of the penetration behaviour in GFRP laminates when one delamination has already been initiated and developed. Total force is the sum of F andF; 163 h Figure 6.4 Curves of delamination initiation force versus penetration depth for a CFRP laminate 163 Figure 6.5 Curves of transverse shear plugging initiation force for CFRP laminates versus the penetration depth. When a pair of contact force and penetration depth falls above the curve corresponding to a given indenter cone angle, plugging mechanism dominates 164 Figure 6.6 Hole expansion force, delamination initiation, and plugging initiation curves of CFRP laminates when penetrated by an indenter with a cone angle of 37° 164 Figure 6.7 Hole expansion force, delamination initiation curve, and plugging initiation curve of CFRP laminates when penetrated by an indenter with a cone angle of 120° 165 Figure 6.8 Comparison between the predicted and measured static force-displacement curves for the CFRP laminate defined in Table 5.2 penetrated by an indenter with an included cone angle of 37° 165 Figure 6.9 Comparison between the predicted and measured energy-displacement curves for the CFRP laminate defined in Table 5.2 penetrated by an indenter with an included cone angle of 37° 166 Figure 6.10 Comparison between the predicted and measured static force-displacement curves for the CFRP laminate defined in Table 5.2 penetrated by an indenter with an included cone angle of 60° 166 Figure 6.11 Comparison between the predicted and measured energy-displacement curves for the CFRP laminate defined in Table 5.2 penetrated by an indenter with an included cone angle of 60° 167 Figure 6.12 Comparison between the predicted and measured static force-displacement curves for the CFRP laminate defined in Table 5.2 penetrated by an indenter with an included cone angle of 120° 167 Figure 6.13 Comparison between the predicted and measured energy-displacement curves for the CFRP laminate defined in Table 5.2 penetrated by an indenter with an included cone angle of 120° 168 xix List of Figures Figure 6.14 Comparison between the predicted and measured static force-displacement curves for the CFRP laminate defined in Table 5.2 penetrated by an indenter with an included cone angle of 180° 168 Figure 6.15 Comparison between the predicted and measured energy-displacement curves for the CFRP laminate defined in Table 5.2 penetrated by an indenter with an included cone angle of 180° 169 Figure 6.16 Comparison between the predicted and measured static force-displacement curves for the CFRP laminate defined in Table 5.2 penetrated by indenters with included cone angles of 15°, 37°, 60°, 90°, 120°, 150°, and 180° 169 Figure 6.17 Predicted forces at the onset of delamination and transverse plugging for IM7/8551-7 CFRP laminates 170 Figure 6.18 Comparison between the predicted and measured maximum penetration force as a function of the indenter included cone angle 170 Figure 6.19 Comparison between the predicted and measured static force-displacement curves for 6.35 mm thick GFRP laminate penetrated by an indenter with cone angle of 37° 171 Figure 6.20 Comparison between the predicted and measured energy-displacement curves for 6.35 mm thick GFRP laminate penetrated by an indenter with cone angle of37° 171 Figure 6.21 Comparison between the predicted and measured static force-displacement curves for 12.7 mm thick GFRP laminate penetrated by an indenter with cone angle of 37° 172 Figure 6.22 Comparison between the predicted and measured energy-displacement curves for 12.7 mm thick GFRP laminate penetrated by an indenter with cone angle of37° 172 Figure 6.23 Comparison between the predicted and measured static force-displacement curves for 19.05 mm thick GFRP laminate penetrated by an indenter with cone angle of 37° 173 Figure 6.24 Comparison between the predicted and measured energy-displacement curves for 19.05 mm thick GFRP laminate penetrated by an indenter with cone angle of37° 173 Figure 6.25 Comparison between the predicted and measured static force-displacement curves for a 6.35 mm thick GFRP laminate penetrated by an indenter with a cone angle of 120° 174 List of Figures Figure 6.26 Comparison between the predicted and measured energy-displacement curves for a 6.35 mm thick GFRP laminate penetrated by an indenter with a cone angle of 120° 174 Figure 6.27 Comparison between the predicted and measured static force-displacement curves for a 12.7 mm thick GFRP laminate penetrated by an indenter with a cone angle of 120° 175 Figure 6.28 Comparison between the predicted and measured energy-displacement curves for a 12.7 mm thick GFRP laminate penetrated by an indenter with a cone angle of 120° : 175 Figure 6.29 Comparison between the predicted and measured static force-displacement curves for a 19.05 thick GFRP laminate penetrated by an indenter with a cone angle of 120° 176 Figure 6.30 Comparison between the predicted and measured energy-displacement curves for a 19.05 mm thick GFRP laminate penetrated by an indenter with a cone angle of 120° 176 Figure 7.1 Flow chart of the solution scheme for the dynamic penetration model in SENACS 206 Figure 7.2 Comparison of the predicted and experimental impact force versus projectile displacement when a 0.32 kg steel projectile with a 37° cone angle penetrates through a 6.15 mm thick CFRP laminate with a striking velocity of 30.2 m/s 207 Figure 7.3 Comparison of the predicted and experimental impact energy absorption of the target versus projectile displacement curves when a 0.32 kg steel projectile with a 37° cone angle penetrates through a 6.15 mm thick CFRP laminate with a striking velocity of 30.2 m/s 207 Figure 7.4 Time histories of the predicted impact force and energy absorption during penetration of a 6.15 mm thick CFRP laminate by a 0.32 kg steel projectile with a 37° cone angle and a striking velocity of 30.2 m/s 208 Figure 7.5 Time histories of the predicted projectile velocity and displacement when a 0.32 kg steel projectile with a 37° cone angle and a striking velocity of 30.2 m/s penetrates a 6.15 mm thick CFRP laminate 208 Figure 7.6 Time histories of the predicted target deflection during penetration of a 6.15 mm thick CFRP laminate by a 0.32 kg steel projectile with a 37° cone angle and a striking velocity of 30.2 m/s. The impact force is also plotted here as a reference 209 List of Figures Figure 7.7 Comparison of the predicted and experimental impact force versus projectile displacement curves when a 0.32 kg steel projectile with a blunt-ended head penetrates through a 6.15 mm thick CFRP laminate with a striking velocity of 30.2 m/s 209 Figure 7.8 Comparison of the predicted and experimental impact energy absorption versus projectile displacement curves when a 0.32 kg steel projectile with a blunt-ended head penetrates through a 6.15 mm thick CFRP laminate with a striking velocity of 30.2 m/s 210 Figure 7.9 Comparison of the predicted and experimental results when a steel projectile with a 37° included cone angle impacts a 6.35mm thick GFRP laminate with a striking velocity of 104 m/s. Both the predicted data and experimental observation show that the projectile stops in the target 211 Figure 7.10 Comparison of the predicted and experimental results when a steel projectile with a 120° included cone angle penetrates through a 6.35mm thick GFRP laminate with a striking velocity of 211 m/s. Both the predicted data and experimental measurement show the non-zero projectile exit velocity 212 Figure 7.11 Comparison of the predicted and experimental results when a steel projectile with a 37° included cone angle completely perforates a 12.7 mm thick GFRP laminate with a striking velocity of 304 m/s. Predicted projectile exit velocity is very close to the measured one 213 Figure 7.12 Comparison of the predicted and experimental results when a steel projectile with a 37° included cone angle impacts a 12.7 mm thick GFRP laminate with a striking velocity of 170 m/s. Both the predicted data and experimental observation show that the projectile finally rests in the target 214 Figure 7.13 Comparison of the predicted and experimental results when a 120° steel projectile penetrates through a 12.7 mm thick GFRP laminate with a velocity of 278 m/s. The predicted projectile residual velocity is 134 m/s. Experimental observation shows the projectile has penetrated through the target but fails to catch its exit velocity 215 Figure 7.14 Comparison of the predicted and experimental results when a steel projectile with a 37° included cone angle impacts a 19.05 mm thick GFRP laminate with a striking velocity of 318 m/s. Both the predicted data and experimental observation show the partially perforation in the target 216 Figure 7.15 Comparison of the predicted and experimental results when a steel projectile with a 37° included cone angle impacts a 19.05 mm thick GFRP laminate with a striking velocity of 211 m/s. Both the predicted data and experimental observation indicate that the projectile finally rests in the target 217 xxii List of Figures Figure 7.16 Comparison of the predicted and experimental results when a 120° steel projectile penetrates through a 19.05 mm thick GFRP laminate with a velocity of 384 m/s. The predicted projectile residual velocity is 243 m/s, comparing with the measured residual velocity of 160 m/s 218 Figure 7.17 Predicted family of impact force versus projectile displacement curves when 0.32 kg steel projectiles with varied included cone angles penetrate through a 6.15 mm thick CFRP laminate with a striking velocity of 30.2 m/s 219 Figure 7.18 Predicted perforation energy when 0.32 kg steel projectiles with various included cone angles penetrate through a 6.15 mm thick CFRP laminate with a striking velocity of 30.2 m/s 219 Figure 7.19 Predicted peak impact force when 0.32 kg steel projectiles with various included cone angles penetrate through a 6.15 mm thick CFRP laminate with a striking velocity of 30.2 m/s 220 Figure 7.20 Predicted family of impact force versus projectile displacement curves when 4.2 g steel projectiles with various included cone angles penetrate through a 6.35 mm thick GFRP laminate with a striking velocity of 600 m/s 220 Figure 7.21 Predicted impact energy absorption (perforation energy) when 4.2 g steel projectiles with various included cone angles penetrate through a 6.35 mm thick GFRP laminate with a striking velocity of 600 m/s 221 Figure 7.22 Predicted peak impact force when 4.2 g steel projectiles with various included cone angles penetrate through a 6.35 mm thick GFRP laminate with a striking velocity of 600 m/s 221 Figure 7.23 Predicted projectile residual versus striking velocity curve for a 0.32 kg projectile with a 37° cone angle penetrating through a 6.15 mm thick CFRP laminate. The experiment data from a gas gun test is also shown here 222 Figure 7.24 Predicted projectile residual versus striking velocity curves for 4.2 g projectiles with 37° and 180° cone angles penetrating through a 6.15mm thick CFRP laminate. The experiment data from gas gun tests are also shown here 222 Figure 7.25 Predicted projectile residual versus striking velocity curves for 4.2 g projectiles with 15°, 37°, 120°, and 180° cone angles penetrating through a 6.15mm thick CFRP laminate 223 Figure 7.26 Predicted projectile residual versus striking energy for 4.2 g projectiles with 37° and 180° cone angles penetrating through a 6.15 mm thick CFRP laminate. The experimental data are also shown in the graph 223 xxiii List of Figures Figure 7.27 Predicted projectile residual versus striking energy curves for 4.2 g projectiles with 15°, 37°, 120°, and 180° cone angles penetrating through a 6.15mm thick CFRP laminate 224 Figure 7.28 Schematic illustration of three parts of impact energy absorption on forcedisplacement curves: in phase I, energy dissipates in the form of hole expansion; in phase II, energy dissipates in the form of: (a) hole expansion and flexural deformation of a split-plate or (b) transverse plug initiation an push-out; in phase III, energy dissipates in the form of friction 225 Figure 7.29 Composition of impact energy absorption of a 6.15 mm thick CFRP laminate perforated by a 4.2 g projectile with various cone angles: (a) 15°, (b) 37°, (c) 120°, (d) 180° 226 Figure 7.30 Predicted projectile residual versus striking velocity curves for 4.2 g projectiles with 15°, 37°, 120°, and 180° cone angles penetrating through a 6.35 mm thick GFRP laminate 227 Figure 7.31 Predicted projectile residual versus striking energy for 4.2 g projectiles with 15°, 37°, 120°, and 180° cone angles penetrating through a 6.35 mm thick GFRP laminate. The results of powder gun tests are also shown in the figure 227 Figure 7.32 Composition of impact energy absorption of a 6.35 mm thick GFRP laminate perforated by a projectile with various cone angles: (a) 15°, (b) 37°, (c) 120°, (d) 180° 228 Figure 7.33 Predicted projectile residual versus striking velocity curves for 13.2 g projectiles with 37° and 120° cone angles penetrating through a 12.7 mm thick GFRP laminate. The experimental result in the powder gun test is also shown in the graph 229 Figure 7.34 Predicted projectile residual versus striking energy for 13.2 g projectiles with 37° and 120° cone angles penetrating through a 12.7 mm thick GFRP laminate. The experimental results in powder gun test are also shown in the figure 229 Figure 7.35 Composition of impact energy absorption of a 12.7 mm thick GFRP laminate perforated by a 13.2 g projectile with various cone angles: (a) 37°; (b) 120°. .230 Figure 7.36 Predicted projectile residual versus striking velocity curves for 13.2 g projectiles with 37° and 120° cone angles penetrating through a 19.05 mm thick GFRP laminate. The experimental results in the powder gun test are also shown in the graph 231 Figure 7.37 Predicted projectile residual versus striking energy for 13.2 g projectiles with 37° and 120° cone angles penetrating through a 19.05 mm thick GFRP laminate. The experimental results in powder gun test are also shown in the figure 231 xxiv List of Figures Figure 7.38 Composition of impact energy absorption of a 19.05 mm thick GFRP laminate perforated by a 13.2 g projectile with various cone angles: (a) 37°; (b) 120°. .232 Figure 7.39 Comparison between the predicted ballistic resistance (perforation energy) of a CFRP laminate (6.15 mm thick) and a GFRP laminate (6.35 mm thick). The projectiles used are defined in Table 7.1 ..233 Figure 7.40 Predicted force-displacement curves for a 13.2 g projectile with a 37° cone angle penetrating through 6.35 mm, 12.7 mm, and 19.05 mm thick GFRP laminates with a velocity of 384 m/s 233 Figure 7.41 Predicted force-displacement curves for a 13.2 g projectile with a 120° cone angle penetrating through 6.35 mm, 12.7 mm, and 19.05 mm thick GFRP laminates with a velocity of 384 m/s 234 Figure 7.42 Predicted force-displacement curves for 0.32 kg projectiles with 37° and 120° cone angles penetrating a 6.15 mm thick CFRP laminate with simply-supported and clamped boundary conditions. The projectile velocity is 30.2 m/s 234 Figure 7.43 Predicted force-displacement curves for a 0.32 kg projectile with a 37° cone angle penetrating a 6.15 mm thick CFRP laminate with different planar dimensions. The projectile velocity is 30.2 m/s 235 Figure 7.44 Predicted ratio of global deformation and motion energy to target perforation energy versus the projectile striking velocity for a 0.32 kg projectile with a 37° cone angle penetrating a 6.15 mm thick CFRP laminate with different planar dimensions 235 Figure 7.45 Predicted force-time histories for projectiles with three different masses penetrating a 6.35 mm thick GFRP laminate. All projectiles have the same cone angle of 37° and the same striking energy of 343 J 236 Figure 7.46 Predicted family of target deflection versus projectile displacement at the impact site for projectiles with different masses penetrating a 6.35 mm thick GFRP laminate. All projectiles have the same cone angle of 37° and the same striking energy of 343 J 236 Figure 7.47 Predicted family of impact energy absorption versus projectile displacement at the impact site for projectiles with different masses penetrating a 6.35 mm thick GFRP laminate. All projectiles have the same cone angle of 37° and the same striking energy of 343 J 237 Figure C l A cantilever laminated composite cylindrical shell with eccentrically mounted composite stiffeners in axial and circumferential directions (Venkatesh et al, 1983) 274 List of Figures Figure C.2 Comparison of the present relationship of load and deflection with experimental data in Zaghloul et al. (1975) for large deflection analysis of a simply-supported plate [0] with in-plane moveable or immoveable boundary conditions 274 Figure C.3 Comparison of the present relationship of load and deflection with experimental data in Zaghloul et al. (1975) and finite element results in Ochoa et al. (1992) for large deflection analysis of an fully-clamped plate [0/90/90/0] 275 Figure C.4 Schematic illustration of laminated composite cylindrical shell panel with a layup of [0 /90 /0 ] subjected to a point load (Lee et al, 1993) 275 4 4 4 Figure C.5 Comparison of the present results with other finite element results by Lee et al.'s (1993) and Saigal et al.'s (1986) for linear-elastic material with large deflection analysis of laminated composite cylindrical shell with lay-up of [0 /90 /0 ] 276 4 4 4 Figure C.6 Comparison of the present results with predictions of micro-mechanics models in Arenburg et al. (1989) and Rahman et al. (1992) for elasto-plastic material in small deflection analysis 276 Figure C.7 Comparison of relationship of non-dimensional load vs. non-dimensional deflection between the present analysis and Owen et al. (1983) for a fullyclamped laminated square plate [0 ] in isotropic or anisotropic elasto-plastic material and small deflection analysis 277 8 Figure C.8 Comparison of the predicted load-deflection relationships using isotropic and anisotropic hardening rule in the case of high anisotropy in plastic tangent modulus 278 Figure C.9 Comparison of the predicted load-deflection relationships using isotropic and anisotropic hardening rule in the case of high anisotropy in initial yielding stress 278 Figure CIO Comparison of the present results with Owen et al (1983) for laminated cylindrical shell panel [0 ] in elasto-plastic and large deflection analysis, (a). Schematic illustration of cylindrical shell panel, (b-f). Relationship of load vs. downward deflection at midside of free edge for one isotropic and four anisotropic cases of E =2E , E =2E , CT =2CT , CT =2CT 279 6 X 0 2 0 1 0 2 0 Figure C l 1 Comparison of the present results with Owen et al (1983) for laminated cylindrical shell panel [0 ] in elasto-plastic and large deflection analysis: (a) 6 E =2E , (b) oi=2<7 X 0 0 280 Figure C.12 Comparison of the present deflection-time history with Tsai et al. (1991) for transient response of an isotropic cylindrical shell roof subjected to a uniformlydistributed vertical half-sinusoidal impulsive load in linear-elastic and (a) small deflection and (b) large deflection analyses 281 xxvi List of Figures Figure C.13 Comparison of the present results with finite strip analysis by Jiang et al (1991) for transient response of an isotropic stiffened cylindrical shell panel, (a-b) Schematic illustration of the stiffened shell panel, (c) Results in small deflection and linear-elastic analysis, (d) Results in large deflection and linear-elastic analysis 282 Figure D.l Two degree of freedom, spring-mass system used in Bucinell et al. (1991) to investigate the nonpenetrating impact responses of composite plates 287 Figure D.2 Contact force history for 6.14 kg, 1.76 m/s impact on the laminate defined in Chapter Three; the predicted results with and without artificial damping are compared with drop-weight test results (Delfosse et al, 1993) 288 Figure D.3 Contact force history for 6.14 kg, 2.68 m/s impact on the laminate defined in Chapter Three; the predicted results with and without artificial damping are compared with drop-weight test results (Delfosse et al, 1993) 288 Figure D.4 Contact force history for 0.314 kg, 7.70 m/s impact on the laminate defined in Chapter Three; the predicted results with and without artificial damping are compared with drop-weight test results (Delfosse et al, 1993) 289 Figure D.5 Contact force history for 0.314 kg, 11.85 m/s impact on the laminate defined in Chapter Three; the predicted results with and without artificial damping are compared with drop-weight test results (Delfosse et al, 1993) 289 Figure D.6 Predicted relationship between energy absorbed in the deformation and motion of the target and the non-dimensional frequency ratio 290 Figure E. 1 Illustration of a circular ring fully clamped along its outer edge. A transverse uniformly distributed shearing force Q acts along the inner edge 294 0 xxvii No menclature Nomenclature a Length of a plain or stiffened cylindrical shell panel in the x axis Gj Outer radius of the f th split in a split-plate (delaminated plate) A Mid-plane area of super shell and beam elements [A] In-plane stiffness matrix of a laminate A b Cross-sectional area of a super beam element A p Projected area of the projectile (indenter) in a target Ay Anisotropic yielding and hardening parameters which describe the shape of the initial yield and subsequent hardening surface [B] Coupling (bending-stretching) matrix in the strain-displacement relationship of a laminate [B ] Strain-displacement matrix for infinitesimal bending strain Z{K} [B ] Strain-displacement matrix for infinitesimal membrane strain |V} [B ] Strain-displacement matrix for infinitesimal membrane strain of a straight super beam element [.By] Strain-displacement matrix for infinitesimal membrane strain of a curved c m x super beam element b Circumferential length of a plain or stiffened cylindrical shell panel in the y axis b Bluntness of a projectile (indenter) cone bj Inner radius of the / t h split in a split-plate (delaminated plate) [C] Strain-displacement matrix of the von-Karman strains in a super shell element. [Cj Strain-displacement matrix of the von-Karman strains of a straight super beam element 0 xxviii Nomenclature [CyJ Strain-displacement matrix of the von-Karman strains of a curved super beam element D A reduction parameter for the hole expansion pressure {D} Assembled global equivalents of {d } and \d j [D] Bending stiffness matrix of a laminate D Bending rigidity of the/th split in a split-plate (delaminated plate) dJ e e {d } Nodal displacement vector of a super shell element \d } Nodal displacement vector of a super beam element E Elastic modulus of a laminate E^ Perforation energy of a target E Energy dissipated in indenting a target E Energy dissipated in delamination E f Maximum strain energy absorbed by a target E p Energy dissipated in plugging of a target E Projectile residual (exit) kinetic energy E Projectile striking (impact) kinetic energy e Eccentricity of the beam centroidal axis from the mid-plane of a shell panel F Contact (impact) force Fj Friction force between the projectile (indenter) shaft and the target material F Maximum contact (impact) force Fj Friction force between the plug and the rest of the target material F Force due to flexural deformation of a split-plate (delaminated plate) F Penetration force when projectile (indenter) shaft reaches the frontal surface of the target (a = L ) e e c d r s m s s0 c xxix Nomenclature {F° j Assembled global equivalents of [f ]. JF'} Assembled global equivalents of {/^'} [F ] Effective load vector during the Newmark-p time integration / Total compliance of a split-plate (delaminated plate) fj Compliance of the / th split in a split-plate (delaminated plate) c# c e {f } Elemental contact (impact) force vector j^'} External force vector of a super shell or beam element G Shear modulus c e G Transverse shear modulus of a single layer of the target h Thickness of a cylindrical shell panel hj Distance in thickness direction between the fth and (/+i)'th delaminations 1^, Moment of inertia about the z axis J Torsional constant 2 J Polar moment of inertia about the centroid of a super beam element [A^] Effective stiffness matrix in Newmark-p time integration [K ] Tangent stiffness matrix assembled from elemental stiffness matrix [k ] in a Newton-Raphson iterative scheme for non-linear problems [K ] Stiffness matrix of a beam element due to the beam torsion and lateral bending c T TB e T effects k Stiffness coefficient of the Hertzian contact law k Spring stiffness due to hole expansion k Spring stiffness due to transverse plugging k Spring stiffness due to flexural deformation of a split-plate (delaminated plate) h s Nomenclature k Effective size of the yield surface y [& ] Elemental stiffness matrix of a super shell or beam element [k ] Elemental tangent stiffness matrix in Newton-Raphson iterative scheme for non-linear problems Z c Length of projectile (indenter) conical head fl Maximum length in the target over which friction is operative. e e L L /2 A length used to check the start of friction and perforation of the target L p Projectile length / Length of a beam stiffener in a stiffened cylindrical shell panel [M] Assembled global equivalents of [m ] {M} Bending moment resultant [M ] Elemental mass matrix due to the effects of beam torsion and lateral bending ra e for a super beam element m Projectile mass [m ] Elemental consistent mass matrix of a super shell or beam element N Number of orthotropic layers in a laminated composite shell panel e N Number of total time steps [N] Matrix of shape functions of a super shell element {N} Stress resultant [/y] Matrix of shape functions of a super beam element n Total number of splits in a split-plate (delaminated plate) n Time step number {P} Assembled global equivalents of {p } 0 s e Nomenclature {p } Internal resistance force vector of a super shell or beam element Q Transverse shear force per unit length [Qef Matrix of elastic constitutive relation of the &'th layer in a laminate [Qp\ Matrix of plastic constitutive relation of the /c'th layer in a laminate [Q* j Matrix of elastic constitutive relation in the laminate co-ordinates ^Qp j Matrix of plastic constitutive relation in the laminate co-ordinates e [£%p\ Elasto-plastic constitutive matrix in the laminate co-ordinates q Exponent of Hertzian contact law during the unloading phase R Shell radius of curvature R Radius of curvature of a projectile head R Radius of curvature of a target R Radius of projectile (indenter) shaft R Radius of the curvature of a super beam element x 2 p b {R } Residual force vector in Newton-Raphson iterative scheme S Boundary surface area of super shell and beam elements T Kinetic energy due to the effects of beam torsion and lateral bending of a super beam element {T] Vector of applied boundary surface tractions Tol Tolerance of the solution t Thickness of one layer in a laminated composite shell panel (ply thickness) / Time U Strain energy of a beam element due to torsion, lateral bending, and warping for a super beam element err 0 xxxii Nomenclature u In-plane displacement of the mid-plane of a laminated shell panel in the x direction u x 1 order derivative of u with respect to x u y In-plane rotation of axial displacement u to the y axis {u} Displacement vector of a super shell element {u} Displacement vector of a super beam element V Volume of super shell and beam elements st V Ballistic limit of a target cr V n Projectile velocity at time t V r Projectile residual (exit) velocity from a target V Projectile striking (impact) velocity on a target v In-plane displacement of the mid-plane of a laminated shell panel in the y direction v, In-plane rotation of circumferential displacement v to the x axis v 1 order derivative of v with respect to y xv Out-of-plane displacement of the mid-plane of a laminated shell panel or of the centroidal line of a beam stiffener in the z direction n s st xv 0 xv s xv s0 Transverse deflection of a target at the impact site Flexural deflection of a split-plate (delaminated plate) at the impact site Flexural deflection of a split-plate (delaminated plate) at the impact site when a projectile (or indenter) shaft reaches the frontal surface of a target xv Bending slope of transverse displacement xv to the x axis xv ^ Curvature of a super shell element in the x axis x xv Bending slope of transverse displacement xv to the y axis xv Curvature of a super shell element in the y axis Nomenclature w Bending twist of transverse displacement w x Longitudinal axis of curvilinear co-ordinate system for a cylindrical shell y Circumferential axis of curvilinear co-ordinate system for a cylindrical shell z Radial axis of curvilinear co-ordinate system for a cylindrical shell a Local indentation (or penetration depth) in a target xy a Penetration depth in a target when delamination or transverse plugging initiates a Constant in calculating permanent indentation by Chen and Sun (1985) a Maximum indentation in a target a Permanent indentation in a target /3 Constant in Newmark-/? method 0 c m p 2/3 Included cone angle of a projectile (indenter) P Constant in calculating permanent indentation by Chen and Sun (1985) A Projectile displacement At Time step {s} Strain vector of a super shell element {s} Strain vector of a super beam element {f } Strain vector on mid-plane of a super shell element s Normal component of strain vector {e} in the x axis of a super shell element 0 0 x e Normal component of strain vector {s} in the x axis of a super beam element x s° Normal component of strain vector {s } in the x axis of a super shell element s Normal component of strain vector {e} in the y axis of a super shell element 0 x s° y Normal component of strain vector {s°} in the y axis of a super shell element xxxiv Nomenclature s Normal component of strain vector {s} in the y axis of a super beam element r Warping constant y Constant in Newmark-/? method y y Shear component of strain vector {e} of a super shell element y Shear component of strain vector {f } of a super shell element \K] Curvature vector on the mid-plane of a super shell element xy 0 xy Curvature of a super shell element in the x axis Curvature of a super shell element in the y axis Bending twist of the transverse displacement w LI Coefficient of friction between a projectile (indenter) head and the local damage area of a target due to hole expansion v Transverse Poisson's ratio of a single layer of a target (p Fibre orientation angle of one layer to the JC axis in a laminated composite shell panel 2 Angle of local deformed zone 0 0 0 Rotation of the cross-section of a super beam element Slope of torsion angle 0to the x axis x 0 Slope of torsion angle 0 to the y axis p Mass density p Mass density per unit length for a super beam element {cr} Stress vector CT Hole expansion pressure of a laminate y L 0 cr. cr Stress components of { c r } m Mean indentation pressure of a laminate Modified mean indentation pressure of a laminate Yield stress of target material Transverse shear stress Natural circular frequency Acknowledgement Acknowledgement I wish to express my greatest appreciation and thanks to my supervisors, Dr. R. Vaziri, and Dr. M . D . Olson, for their constant guidance and encouragement both technically and personally. I remember when I did the first presentation on the thesis proposal, how nervous I was and how the instructions from Dr. Vaziri helped me to make a good presentation. I also won't forget that in the first month I arrived at U B C from the other side of the Pacific Ocean, how the personal check from Dr. Olson helped me to overcome my financial difficulty in that month. I would also like to thank my supervisory committee members, Dr. A . Poursartip, Dr. R.O. Foschi, and Dr. D.L. Anderson, for their precious time and helpful suggestions. Without the enlightenment on the physical nature of impact phenomenon and many good discussions with Dr. Poursartip, this work would not have been as successful. The financial support provided by the Natural Sciences and Engineering Research Council of Canada and the Canadian Department of National Defence is also appreciated. I also give my thanks to the people whose excellent work created the experimental data for this thesis: Mr. R. Bennett, Dr. D. Delfosse, Tim Sanders, Elvis Cepus, Dan Ursenbach, Mike Pierson, Walter Lau, Edward Fidler, and Etinne Lecomte. M y thanks extend to all the current and previous members of the composites group for their help and friendship. Finally, I would like to thank my wife, Ru Sun, and my son, Wei Quan, for their loving and understanding. I would also like to thank my parents for bringing me into this mysterious world full of unknowns for PhDs to write their theses on. xxxvii Chapter One: Introduction Chapter One: Introduction 1.1 Impact Events in Applications of Composite Materials Laminated composite materials have been used extensively in many engineering applications as light-weight structures to provide high specific strength and stiffness. For example, the wings and fuselages in aeroplanes are made of composite materials in order to reduce their weight and increase load-carrying capability. However, during their service life these structures will likely encounter many kinds of impact loadings, such as dropped tools, flying birds, fired projectiles, and so forth. Impact on composite structures by foreign objects will cause significant internal and external damages and both types of damage will reduce the structural strength and stiffness. For example, dropped tools will cause delaminations in composites which sometimes cannot be detected by visual inspection and thus may be neglected. If such structures are still in service and not repaired or replaced, the growth of damage will trigger the ultimate failure of the structures and cause financial and human loss. Fired projectiles with high kinetic energy will perforate the structure and lead to its ultimate failure. The projectiles with low kinetic energy may not perforate the structure but may still lead to some other forms of undesirable damage. Therefore, it is necessary to study the initiation and development of damage in composite structures during foreign object impact and in doing so understand the manner in which they Chapter One: Introduction absorb the impact energy, and finally to predict the structural behaviour of damaged composites to subsequent loading. 1.2 General Approaches to Impact Problems There are three ways to approach impact problems. The first way is experimental investigation, where the structures are impacted by instrumented or free-flying projectiles. The loss of projectile kinetic energy and impact damage in the targets are measured during and after the projectile penetration. Simple empirical curves or equations are generated for the tested targets and projectiles. These empirical results are only accurate when they are applied to similar impact conditions and similar structures to those in the original tests. Their predictive capability are doubtful if the structure type, material type, geometric sizes of the targets, and projectile size and head shape are different from the original test conditions. Therefore, they have limited applications. Some empirical equations can be found in Zukas et al. (1982) and Backman and Goldsmith (1978). The second way is analytical modelling. Based upon careful examinations of the initiation, development, and progression of impact damage in the structures during projectile penetration, the associated damage modes and penetration mechanisms are found from these impact tests. Then, analytical models for penetration problems are developed using these penetration mechanisms by applying mechanistic theories. This kind of modelling is the most versatile and effective way when evaluating the ballistic performance of a wide range of projectiles and targets, because the damage mechanisms used in the models allow for many Chapter One: Introduction impact and structural parameters to be considered. However, the validity of the analysis is highly dependent on the reliability of original assumptions, or the appropriateness of penetration mechanisms used in the models. The third approach is numerical modelling. Finite element or finite difference analyses are applied to penetration problems. The impact response, and damage initiation and growth in structures are calculated using computer codes. This kind of modelling provides detailed stress and strain distributions in the structures and thus provides more information than experimental investigation and analytical modelling. However, they are dependent on reliable constitutive models that are often difficult to formulate and calculate. In the present approach, the above three methods are combined together in the way which the advantages of each method are used to complement the disadvantages of another method or methods. The objective is to arrive at efficient ways of analysing impact behaviour of composite structures. At first, impact tests are conducted by Delfosse (1994b) and Sanders (1997) to identify the damage modes and penetration mechanisms in the composite targets during projectile penetration, where projectiles are assumed to be rigid so that they do not deform throughout the whole penetration process. Due to the ease with which one can run tests statically, static penetration tests on the same kind of targets are also performed by Delfosse (1994b) and Sanders (1997) to discover the initiation, evolution, and progression of damage in the targets. Then, penetration mechanisms and damage modes are proposed based upon these experimental studies to model the initiation and growth of impact damage in the structures. 3 Chapter One: Introduction Material characterisation tests are also conducted to obtain the required empirical parameters. Finally, the analytical model is developed and implemented into the finite element program developed in Chapter Two and Three, which can take into account the different structural shapes, boundary conditions, sizes, and material properties. 1.3 Objectives and Scope of the Present Research There are two kinds of targets in impact problems: semi-infinite and finite thickness targets (Zukas et al., 1982). In finite thickness targets, the distal surface of the targets have significant effects on the penetration mechanisms and damage modes, which leads to a more complicated problem and for which it is difficult to find closed-form solutions (Corbett et al, 1996). However, finite thickness targets are of far greater practical interest and therefore are studied in the present work. The objective of this thesis is to develop an efficient approach to modelling the impact of laminated composite plates and cylindrical shells. In this approach, the local damages due to projectile penetration in the impacted structures are modelled by simple analytical methods, while the finite element modelling takes into account the variations in structural dimensions, geometric and boundary conditions. The whole thesis is presented in eight chapters and five appendices. The finite element formulation, time integration method, and Newton-Raphson iterative scheme for non-linear problems in numerical modelling are presented in the second chapter. The numerical verifications and applications of the associated computer code in calculating 4 Chapter One: Introduction the non-linear structural response of composite plates and cylindrical shells are presented in Appendix C. The Hertzian contact law and the iterative scheme used for solution of nonpenetrating impact problems for laminated composite structures are discussed in Chapter Three. The present approach is verified by comparing the predicted results with available experimental measurements, analytical solutions, and other finite element results in the literature. The prediction of nonpenetrating impact response of stiffened laminated composite plates and shells is also presented there and parametric studies are performed for stiffened structures. A literature review on penetration mechanisms and models is presented in Chapter Four. Then guided by experimental observations, a penetration model for laminated composite materials is presented in the fifth and sixth chapters. In addition to predicting the energy absorption in impacted composite structures, this model is aimed at capturing the essence of damage patterns in composites during projectile penetration and considers the transitions of penetration mechanisms with different head shapes and sizes of projectiles and different target thicknesses. Two different material systems, IM7/8551-7 Carbon Fibre-Reinforced Polymer (CFRP) and S2-glass/phenolic resin Glass Fibre-Reinforced Polymer (GFRP) laminates, are studied and provide the experimental verifications for the present model. In Chapter Seven, the penetration model developed in Chapter Six is implemented into the finite element code. The dynamic penetration study on IM7/8551-7 CFRP and S2-glass/resin GFRP laminates is presented in this chapter. The energy absorption and penetration force histories of different laminated composites under different impact conditions, geometric sizes 5 Chapter One: Introduction of targets, and geometric shapes of projectiles are calculated using this code and compared with the experimental measurements. The analytical and numerical studies in the present thesis are summarised in Chapter Eight. The major achievements and contributions are highlighted along with the weaknesses of the present approach. A proposal for future improvement of the present model is finally presented. 6 Chapter Two: Structural Finite Element Formulation Chapter Two: Super Finite Element Formulation 2.1 Introduction Finite element method (FEM) has proved to be an effective numerical tool for solving structural problems over the last few decades. However, in order to model real engineering structures traditional FEM generally requires many elements with a huge amount of data preparation and computing time, thus rendering the analysis impractical for preliminary engineering design. To improve this situation, super finite elements (SFE) have been introduced in the conventional finite element formulation, e.g. Olson (1991) and Koko et al. (1991a) among others. Through the use of SFE, engineering design accuracy can be achieved by only a few elements. The inherent accuracy arises from the displacement functions adopted for these elements. These displacement functions are typically polynomial functions that are selected from the basic analytical solutions to provide a good approximation of the linear and nonlinear deformation modes that may occur in beam, plain and stiffened plate and cylindrical shell structures. Super finite elements have been applied to isotropic materials for non-linear structural analysis in Koko et al. (1991 a-b), Koko et al. (1992), and Jiang et al. (1993). Recently, failure and post-failure models have been proposed for plain and stiffened plates by incorporating interactive failure criteria in SFE analyses in Rudrapatna et al. (1996). 7 Chapter Two: Structural Finite Element Formulation In the present study, the super finite element method developed previously is extended to take into account the structural response of orthotropic laminated composite materials. This code, called SENACS, will be used as a test bed for all the theoretical developments discussed in this thesis. 2.2 Finite Element Equations Formulating finite element equations for super elements follows the same procedure as in conventional FEM for the purposes of this study. Laminated composite plates and cylindrical shells are assumed to be thin and in the case of stiffened structures the beam stiffeners are considered to be slender so that the effect of transverse shear deformation is negligible. The von-Karman assumptions for strains are employed to take account of the effect of large deflections in the structures. When ductile materials, such as metals, are used as the matrix material in composites, the plasticity effects in the matrix cannot be ignored in formulating the finite element equations. Due to anisotropy of composite materials, the elasto-plastic constitutive relations with anisotropic hardening rule (Vaziri et al., 1992) has been applied instead of the conventional rule of isotropic hardening. 2.2.1 Strain-Displacement Relations A stiffened laminated composite cylindrical shell panel as shown schematically in Figure 2.1 is stiffened by a set of orthogonally placed beams, which can be attached from above or 8 Chapter two: Structural Finite Element Formulation below the panel. The beams are usually placed along the axial and circumferential directions of the shell. Consider an isolated panel bay with two adjacent stiffeners (one straight and one curved) shown in Figure 2.2. The local curvilinear co-ordinate system (x,y,z) is attached to the midsurface of the shell panel. For simplicity, it is also assumed in a shell and beam element that the local x-axis is always parallel to the shell axis, while the y and z axes are defined in the circumferential and radial directions, respectively. Figure 2.3 shows the nodal configuration of super elements. The shell panel with thickness h is composed of N orthotropic layers with their reinforced directions oriented at different angles <p to the laminate co-ordinate x as shown in Figure 2.4. In the laminated panels, it is assumed that all the layers are perfectly bonded together and each layer is of uniform thickness (ply thickness) to. Thus the strains are continuous through the shell thickness, including the interfaces between each layer. For linear-elastic materials, this assumption will allow us to replace a laminated panel with an equivalent single layer whose material properties are averaged over the thickness (see section 2.2.2). Thus this will simplify the analysis and still give good results in modelling the global response of the laminated panel. In practical structural design, a laminate is attached to the shell panel as beam stiffeners on its lateral surface as shown in Figure 2.5 (Dost et al, 1991). Therefore, a layered approach is not suitable for formulating finite element equations of beam elements. The beam stiffeners will be treated as isotropic materials. Their elastic modulus and Poisson's ratio are calculated from lamination theory. 9 Chapter Two: Structural Finite Element Formulation 2.2.1.1 Element Displacement Fields The displacement fields within a super shell element can be written as u (2.1) where u, v are the in-plane and w is the out-of-plane displacements of the mid-plane of a laminated shell panel in the x, y, and z directions, respectively; [A ] is the matrix of shape 7 functions with details shown in Appendix A ; {d } is the elemental nodal displacement vector e which has 59 components as shown in Table 2.1, where u and v are the in-plane rotations, >y w and vv are the bending slopes, and w x y x y x is the twist. The displacement fields within a straight super beam element can be written as (2.2) where u, w, and v represent the axial, vertical bending (in z direction), and lateral bending (in y direction) displacements, respectively; and 9 is the rotation of the cross-section. The beam shape function matrix [ N ] is also shown in Appendix A . The total number of beam nodal displacement components in is 19 and shown in Table 2.2, where 6> and 9 are slopes x y of the torsion angle, which denote the rates of twist. The element formulation of a curved super beam element can be obtained by a proper transformation of coordinates. 10 Chapter Two: Structural Finite Element Formulation 2.2.1.2 Strain-Displacement Relations For Super Shell Elements The strain-displacement relations employed in the present formulation are based upon Love and Timoshenko's shell theory: M 2 w v (2.3) where R is the shell radius of curvature and z is the radial co-ordinate measured from the mid-plane of the shell. In terms of the mid-plane strains j^ } and curvatures \K] , we can write Equation (2.3) as 0 follows (2.4) \7xyJ where >,) v ' v y 2 + — + - ^ ^ - R 2 (2.5a) 7. II Chapter Two: Structural Finite Element Formulation -XV, (2.5b) K„ Kxy After Equation (2.1) is substituted into Equation (2.4), the relations between strains {e} and nodal displacement {d } will be in the form of e ^}-{[B hkc]^[B ]]{d } m \ 2 c e (2.6) J where [B ] and [B ] denote strain-displacement matrices for infinitesimal membrane and m c bending strains, respectively; and strain-displacement matrix [C] denotes the von-Karman components. 2.2.1.3 Strain-Displacement Relations for Super Beam Elements The strain-displacement relations for super beam elements are based upon Bernoulli-Euler's beam theory. For example, for a straight beam in the x-direction, (2.7) and for a curved beam in the y-direction, xv v (xv f (2.8) 12 Chapter In Equation (2.8), R = R-e Two: Structural Finite Element Formulation is the radius of the curvature of the beam element with e b representing the eccentricity of the beam centroidal axis from the mid-plane of the shell panel. Also, overbar denotes quantities associated with the beam. After Equation (2.2) is substituted into Equation (2.7) and (2.8), the relations between strains {f} and nodal displacement {J } will be written in the form of c **=(PJ + PJ)P.} *, = (PJ + " (2 PJ)R} 9) (2-10) Details of the matrices of strain-displacement relations [2?,], [B ], [C,], and \C ~\ for beam y y elements are shown in Jiang et al. (1993). The geometric compatibility is applied to satisfy the displacement continuity on the interface between the shell panel and beam stiffener (Jiang et al., 1993). 2.2.2 Constitutive Relations The stress and bending moment resultants {N} and {M} can be written in terms of the stress vector {cr} as h/2 {N}= ff z t+1 \{cr}dz = ^\{cj} dz k -h/2 h/2 k=\ fj Zk z , i+ {M} = j {a]zdz = ]T J {afzdz -A/2 *=1 (2.11) z k (2.12) Chapter Two: Structural Finite Element Formulation There are two ways to evaluate the integrations in the above equations. One method is to calculate the stress components in each layer and then integrate them through the thickness of that layer. The other method, is used for linear-elastic material behaviour, to express the stresses in terms of the displacements where the integral before the displacement vector can be integrated through the laminate thickness in the first step and then saved for the subsequent steps. In the analysis of elasto-plastic materials, the material constitutive relations vary with the stress state. The first method is necessary in order to capture the plastic stress components in each layers through the thickness. For linear-elastic materials, where the material constitutive relations are constant in the analysis, the second method will improve the computing efficiency and save the CPU time in integrations through the shell thickness. Therefore a linear-elastic material, whether it involves small deflection or large deflection analysis, the stress and moment resultants {iV} and { M } will be written in the following form (2.13) where [A], [E], and [D] are the usual in-plane, coupling (bending-stretching) and bending stiffness matrices of the laminate [e.g., Ochoa etal. (1992) and Whitney (1987)]. For elasto-plastic materials, the yield function for anisotropic yielding and hardening under plane stress conditions can be written as (2.14) 14 Chapter Two: Structural Finite Element Formulation where A\j denotes the anisotropic yielding and hardening parameters which describe the shape of the yield surface, o; denotes the stress components, and k is the effective size of the y yield surface. Both k and Ay are updated with the stress state cr, as plasticity progresses. y The subscripts i,j in Equation (2.14) are numbered as 1,2, and 6, referring to the three inplane stress components in the principal material directions. The mathematical derivation from yield criterion and hardening rule to stress-strain relationship was shown in Vaziri et al. (1992) and has been briefly summarised in Appendix B. Thus, the incremental stress-strain relationship in the k-th layer of the laminate can be written as d{cr} =([Q ] -[Q ] )d{s} K K E k (2.15) k p where [Q f and \Q \ are matrices of elastic and plastic constitutive relations, respectively; e P and {e] is the strain tensor in the principal material directions of the shell panel. The details k of matrices [Q ] and [Q ~\ are given in Appendix B. K E P 2.2.3 Virtual Work Principle Neglecting structural damping, the principle of virtual work for super elements can be written as j{du} p{u}dV+^{Se°} {N} T V T A + {SK} {M}yA T = J" {Su} {T}dS T (2.16) S where p is the mass density; V, A, and S are, respectively, the volume, mid-plane area, and the boundary surface area of the shell and beam elements; {J} is the vector of applied 1 15 Chapter Two: Structural Finite Element Formulation boundary surface tractions; overdots denote differentiation with respect to time; and 8 denotes a virtual change in the quantity to which it is attached. Substituting the strain-displacement relations [Equations (2.6), (2.9), and (2.10)] into Equation (2.16), and noting that the virtual changes in the nodal displacements {Sd } are e arbitrary, the following set of discretized equations of motion for a single element can be obtained: k]R}+k}={/;} (2.17) where [wj, {p }, and JX'} are the elemental consistent mass matrix, internal resistance e force vector, and external force vector, respectively. The elemental tangent stiffness matrix can be determined as the derivative of the internal force with respect to the nodal displacements. For linear-elastic materials and a shell element, it is <?{Pe} ^{^VJ +M ' M ^ [K] =-m =^ We\ T (' > 2 18a We) = After substituting Equations (2.5a), (2.5b), and (2.13) into (2.18a), the stiffness matrix can be written as [K] T KM MB.] A +fe]>PJ)^ + j([^][4C] + [Cr[45]+ r m (2.18b) 16 Chapter [cn4C] Two: [5j [5][C] r + + Structural Finite Element Formulation [C]^PJ)^ where the first integration on the right-hand side of the above equation is the linear part of the tangent stiffness matrix while the second integration is its non-linear part taking account of large deflections. For elasto-plastic materials, it is W r = | R } = J { ( ^ + where [gj,] is the elasto-plastic ^ } ^ W ( 2 A 9 ) constitutive matrix in the laminate co-ordinates ([fij,] = [fif ] - [ « ] ) • Adding the element contributions in the usual manner, the discretized equations of motion for the shell structure can be written as (2.20) [M]{D} + {P} = {F'} where [M], [D], {P}, and {f} and are the assembled global equivalents of [m ], [d ], e e {p }, e {/:}. For a super beam element, elemental mass and stiffness matrices are calculated first on an element basis in the same way as a shell element and then added into the global mass and stiffness matrices. The virtual work principle shown above in Equation (2.16) is based solely on beam bending theory so that the effects of beam torsion and lateral bending have not yet 17 Chapter Two: Structural Finite Element Formulation been taken into account. In order to consider these effects, the formulation of mass and stiffness matrices developed in Jiang et al. (1993) is used in the present analysis. Considering a beam element vibrating at a circular frequency a>, when the effect of rotary inertia is ignored, the kinetic energy due to the effects of beam torsion and lateral bending can be written as T = -oo p \{A v +J 0 )dx ^ o 2 2 L b (2.21) 2 c where PL is the mass density per unit length, is the cross-sectional area, J is the polar C moment of inertia about the centroid, and / is the length of the beam. Using Equation (2.2), the matrix form of Equation (2.21) can be written as T = ±G) {d } [M ]{d } 2 (2.22) T e w e Thus the total beam mass matrix is obtained by combining [M ] with elemental mass matrix TB due to beam bending and adding them together into the global mass matrix. The strain energy of a beam element due to torsion, lateral bending, and warping for a straight beam element in the x-direction, is calculated from (2.23) while for a curved beam element it is U = -GJ\ (— + ) dx +—EI \ ( 2 Jo dx R„dx 2 o R ) dx 77 T zzJ dx 2 b Chapter Two: Structural Finite Element Formulation (2.24) where G, E are, respectively, the shear and elastic moduli; J is the torsional constant; is the moment of inertia about the z axis; and ris the warping constant. After Equation (2.2) is applied, the matrix form of equation (2.23) and (2.24) will become (2.25) where [/C J is the stiffness matrix of a beam element due to the beam torsion and lateral ra bending effects and will be assembled in the same way as the beam mass matrix. 2.3 Solution Scheme The Gauss integration method is used to evaluate the mass and stiffness matrices and all the consistent force vectors. A 5x5 Gauss integration rule is applied to the mid-plane of the shell element and a minimum of one Gauss point per layer is used for integrations through the thickness of the laminate for elasto-plastic analysis. The Gauss integration rule for a beam element is 5x1 where 5 Gauss integration points are placed along the beam axis. The number of Gauss integration points through the beam thickness depends on the cross-sectional shape of beams as shown in Jiang et al. (1993). Because the number of layers making up a laminate panel may be very large, it is necessary to use different number of Gauss integration points through the thickness of each layer for 19 Chapter Two: Structural Finite Element Formulation elasto-plastic problems to improve the computational efficiency. The Gauss integration rule used in the present formulation and computer program SENACS, is shown in Table 2.3. 2.3.1 Time Integration Method The general Newmark-P method is used to integrate the global equations of motion [Equation (2.20)] in the time domain. Accordingly, the generalised nodal velocity and displacement vectors at time t n+l =t +At can be written as n {^L={^+M(i-^} n + ^}„ + 1 ] (2.26) {D} n+l = {D} At{D} ^[(l-2/J){D} n+ n+ n +2/3{D} J n where At is the integration time step, and the subscripts n and n+\ denote quantities evaluated at time t n and t , respectively. B+i When B=0.25 and y = 0.5, Equation (2.26) will be degenerated to the case which has no artificial damping. When y > 0.5, artificial damping will be introduced to filter out the high-frequency vibrations in the structure. The highfrequency dissipation will be maximised for y > 0.5 when is taken to be related to y as [Cook etal. (1989)] (2.27) For any values of /? and y satisfying 2j3>y>0.5, the solution of Equation (2.20) will be unconditionally stable when its velocity and acceleration are expressed as in Equation (2.26). It should also be noted that for a structure which has a maximum naturalfrequency0) , the max 20 Chapter Two: Structural Finite Element Formulation time step At chosen in the analysis should satisfy the convergence criterion in Newmark (1959) as follows At< -= 4fi (2.28) l ^max 2.3.2 Newton-Raphson Iteration Scheme The non-linearities arising from the combination of large deflections and plasticity lead to a set of non-linear algebraic equations that need to be solved at each time or loading step using an iterative scheme. In the current formulation, the equation of motion [Equation (2.20)] at time t n+l is solved by forming an effective static problem which is then solved using a Newton-Raphson solution procedure. Thus, solution of the following equation is sought M^L,={^L (2-29) where {A£}„ + 1 = {£}„ + 1 -{/>}., In Equation (2.30), [K ] is the tangent stiffness matrix resulting from the non-linear strainT displacement or stress-strain relations, and can be assembled from elemental stiffness matrix [k ] . e T For linear-elastic and small deflection analysis, [K ] T will remain a constant throughout the solution process. The residual force vector in the structure is calculated as 21 Chapter Two: Structural {R } = {F'}-[M]{D}-{P} err Finite Element Formulation (2.31) at each iteration within the time or loading step. When the required tolerance Tol given by Tol = 11*' " F' (2.32) is reached at an iteration, the iteration will end and then the analysis proceeds to the next step. In the present study Tol = 0.01. The tangent stiffness matrix for large deflection and/or elasto-plastic analysis can be updated in each iteration or just the first iteration of a given time or loading step. If the non-linearity in the problem is not very severe, [K ] is suggested to be updated only in the first iteration of T each time step and maintained constant for each subsequent iterations to maintain computational efficiency. 2.4 Summary A computer program, SENACS (Super Elements for Non-linear Analysis of Cylindrical Shells-Version for Composite Materials), has been written based upon the above finite element formulations using FORTRAN language. Figure 2.6 shows the flow chart of the program. The structural geometry, element mesh, boundary conditions, material properties, and external load condition are inputted first. Because only a few elements are needed to model Chapter Two: Structural Finite Element Formulation the problem, it is very easy to generate the input of nodal and element data to the program without any pre-processor. After initialising all the arrays and parameters used in the analysis, the program starts the loops for time or load steps and iterations. The elemental mass (only for eigenvalue and transient problems) and stiffness matrices, as well as force vector are generated first and then assembled into the global mass matrix, stiffness matrix, and force vector. The effective stiffness matrix and load vector are calculated if a transient analysis is required. The solutions are attained by solving the finite element equations. If the solutions satisfy the required convergence tolerance, the iteration loop ends and the final solutions are written into pre-assigned files. Otherwise, more iterations are performed until the solution converges. After all the required time or load steps are looped over, the program stops. The numerical verifications of SENACS have been conducted by comparing the predicted results with available experimental, analytical, and numerical results. The comparisons show good agreement [see Appendix C] and thus verify the program. This instils confidence in using the program SENACS for linear and non-linear static and transient structural analysis of laminated composite plates and shells. In the next chapter, details of adding impact analysis capability to SENACS will be presented. 23 Chapter Table 2.1 Finite Element Formulation Degrees of Freedom Ui, V i , W l , U y l , V x l , Wxl, W y l , W , i x y U , V ,W ,U , Vx2, W , , W , , W , x y 2 u , v , w , u , y , w , w, , w, xy3 2 3 2 2 3 3 U4, V , W , U 4 4 U , W ,W 5 y 2 x 2 y 3 5 x3 , Vjrf, W y 4 y 2 >x3 ] X 4 y3 ,W > y 4 , W j X y 4 , U8, U n y 5 v , w , w, , v , V12 5 6 x6 9 U , W , W , , U , U12 6 7 V6, Wg, W y 7 > x 8 9 , V , V n 8 U , V , W , Uio, Vio, U i , V i 7 7 9 3 3 Nodal degrees of freedom for a super beam element (Jiang et al, 1993). Type Direction straight X curved y Node Number 1 2 3 1 Degrees of Freedom U l , V i , W i , V i , W , i , Gi, 0 i x x x U , V , W , V , W , 02, 0 U , W , 0 , U4, U V l , U l , W l , U y l , W y l , 01, 0,yl 2 2 3 2 3 ;X2 3 V , U ,W ,U J V3, W , 2 2 2 3 0, 3 ;X2 iX2 5 z -5 Table 2.3 Structural Nodal degrees of freedom for a super shell element (Jiang et al, 1993). Node Number 1 2 3 4 5 6 7 8 9 Table 2.2 Two: j y 2 , W V , V 4 > y 2 , 02, 0, y2 5 Gauss integration rule used in SEN ACS for a super shell element. Axial Circumferential 5 5 5 5 5 5 Through the thickness of each layer Linear Elastic Elasto-Plastic No. of Layers none 4 1 none 2 2-10 none 1 >10 24 Chapter Two: Structural Finite Element Formulation X, U Figure 2.2 Illustration of element connection between a super shell element and a super beam element (straight and curved). 25 Chapter Two: Structural Finite Element Formulation X, u 1 (b) Curved Beam Element • (c) Straight Beam Element Figure 2.3 Nodal configuration of super elements. Figure 2.4 Schematic illustration of a laminated composite plate. 26 Chapter Two: Structural Finite Element Formulation 2A£ Figure 2.5 Schematic illustration of a stiffened laminated composite plate where a laminated stiffener is attached to a laminated skin (Dost et al, 1991). 27 Chapter Two: Structural Finite Element Formulation Input and Initialization Time Step Loop Iteration Loop Element Loop * Calculate Element Mass, Stiffness, and Force Assemble Global Mass, Stiffness, and Force no yes for all the elements > r Calculate Effective Stiffness and Load i Solve the System Algebric Equations Check the Convergence of the Solutions no solution converges > r Write Results into Separate Files yes for all the steps required ^ no End Figure 2.6 Flow chart of structural response calculation using SENACS. 28 Chapter Three: Nonpenetrating Impact Response Chapter Three: Nonpenetrating Impact Response 3.1 Literature Review The transient response of structures subjected to impact can be assumed to occur at two levels: a global response of the overall structure and a local response under the point of impact (Pierson et al, (1993). Thus, the first step in understanding the impact response of structures is to accurately predict their transient global response (i.e., nonpenetrating impact response). Numerous works have been done in the area of impact of laminated composite materials. They have been carefully reviewed in Abrate (1991 and 1994). Only selected work on analytical modelling and finite element modelling of nonpenetrating impact response of plain and stiffened laminated composite plates and shells will be reviewed here. 3.1.1 Analytical Work on Plates and Shells As one of the earliest work on the analytical treatment of the impact behaviour of composite plates, Sun and Chattopadhyay (1975) used a Mindlin-type laminated plate theory originally developed by Whitney and Pagano (1970) to solve the resulting equations of motion by a modal analysis technique. A similar approach was also used by Dobyns (1981), and Qian and Swanson (1990) to study the impact response of laminates. Matsuhashi et al (1993) included the effects of large deflections within a Rayleigh-Ritz procedure to investigate the 29 Chapter Three: Nonpenetrating Impact Response sensitivity of the impact response to partial in-plane constraints at the supports. Pierson and Vaziri (1996) presented an efficient modal analysis procedure for predicting the impact response of simply-supported composite plates. Their analytical formulation took into account the combined effects of transverse shear deformation, rotary inertia, and the nonlinear Hertzian contact law. The analytical technique to solve nonpenetrating impact problems for laminated composite shells generally employs Fourier series expansion. Ramkumar and Thakar (1987) predicted the transient response of a cylindrical shell under an impact loading, which was assumed to be a known input. Christoforou and Swanson (1990) applied a double Fourier series expansion to the time histories of the impact load and transverse deflection to obtain a closed-form solution for the impact response of cylindrical shells under simply-supported boundary conditions. Their analysis neglected the effects of in-plane and rotary inertia as well as the contact stiffness. High-order shear deformation theory was utilised in Gong et al. (1995) to predict the transient response of simply-supported, doubly curved shells excited by an impact from a blunt-ended projectile. The impact force history was calculated based on the spring-mass model developed in Gong et al. (1994) and the equations of motion were solved by Fourier series expansion of in-plane orthogonal curvilinear co-ordinates and time. 3.1.2 Numerical Work on Plates and Shells For more complex structures with more general boundary conditions, one has to resort to numerical methods such as the finite element method to model the impact response of these structures. 30 Chapter Three: Nonpenetrating Impact Response Sun and Chen (1985) developed a two-dimensional finite element procedure to analyse the impact response of initially stressed laminated composite plates. The plates were modelled by a nine-node isoparametric plate element accounting for the transverse shear deformations. The contact phenomenon was simulated using an experimentally determined static indentation law proposed by Tan and Sun (1985). The Newmark constant-acceleration time integration scheme was used to solve the equilibrium equations of both projectile and plate. Furthermore, based on the above finite element formulation and contact law, the large deflection response of composite laminates subjected to impact was investigated by Chen and Sun (1985). Montermurro et al. (1995) studied the impact response and damage progression of laminated plates and shells using isoparametric Mindlin plate elements. The Newton- Raphson iterative method was used to solve the non-linearities caused by the contact effects and large deflections. Bachrach and Hansen (1989) developed a mixed finite element method for a composite cylinder subjected to impact. The projectile displacement and the deflection at the impact point of the cylinder were approximated using the Wilson-0 method and a finite difference method, respectively, to solve the non-linear equations arising from contact effects. Lee et al. (1993) performed finite element analyses of cylindrical shells subjected to impact using a non-linear contact law that took into account the curvatures of the contacting bodies. The effect of large deflections was also investigated in their analysis. The large deflection response of laminated cylindrical and doubly curved shells to impact was also investigated in Chandrashekhara and Schroeder (1995) using a shear flexible finite element method. The Chapter Three: Nonpenetrating Impact Response time histories of impact force and central deflection were presented for various projectile parameters, shell geometries, and boundary conditions. 3.1.3 Experimental, Analytical, and Numerical Work on Stiffened Structures The research on the impact response of stiffened composite structures initiated in recent years due to extensive applications of these structures. Madan and Shuart (1990), and Madan (1991) studied the influence experimentally. of impact on blade-stiffened graphite/epoxy plates In their tests, the stiffened flat panels were impacted by a drop-weight projectile at various energy levels. The impact points were either at the midbay or at the middle of the panel over the centreline of the stiffener. Damaged specimens were then loaded uniaxially and tested to failure by slowly applying a compression load. A simple analytical model was proposed to calculate the residual strength of the stiffened plates in post-impact compression. Basque and Aoki (1993) presented a phenomenological study on the impact damaged CFRP blade-stiffened flat panels. Their experimental results showed that the impact damage occurred first in the region of the blade stiffener and panel joint. Then the impact on the plates was approximated by a transverse load neglecting the dynamic effects and plugged into finite element analysis to model damage development in the joint region between the panel and the blade with more than two thousand 3-D elements. Bucci and Mercurio (1994) studied the static and fatigue behaviour of CFRP panels with bead stiffeners under compression loads. The influence of impact damage from low velocity impact on buckling threshold load was also investigated. Scott and Rees (1995) analysed the post-buckling behaviour of carbon/epoxy panels with blade stiffeners experimentally and 32 Chapter Three: Nonpenetrating Impact Response numerically. The experimental results of damage development in CFRP stiffened plates with impact damage under compression loads was also presented in Aoki (1995). Although these studies dealt with the impact response of stiffened composite plates, their emphasis was on the damage development or buckling threshold of the stiffened structures with impact damages. No contact and dynamic effects were considered in their analytical models and numerical analyses. 3.2 Contact Law According to the indentation law developed by Tan and Sun (1985), the contact force F is given by the following modified non-linear Hertzian-type relations ka 3/2 F= loading ' a-a 1 p ( a —a. unloading (3.1) .3/2 reloading where a denotes the local indentation in the target, £ is a contact stiffness coefficient, and F m denotes the maximum contact force reached before unloading, k depends on the material properties and interface curvature between the two contacting bodies, which are assumed as cylindrical or spherical. The permanent indentation in the target, a , was found to vary with p the material systems and in a linear relationship with the maximum indentation a . m In Equation (3.1), q is the exponent of contact law during the unloading phase which is 33 Chapter Three: Nonpenetrating generally taken as 2.5 based upon experimental measurements. Impact Response Note that because the indentation law is based on static test results, the target inertia due to the local indentation is ignored here. Various analytical and experimental techniques have been proposed for determination of k in the literature. In the present approach, the following explicit expressions taken from Hills et al. (1992) for contact between a spherical indentor and a transversely isotropic medium are employed, k= 8 1 (3.2) where G and v are the transverse shear modulus and Poisson's ratio of a single layer of the z z target; and R and R are the radii of curvature of the projectile and the target, respectively. x 2 For simplicity, the transverse Poisson's ratio and shear modulus are assumed to be equal to their corresponding in-plane values. The projectile is assumed to be rigid in the present approach and the effect of its elastic properties on contact stiffness coefficient is thus ignored in Equation (3.2). 3.3 Solution Scheme Similar to Equation (2.16) in Section 2.2.3, the virtual work principle for the target can be written as 34 Chapter \{5u} p{u^dV J V + \\{5e°} {N} Three: + {8K} {M} T T A Nonpenetrating Impact dA-j{Su} {T}dS-FSA Response =0 T (3.3) S where A and m are the projectile displacement and mass, respectively. The local indentation of the target, a, satisfies the following constraint condition at the contact point (x ,y ), c A=a + w c (3.4) 0 where w = w(x ,y ) is the transverse displacement of the target at the impact point. The 0 c c other parameters and symbols, p, V,A,S, {T}, and 5 are the same as those used in Chapter Two. Applying the same derivations in Chapter Two for the target and the Newton's 2nd law for the projectile, the discretized equations of motion for a single element and the projectile can be written as: [m ]{d } + { } = {/:} + {/:} (3.5) mA + F = 0 (3.6) e where {^} e e Pe denotes the elemental contact force vector. It should be noted that Equations (3.5) and (3.6) are coupled through the compatibility condition of Equation (3.4). The assembled discretized equations of motion for the target can be written as [M]{D} where + {P} = {F'} + {F<} (3.7) j , and {F } are the assembled global equivalents of [m ], {p }, {f '}> c e e e and{X }. c 35 Chapter Three: Nonpenetrating Impact Response In the differential equation (3.7), the contact force term { F } contains the deflection at the e impact site of the structure. However, to solve for the deflection of the structure we need to know the magnitude of the contact force {F }. c Therefore, Equation (3.7) is a non-linear equation even for small deflection analysis and linear-elastic materials and an iterative scheme should be used to solve it. Generally, the Newton-Raphson iterative method is employed to perform this kind of task. Updating the stiffness matrix at each time step (for initial stiffness method) or at each iteration (for tangent stiffness method) reduces the computational efficiency of super finite element analysis. Thus, the contact non-linearity is linearized in the'present analysis using the finite difference method. At time r n+1 , the projectile displacement A is given by the solution of Equation (3.6) as n + 1 F At 2 A„ =A„ + F „ A r - ^ — 2m (3.8) +1 where V and F are the projectile velocity and the contact force evaluated at time t . The n n B deflection at the impact point of the target at time / n+1 is approximated by using the finite difference method, w Ar + | w „ (AO o„ = Wo. + \ +1 The indentation of the target at time t D+l 0 2 (3-9) is thus given by =A , -w„ J +1 (3.10) 36 Chapter Therefore, the contact force at time t n+l Three: Nonpenetrating Impact Response can be explicitly written in terms of known quantities at time t and thus there will be no need for any iterations. n 3.4 Results for Laminated Plates and Shells In this section, nonpenetrating impact model is verified by comparing the predicted results with experimental, analytical, and other numerical results available in literature. Some of them have already been published in Vaziri et al. (1996). Numerical experiments are also carried out for parametric studies. 3.4.1 Small Deflection Analysis 3.4.1.1 Impact Response of [0/90/0/90/0] CFRP Plate s The response of a simply-supported T300/934 CFRP square plate with a lay-up of [0/9070/90/0] impacted by a 8.54 g steel ball with a striking velocity of 3.0 m/s will be s considered here. The orthotropic material properties of the constituent layers and the geometric sizes of the plates are defined in Table 3.1. Fundamental frequency The fundamental frequency of the plate calculated using different mesh densities as shown in Table 3.2 is compared with the analytical solution by Pierson and Vaziri (1996) which accounts for transverse shear deformations. Present results, even using the coarsest mesh (lxl for the whole plate), give a good estimation of the fundamental frequency of the plate. 37 Chapter Three: Nonpenetrating Impact Response Impact analysis The transient analysis of the same laminate defined in Table 3.1 involving the impact of a 12.7 mm diameter steel ball with a mass of 8.54 g and velocity of 3.0 m/s is carried out using different mesh densities. The predicted force-time histories are shown in Figure 3.1. It can be seen that the results obtained from 4x4 and 5x5 grids for a quarter plate are almost indiscernible. The contact stiffness coefficient used in the present analysis is the same as that used in Pierson and Vaziri (1996), namely, k = 3.14xl0 N/m . The time step is taken to be 8 3/2 At = 0.3 JJS. Figure 3.2 and Figure 3.3 show, respectively, the predicted time histories of the impact force and central plate deflection using a 4x4 mesh for a quarter plate. No artificial damping is introduced in this case in order to verify the present approach. The present prediction is seen to be reasonably close to the modal analysis results of Pierson and Vaziri (1996) and the finite element results of Sun and Chen (1985). The discrepancies can be attributed to the fact that, unlike the other two approaches, the present model does not include the effect of transverse shear deformations and thus leads to somewhat stiffer results. 3.4.1.2 Impact Response of [45/90/-45/0] T800H/3900-2 CFRP Plate 3s Comparison with instrumented impact tests To validate the predictive capability of the present approach, present results are compared with instrumented impact test data reported in Delfosse et al. (1993a). Comparisons between the predicted and measured results are shown in Figure 3.4 to Figure 3.7, inclusive. The 38 Chapter Three: Nonpenetrating Impact Response properties of the target plate used here are shown in Table 3.3. The numerical simulations are conducted using the option of small deflection analysis and a uniform 4x4 mesh for a quarter-plate. The fundamental frequency of the plate is calculated as 2.95 kHz and the time step used is At = 1 ps. Also, a contact stiffness value of k = 6.0x10 N/m 8 3/2 is used in the simulations. This value was found to fit the static indentation test results quite accurately (Delfosse etal, 1993a). The trends of the predicted contact force histories are seen to be in fair agreement with the experimental results for both the low-velocity/large mass (Figure 3.4 and Figure 3.5) and high-velocity/small mass (Figure 3.6 and Figure 3.7) impact events. In most of the cases considered, peak values of the impact force are overpredicted while the impact durations are underpredicted; all indications of the response of a stiff system. It should be remarked that the multiple oscillations in the predicted force-time histories caused by high frequency vibrations of the projectile/target system have been smoothed by introducing artificial damping with ^ = 5.0 and then fi= 15625 from Equation (2.27). In the experiments presented here, some of these oscillations had also been filtered out as high frequency noise (see Delfosse et al, 1993a). Bucinell et al. (1991) proposed a two degree of freedom spring-mass system to model the nonpenetrating impact response of plates. The time periods of the high frequency oscillations (without artificial damping) in the present force time history are found to be close to the second vibration period of the Bucinell's spring-mass system. More details on how artificial damping affects the impact force history in the present approach is presented in Appendix D along with the Bucinell's spring-mass model. 39 Chapter Three: Nonpenetrating Impact Response Response under different impact velocities In this section, the same laminate is used to investigate the impact response under different striking velocities. The impact energy is taken to be the same as that in the instrumented impact tests shown in Figure 3.5 and Figure 3.7, i.e., 22 J. Experimental studies [see Delfosse et al, (1994b)] indicated that the force-displacement curves generated from high velocity impact events initially followed the same curve as the static indentation curves and then deviated from it when flexural waves that had initiated at the impact site had time to travel to the boundary and reflect back. Figure 3.8 shows the relationship between the impact force and the projectile displacement under impact velocities ranging from 59.25 m/s to 296 m/s. Superposed on the same graph is the static indentation curve. It can be seen that for high velocity impacts, the initial part of the force-displacement curve coincides with the static indentation curve. Therefore at high velocity impacts, the target has very little time to bend and the impact behaviour resembles that of a projectile striking an infinite or semi-infinite target. For very high velocity impact events, such as 132.5 m/s and 296.3 m/s, there is only one peak in the force-displacement curve, which corresponds to the maximum indentation depth. For intermediate impact velocities, such as 59.25 m/s, there are two more peaks in the forcedisplacement curve, which are caused by interaction of the projectile motion and the plate vibration (also see Figure 3.9 and Figure 3.10). It should be noted that the projectile continues to decelerate during the impact. After the first peak of the impact force, both the 40 Chapter Three: Nonpenetrating Impact Response projectile and plate decelerate. Their rates of deceleration are different and vary with the time. Near to the time when the plate velocity is equal to the projectile velocity again (t = 0.095 ms), there is a second peak of the impact force. Shortly after that, the projectile starts to retreat, moves backward and its velocity increases negatively. At the same time, the plate velocity decreases to zero and continues to increase in the opposite direction. The plate moves faster than the projectile and results in an increase of the impact force. When the projectile moves with the same velocity as the plate (i.e., the velocity of indentation depth cc = A - WQ is zero), the third peak force is reached. Then, the projectile moves away from the plate and the impact force decreases and reduces to zero. The time at which peak forces occur are indicated in Figures 3.8 and 3.9. For low velocity impacts in Figures 3.4 and 3.5, the target deforms in a way that is similar to a quasi-static deformation. Thus there is only one peak force in their force-displacement curves. Figure 3.11 shows the deflection profiles at the instant when maximum deflection occurs for three typical impacts: low velocity impact (5.30 m/s), intermediate velocity impact (59.25 m/s), and high velocity impact (296.3 m/s). As expected, the low velocity impact will cause much more deformation in the target than the intermediate and high velocity impacts. 3.4.1.3 Impact Response of [90/±22] IM7/55A 2T CFRP Cylinder Christoforou et al. (1989) experimentally studied the transverse impact behaviour of [90/±22] IM7/55A CFRP laminated composite cylinder impacted by a 32 g steel ball at 3 2T 41 Chapter Three: Nonpenetrating Impact Response m/s. In a subsequent paper, Christoforou and Swanson (1990) presented analytical solutions for some simple boundary conditions that were different from those used in their experimental set-up. The cylindrical shell problem is shown in Figure 3.12(a) and the material constants, geometric dimensions, and laminate lay-up are listed in Table 3.4. A series of transient analyses for a prescribed load-time history have been performed first to study the convergence of the numerical results and hence arrive at a desirable mesh geometry. Three uniform meshes (2x2, 2x4, and 4x4) and one non-uniform mesh (4x4) in which the elements were gradually refined near the impact site [Figure 3.12(b)] are considered for discretization of a quarter-cylinder. The transient response of the shell to a given force-time history [Figure 2 in Christoforou and Swanson (1990)] is carried out using the above mesh configurations. It has been found that the 4x4 non-uniform mesh arrangement yields results that agree more closely with the analytical predictions and is therefore used in subsequent impact simulations. Before conducting the impact analyses, the contact stiffness coefficient between the target and the projectile have to be determined. However, this information was not available from the paper by Christoforou and Swanson (1990) as their analysis neglected the local contact deformation in the target and only considered its bending response. Using a projectile diameter of 15.9 mm (taken from Christoforou et al, 1989), the contact stiffness coefficient is found to be k = 1.58 xlO N/m which is calculated from Equation (3.2). The first natural 9 3/2 frequency of the cylinder is 0.99 kHz and the time step size used is Ar = 1 ps. Chapter Three: Nonpenetrating Impact Response An impact simulation with the conditions described in Table 3.4 was performed. The time history of the normalised transverse deflection of the shell at the impact point is shown in Figure 3.12(c). The predicted deflection-time history agrees well with the analytical solutions. As noted before, the analytical model did not consider the indentation response and therefore resulted in a higher peak value of central deflection. 3.4.2 Large Deflection Analysis 3.4.2.1 Impact Response of [0/45/0/-45/0] T300/934 CFRP Laminated Plate 2s Large deflection response of laminated composite plates subjected to impact loads was first investigated by Chen and Sun (1985) using the finite element method. For comparison, the same problem is analysed using the present formulation. The laminated plate considered in this case has a [0745/07-45/0] lay-up with the same basic layer properties as those shown in 2S Table 3.1. The plate dimensions are 15.24 cm x 10.16 cm x 0.269 cm and the boundary conditions are simply-supported and immovable in the plane of the plate along all the edges. The laminate is considered to be impacted by a 8.54 g ball fired at a velocity of 30.0 m/s. The contact stiffness coefficient is the same as that used in the Section 3.3.4.1.1. The permanent indentation is determined by p=Pp(. m-a ), a a c a = 0, p \ia >a m c if aM < a c Chapter Three: Nonpenetrating where fi = 0.094, a = 1.667xl0" cm [Chen and Sun (1985)], 2 p c Impact Response is the maximum indentation depth, and the unloading index of the contact law q = 2.5. A 2x2 mesh for a quarter plate is used in the analysis. The fundamental frequency of the plate is calculated to be 0.78 kHz and the time step used is At = 4 us. No artificial damping is introduced in the time integration scheme. The time histories of the target deflection at the impact point are shown in Figure 3.13, where present results using both small and large deflection analyses are compared with corresponding results in Chen and Sun (1985). The present results agree very well with the predictions of Chen and Sun (1985), thus validating the large deflection analysis capability of the present model. It should be pointed out that the present analysis uses larger time steps than those used in Chen and Sun (1985), leading to more computational efficiency. 3.3.1.1 Impact Response of[±t5/0J s AS4/3501-6 CFRP Plate Matsuhashi et al. (1993) investigated the impact response of a [±45 /0 ] laminated CFRP 2 2 s plate both experimentally and analytically. The same problem is studied here and present results are compared with experimental results reported in Matsuhashi et al. (1993). The material properties, geometrical dimensions, boundary conditions, and laminate lay-up are shown in Table 3.5. The mass of the projectile is 1.53 kg and its velocity is 3.0 m/s. The contact stiffness coefficient, k, used in the present analysis is the same as in Matsuhashi et al. (1993), namely, /c = 5.0xl0 iV7m . 8 3/2 Chapter Three: Nonpenetrating Impact Response A 2x2 mesh for a quarter-plate is used. The fundamental frequency of the plate is 1.33 kHz and the time step used is 1.5 us. No artificial damping is introduced in the time integration scheme. A comparison of the present results with experimental results of Matsuhashi et al. (1993) is shown in Figure 3.14. The present large deflection analysis results are stiffer than the experimental results because the clamped boundary condition cannot be realised in the experiments, especially when the deflection is very large. However, the present small deflection results are on the "soft" side of the experimental results. This indicates that the present small and large deflection results are reasonable bounds on the real behaviour. Also, in the analytical study of Matsuhashi et al. (1993), the fully clamped boundary condition was not applied and a flexibility factor for the in-plane boundary condition was introduced to approach the experimental results. 3.4 Results for Stiffened Plates and Shells After verifying the ability of the computer program in calculating the nonpenetrating impact response of plain plates and shells, the nonpenetrating impact response of stiffened plates and shells is carried out in this section. Following the verifications of the model, some numerical experiments (parametric studies) will be performed here as well. 3.4.1 Verification Consider a stiffened laminated structure, IM7/977-2 CFRP laminated plate with a [45/90745/0/45/90/-45/0/90/0/90/0] lay-up. The plate dimensions are 17.8 cm x 50.8 cm x 0.451 s Chapter Three: Nonpenetrating Impact Response cm. The finite element mesh used in the present analysis consists of 2x2 shell elements and 2x1 beam elements for a quarter-plate as shown in Figure 3.15. The plate is simply- supported along two opposite stiffeners. The other two opposite edges are free. The elastic properties of the lamina are listed in Table 3.6. There are two T-shape stiffeners beneath the skin in the stiffened plate. These stiffeners are made of the same laminate material as the skin with a [22.5/90/-22.5/0] lay-up. The elastic 2s modulus and Poisson's ratio for the stiffeners are evaluated by classical laminate plate theory as E = 68.3 GPaand v = 0.26. The impact point is at the midbay of the stiffened panel. The projectile mass and velocity are m = 6.347 kg, V = 2.74 m / s. The diameter of the spherical projectile is 25.4 mm. Then, the s coefficient of contact stiffness k is calculated using Equation (3.2) to be k = 4.68 GN / m . 3/2 The present contact force history is compared with the experimental data measured from the instrumented impact tests at the University of British Columbia (Dost et al, 1991) in Figure 3.16 and fairly good agreement is attained. The artificial damping (y = 8.0 and = 18.0) is introduced to filter out the spurious high frequency oscillations. 3.4.2 Numerical Experiments Having validated the present approach, a variety of example problems in this section are considered to study the impact response of stiffened laminated plates and cylindrical shells using small and large deflection analysis. These are hypothetical problems which have not been reported in the literature. The purpose of the analyses is to identify the influence of 46 Chapter Three: Nonpenetrating Impact Response various parameters on the impact response of stiffened plates and shells. For all the problems considered, the material properties and geometric sizes are defined in Table 3.6. projectile mass is m = 0.635 kg and projectile velocity is V = 8.7 m/s. s The Unless mentioned otherwise, all the impact sites are taken to be at the midbay of the stiffened panel. 3.4.2.1 Effect of Curvature Consider a stiffened shell panel with the ratio of shell radius to arc length as infinity (i.e., flat plate), 5, and 1. The boundary conditions and stiffener arrangements are the same as in the problem considered in section 3.5.1. Artificial damping was not introduced in the time integration method used here. Figures 3.17 and 3.18 show the effect of shell radius on the time history of the central deflection and the impact force. It is observed from these graphs that the maximum impact force will increase and the central deflection will decrease with increasing curvature of the stiffened shell panel. This stems from the fact that decreasing the shell radius and thus increasing the curvature has a stiffening effect on the shell panel due to membrane actions. This increases the natural frequency and thus reduces the impact duration. The calculated fundamental frequencies of the stiffened shells for various radii of curvature are listed in Table 3.7. In Figure 3.17, the deflection time history of the shallow shell with radius R = 5a is very similar to that of the flat plate. The maximum deflection of the plate and shallow shell (R = 5a) shown in Figure 3.17 is greater than the skin thickness. Therefore large deflection analysis is carried out for these problems and the results are shown in Figure 3.19. It can be seen that the membrane Chapter Three: Nonpenetrating Impact Response deformations accounted for in large deflection analysis will make the panel act stiffer and thus decreases the magnitude of the central deflection. 3.4.2.2 Effect of Impact Site Consider a plate with three stiffeners as shown in Figure 3.20. The material properties and geometric sizes of the plate and stiffeners are the same as in Table 3.6 and Figure 3.15 except that there is an intermediate stiffener added to the plate along the centre line. The projectile mass and velocity are the same as in the above examples. Two typical impact sites, midbay and middle of the central stiffener, are studied using small deflection analysis. The time histories of transverse deflections at the impact point of the stiffened plates are shown in Figure 3.20 and compared with the deflection history for the case of a plate with two-stiffeners. Table 3.8 predicts a comparison between the maximum impact force in these three cases. It can be seen that in the case where the impact is on the stiffener, maximum impact force is higher and transverse deflections are lower (see Figure 3.20). Transverse deflections are highest when the projectile impacts the two-stiffener plate at the midbay. The time histories of the impact force on the two-stiffener and three-stiffener plates are shown in Figure 3.21. 3.5 Summary In this chapter, a numerical analysis procedure using super finite elements has been presented for simulating the nonpenetrating impact response of stiffened and plain laminated composite 48 Chapter Three: Nonpenetrating Impact Response plates and cylindrical shells. The non-linearityfromlarge deflections is taken into account in the analysis as well. The present formulation has been verified by comparing with existing experimental data and other theoretical solutions in the literature. Good agreement between the predicted and measured results shows that the present numerical technique based upon coarse finite element mesh is an efficient way to compute nonpenetrating impact response of composite structures. The use of artificial damping in the time integration scheme to filter out the high-frequency oscillations in impact force time history shows a clear picture of dominant vibration mode in impact response. The success of the present numerical analysis gives us confidence to handle nonpenetrating impact response of composite structures, and furthermore provides a robust framework for computing the structural response under penetrating impact events. 49 Chapter Table 3.1 E„ E 22 v, 2 Response s 5.5 GPa 2.69 mm 1580 kg/m 12 3 Fundamental frequencies of the laminate defined in Table 3.1 obtained from different mesh geometries. Fundamental Frequencies (Hz) 303.116 303.106 303.083 303.082 303.082 302.692 Properties of the laminated plate used in instrumented impact tests. Boundary Condition Size Material System E„ E 22 Vl2 Table 3.4 Impact Simply-Supported 0.2m x 0.2m x 0.00269m [0/90/0/90/0] T300/934 CFRP 120 Gpa G 7.9 Gpa h 0.33 P Super-Finite Element Mesh lxl for the whole plate lxl for a quarter plate 2x2 for a quarter plate 4x4 for a quarter plate 5x5 for a quarter plate Analytical solution Table 3.3 Nonpenetrating Properties of the laminated plate used for convergence studies. Boundary Condition Size Material System Table 3.2 Three: Simply-Supported 0.127m x 0.0762m x 0.00465m [45/90/-45/0] T800H/3900-2 CFRP 129 GPa G 7.5 GPa h 0.33 P 3S 3.5 GPa 4.65 mm 1540 kg/m 12 3 Properties of the cylindrical shell used in the analysis of Christoforou and Swanson (1990). Target Material System Size Projectile Parameters Target Material Constants [90/±22] IM7/55A CFRP Z=41.9cm, i?=4.91 cm, A=1.52mm MassOT=0.032kg, Impact velocity V =3.0 m/s E =169GPa v =0.266 E =6.72 GPa p=1620 kg/m G =5.26 GPa 2T 0 n 12 3 22 12 50 Chapter Table 3.5 Size Material System E E n 22 V12 Clamped - Clamped in width Free -Free in length 0.252m x 0.089m x 0.001608m [±45 /0 ] AS4/3501-6 Graphite/Epoxy 142.0 GPa G 9.81 GPa h 0.30 P 2 2 Response s 6.0GPa 1.608 mm 1540 kg/m ) 2 3 E, = 120 GPa, E = 7.65 GPa, G = 3.44 GPa, v = 0.363, .p= 1540 Kg/m skin [45/90/-45/0/45/90/-45/0/90/0/90/0] stiffener [22.5/90/-22.5/0 ] shape of cross-section: T web spacing: 178.0 mm web thickness: 6.02 mm web height: 28.40 mm flange thickness: 3.0 mm flange width: 72.06 mm 2 1 2 3 12 s 2 Stiffener size 2s Parametric studies: fundamental frequency and maximum impact force of the stiffened laminated cylindrical shells. R = infinity R/arc length = 5 R/arc length = 1 Table 3.8 Impact Material properties and lay-up of the stiffened plate shown in Figure 3.15. Material properties Lay-up Table 3.7 Nonpenetrating Properties of the laminated plate used in Matsuhashi et al. (1993). Boundary Condition Table 3.6 Three: Fundamental frequency (Hz) 274 408 1250 Maximum impact force (KN) 5.77 10.8 18.2 Maximum impact force for a stiffened plate impacted at different sites. Impact sites Midbay of two-stiffener plate Midbay of three-stiffener plate Stiffener centre of three-stiffener plate Maximum impact force (KN) 5.77 10.6 21.7 51 Chapter Three: Nonpenetrating Impact Response 700 1x1 for a quarter plate 4 x 4 a n d 5 x 5 for a quarter plate 2x2 for a quarter plate 0.05 0.1 0.15 0.2 0.25 Time (ms) Figure 3.1 Convergence of the transient response of the laminated plate defined in Table 3.1 subjected to a 8.54 g, 3.0 m/s central impact. 400 4x4 for a quarter plate 300 F E M (Sun & Chen) O lm £ 200 o ra a Modal analysis E 100 r- 0.05 0.1 0.15 0.2 0.25 Time (ms) Figure 3.2 Impact force history of the laminated plate defined in Table 3.1; present analysis compared with modal analysis of Pierson and Vaziri (1996) and finite element analysis of Sun and Chen (1985). 52 Chapter Three: Nonpenetrating Impact Response 0.5 Time (ms) Figure 3.3 Central deflection history of the laminated plate defined in Table 3.1; present analysis compared with modal analysis of Pierson and Vaziri (1996) and finite element analysis of Sun and Chen (1985). Figure 3.4 Impact force history for 6.14 kg, 1.76 m/s impact on the laminate defined in Table 3.3; present model compared with drop-weight test results (Delfosse et al, 1993b). 53 Chapter Three: Nonpenetrating Impact Response 16 Present j m =6.14 kg v=2.68 m/s 12 \ 0 * / Q) O o u. f o re i 1 CL f E / J # M 1 V-/ \ Measured \ \ 1 % \ 1 * \ * * / 1 1t 1 \ * \ \ • 1 • 0.9 1.8 2.7 i \ \ % * 3.6 4.5 Time (ms) Figure 3.5 Impact force history for 6.14 kg, 2.68 m/s impact on the laminate defined in Table 3.3; present model compared with drop-weight test results (Delfosse et al, 1993b). 10 m=0.314 kg v=7.70 m/s Present Measured 0.2 0.4 0.6 0.8 Time (ms) Figure 3.6 Impact force history for 0.314 kg, 7.70 m/s impact on the laminate defined in Table 3.3; present model compared with gas-gun test results (Delfosse et al, 1993b). 54 Chapter Three: Nonpenetrating Impact Response 16 Time (ms) Figure 3.7 Impact force history for 0.314 kg, 11.85 m/s impact on the laminate defined in Table 3.3; present model compared with gas-gun test results (Delfosse et al, 1993b). Projectile Displacement (mm) Figure 3.8 Parametric studies: Predicted impact force-displacement curves for impact events with the same incident energy but with different striking velocities on the laminate defined in Table 3.3. Also see Figure 3.9 for a description of the force-displacement curve with a striking velocity of 59.25 m/s. 55 Chapter Three: Nonpenetrating Impact Response Target Velocity Projectile Velocity 0.12 0.08 Velocity of Indentation Depth; 0.095ms 0.182ms Contact Time (ms) Figure 3.9 Predicted time histories of the projectile velocity, central target velocity, and velocity of indentation depth for the impact event shown in Figure 3.8 with a striking velocity of 59.25 m/s. 0.8 Impact Energy : 22 J T800H/3900-2, [45/90/-45/0]3s 0.127 x 0.0762 x 0.00465 m 11.85 m/s 26.50 m/s 59.25 m/s 132.5 m/s 296.3 m/s Time (ms) Figure 3.10 Parametric studies: Time histories of central deflection of the target for impact events with the same incident energy but with different striking velocities on the laminate defined in Table 3.3. 56 Chapter 0.7 » c « —— Impact Response - - - 5.30 m/s 59.25 m/s 296.3 m/s S S 0.5 | t Nonpenetrating Impact Energy : 22 J T800H/3900-2, [45/90/-45/0]3s 0.127 x 0.0762 x 0.00465 m 0.6 .x Three: s s s —» - s * 0.4 I 0.3 u % 0.2 Q 0.1 s • 0 1 0.01 sV 1 1 1 : 0.02 0.03 0.04 0.05 *> ^ 0.06 Distance From the Center (m) Figure 3.11 Profiles of central deflections under impacts with the same amount of energy but in different velocities on the laminate defined in Table 3.3. 57 Chapter Three: Nonpenetrating Impact Response z, w Time (ms) (c) Figure 3.12 Vertical deflection history at the impact point for the Christoforou and Swanson's cylinder impact problem defined in Table 3.4. (a). Circular cylinder under an impact at the center, (b). Finite element mesh for a quarter cylinder used in present analysis, (c) Present predictions compared with Christoforou and Swanson's analytical results. 58 Chapter Three: Nonpenetrating Impact Response Figure 3.13 Central deflection history for a 8.54 g, 30.0 m/s impact on a simply-supported [0/45/0/45/0] CFRP rectangular plate with the layer properties defined in Table 3.1; present results obtained from a small and large deflection analysis compared with the corresponding results of Chen and Sun (1985). 2S 20 Small Deflection Experimental Time (ms) Figure 3.14 Central deflection history for a 1.53 kg, 3.0 m/s impact on a clamped-free [±45 /0 ] CFRP plate with the layer properties defined in Table 3.5; present small and large deflection analysis results compared with experimental data in Matsuhashi et al. (1993). 2 2 2S 59 Chapter Figure 3.15 Three: Nonpenetrating Impact Response Super finite element mesh in nonpenetrating impact analysis of laminated composite stiffened plates (Dost et al, 1991). Material properties and lay-up are shown in Table 3.6. 1.6 0.01 Contact Time (sec) Figure 3.16 The comparison of the predicted contact force time history with the measured data from instrumented impact tests in Dost et al. (1991). The material properties and geometric sizes are shown in Figure 3.15 and Table 3.6. 60 Chapter Three: Nonpenetrating Impact Response 1.6 Time (ms) Figure 3.17 Parametric studies: The time history of the central deflection at the impact point for stiffened cylindrical shells with various radii of curvature. The material properties and geometric sizes are shown in Figure 3.15 and Table 3.6 with arc length b = 0.508 m. 20 , 0 , 0.5 1 1.5 2 2.5 3 3.5 Time (ms) Figure 3.18 Parametric studies: The time history of the impact force for stiffened cylindrical shells with various radii of curvature. The material properties and geometric sizes are shown in Figure 3.15 and Table 3.6 with arc length b = 0.508 m. 61 Chapter Three: Nonpenetrating Impact Response 1.6 Time (ms) Figure 3.19 Parametric studies: The time history of predicted non-linear central deflection for stiffened cylindrical shells with various radii of curvature. The material properties and geometric sizes are shown in Figure 3.15 and Table 3.6 with arc lengths = 0.508 m. 1.2 Time (ms) Figure 3.20 Parametric studies: Comparison of deflection at different impact sites. The material properties and geometric sizes are shown in Figure 3.15 and Table 3.6. 62 Chapter Three: Nonpenetrating Impact Response 25 Time (ms) Figure 3.21 Parametric studies: Comparison of the impact force at different impact sites. The material properties and geometric sizes are shown in Figure 3.15 and Table 3.6. 63 Chapter Four: Literature Review on Penetration Mechanisms and Modelling Chapter Four: Literature Review on Penetration Mechanisms and Modelling There are numerous papers in the literature addressing penetration problems of metallic materials. These have been carefully reviewed by Johnson (1972), Backman and Goldsmith (1978), and Corbett et al. (1996). The work on modelling the penetration behaviour of laminated composite materials started in the last decade. Some of the work have been reviewed by Abrate (1991 and 1994). The present review will concentrate on the major penetration mechanisms and damage patterns found in penetration tests on metallic and composite materials using an indenter (or projectile). The transitions between these penetration mechanisms with changes in geometric sizes of projectiles and targets will also be discussed. The penetration models developed for laminated composite materials are reviewed as well. Ballistic limit is the critical impact velocity required by the projectile to just perforate (or penetrate through) the target. It is treated as a kind of material property by engineers in their assessment of ballistic performance of materials. The major task of the next four chapters (Chapter Four to Seven) is to predict ballistic limit of a target and evaluate the effects from geometric sizes of a projectile and a target. 64 Chapter Four: Literature Review on Penetration Mechanisms and Modelling 4.1 Penetration Mechanisms The major penetration mechanisms that have been observed in penetration tests of metallic and composite materials by a conical indenter (or projectile) are hole expansion, dishing/petalling, transverse plugging, and delaminations. They are reviewed separately in the following sections. 4.1.1 Hole Expansion 4.1.1.1 Metals When metallic materials are penetrated by conical projectiles, the projectiles push the material to the side, causing the material to yield and flow plastically. This kind of deformation is called "hole expansion" and was first tackled by Taylor (1948) and Hill (1950). In their models, the hole expansion problems were treated as plastic expansion of a circular hole in radial direction under plane stress condition. The material constitutive relation was assumed to be rigid-plastic, and the resulting stress and strain fields around the expansion hole were obtained by solving the differential equations of equilibrium. The total plastic work done in expanding the hole from zero radius to the projectile or indenter radius is equal to the product of target material yield stress CT and the material volume displaced by y the projectile. The plate thickening around the hole, caused by plastic flow and incompressibility of the material, was also taken into account in the calculation of plastic work by Taylor (1948) and Hill (1950). Chapter Four: Literature Review on Penetration Mechanisms and Modelling After measuring the ballistic limits for different thicknesses of metallic plates, Woodward (1978) concluded that the plastic work done in hole expansion was a lower bound on energy absorption provided that the target yield stress value was carefully chosen. The flow stress (i.e. yield stress) defined in Woodward (1978) was the stress at a natural compressive strain of 1.0 in static indentation tests on that material. The natural compressive strain was the natural logarithm of the instantaneous thickness of material between the indenter tip and rigid support divided by the original plate thickness. Woodward also pointed out that when the ratio of projectile diameter to target thickness decreased, the plate thickening around the hole would not be as severe as that predicted by Taylor (1948). In this case, because of a relatively large value of target thickness compared to projectile diameter, a transition was occurring from the plane stress state assumed in Taylor (1948) to the state approaching the plane strain deformation where no plate thickening occurred. 4.1.1.2 Composites Based upon cross-sectional examinations of perforated GFRP laminates (E-glass woven rovings in polyester resin) with a thickness of 15.0 mm, Greaves (1992) indicated that the major penetration mechanism, which was defined as absorbing most of the impact energy, was "compression and displacement of the plate material" by the fragment simulating projectile (FSP). Although Greaves (1992) didn't use the term "hole expansion" to describe this kind of deformation state, it did have the same characteristics of deformation pattern as hole expansion. Chapter Four: Literature Review on Penetration Mechanisms and Modelling In their subsequent penetration tests of GFRP laminates (E-glass/polyster and S2glass/phenolic), Howlett and Greaves (1995) found that the initial part of the measured forcedisplacement curves had a parabolic increase in load with indenter displacement. The curves then levelled off and there was no further increase in load once the indenter shaft started to penetrate into the GFRP laminates. The parabolic increase of load meant that the penetration mechanism in the plate material was hole expansion (see Section 5.2.2). The indenters used in their tests had diameters of 7.62 mm and 12.7 mm, and an included cone angle of 90°. The hole expansion mechanism was also used by Zhu et al. (1992b) as the deformation pattern in the first stage of penetration when modelling the ballistic behaviour of woven kevlar composites impacted by a conical projectile with a cone angle of 60°. 4.1.2 Dishing/Petalling 4.1.2.1 Metals After examining the cross-sections of perforated metallic plates with different thicknesses, Taylor (1948) found that hole expansion deformation was only valid when the projectile diameter was in the same order of or greater than the target thickness. For very thin plates, {e.g. plate thickness to projectile diameter ratio less than 0.1), the plates were always bent out of their planes. This type of deformation was called "dishing" by Thomson (1955). The energy absorption during the dishing type of deformation was calculated in Taylor (1948) and Thomson (1955) to be about one quarter of the impact energy absorbed in hole expansion. 67 Chapter Four: Literature Review on Penetration Mechanisms and Modelling When the dishing area around the hole continues to deform, the circumferential strain will exceed the material ductile limit and the dished material will break and form into several petals. This mechanism was called "petalling" by Zaid and Paul (1957). Johnson et al. (1973) conducted a thorough series of penetration tests on metallic plates using conical indenters with included cone angles of 30°, 45°, 60°, 90°, 150°, and 180°. It was shown that the peak contact force increased with the indenter cone angle, while the total indenter movement for a complete perforation decreased. The contact force increased with the indenter movement first and then dropped when the indenter shaft entered into the plates. Johnson et al. (1973) proved in their analysis that the dishing or petalling deformation would not occur when the half cone angle /? >tan"'(l/ //), where JU is the coefficient of friction 0 between the indenter and the dishing (or petalling) area in the plates. The direction of the plate material movement in hole expansion differs from dishing/petalling. In hole expansion, the plate material is assumed to conform to the conical head shape of projectile and be pushed aside. However, in dishing/petalling mechanism, the plate material does not necessarily conform to the conical shape of projectile head and deforms in stretching or out of plane bending. A typical model for petalling deformation was developed by Landkof and Goldsmith (1985), where the crater deformation was determined by bending of the petals with plastic hinge on one end, instead of conforming to the configuration of projectile head. When the plate thickness decreases, the penetration mechanism changes from hole expansion to dishing/petalling mode. For example, based upon calculated energy absorption during 68 Chapter Four: Literature Review on Penetration Mechanisms and Modelling perforation, Woodward (1978a) showed the transition of energetically favourable penetration mechanism from dishing to hole expansion when the plate thickness increased up to the projectile diameter. This prediction was also verified in his ballistic tests. Dishing/petalling can also be attained from transverse plugging. Woodward and Cimpoeru (1997) observed the transition of penetration mechanisms from transverse plugging to dishing on the distal side of laminated aluminium plates in their ballistic tests, where bluntended and 90° conical projectiles were used. Thick homogeneous plates plugged transversely whereas laminates with the same thickness were indented on the impact side and deformed by a dishing mechanism on the exit side. Thinner laminates favoured more extensive dishing, which mainly involved energy absorption by stretching and bending. 4.1.2.2 Composites Petalling on the distal side of thin laminates was also observed in ballistic tests of woven carbon fibre laminates in Goldsmith et al. (1995) where the ratio of the projectile diameter to target thickness was about 5 and the projectile included cone angle was 60°. 4.1.3 Transverse Plugging 4.1.3.1 Metals When metallic plates are impacted by blunt-ended projectiles, the target material is geometrically constrained to move forward instead of laterally as impacted by conical projectiles. Thus a shear failure (ductile plugging) occurs. This has been shown clearly in 69 Chapter Four: Literature Review on Penetration Mechanisms and Modelling the ballistic tests by Woodward (1978b), Liss and Goldsmith (1984), and Crouch and Woodward (1988). In the instrumented impact tests of thin metallic plates by blunt projectiles, Langseth and Larsen (1990) found that a transverse plug always occurred at maximum impact force and a sudden drop of the force happened after occurrence of the plug. In some cases, depending on the geometric sizes of projectiles and targets, transverse plugging also occurs when metallic plates are impacted by conical projectiles. For example, Recht et al. (1963) indicated that a transverse shearing plug was formed in the plates if the plate thickness h was less than both the projectile radius R and one half of the projectile head p length L , i.e., h<R c p and h<^-, provided that the projectile material was relatively strong and did not deform excessively. In ballistic tests by Woodward (1978b), transverse plugging generally took place in the plates when the projectile cone angle approached 90°. Johnson et al. (1973) conducted static penetration tests on metallic plates by a series of conical indenters. According to their tests and analysis, transverse plugging occurred when the included cone angle 2/? was greater than 90° no matter what friction coefficient LI was 0 applied. Plugging always occurred if the values of 2{3 and ju fell into the area above line AB 0 shown in Figure 4.1 and dishing or petalling for values below line CD. In the region between AB and CD, predicted behaviour was obscure and depended on the precise values of 2/3 and 0 ju. The ratio of the indenter shaft diameter to the plate thickness was about 4 in their studies, which meant that the plates were still relatively thin. Chapter Four: Literature Review on Penetration Mechanisms and Modelling Similar trends were also seen by Woodward (1984) in his qualitative study based upon observations from many ballistic tests. 4.1.3.2 Composites In penetration tests of graphite/epoxy laminates by blunt-ended projectiles, Lee and Sun (1993a) found that transverse plugging was a major damage pattern during penetration of composites. Ursenbach et al. (1995) also observed the same phenomenon in their penetration tests of T300H/F593 CFRP laminates. Similar to metals, for conical-tipped projectiles, transverse plugging also occurs when the projectile cone angle is large. For example, Zhu et al. (1992a) found transverse plugging on the distal side of a 6.35 mm thick woven Kevlar laminate when penetrated by a 120° cone. 4.1.4 Delamination Malvern et al. (1989) conducted a series of nonpenetrating impact tests on different kinds of laminated composites. They found that delamination was a major damage mechanism in the laminates. The total delamination area was found to be a linear function of the energy absorbed by the plates. When the impact velocity was increased beyond the ballistic limit, the relationship between the delamination area and the initial impact energy was still linear but with a much smaller slope as shown in Wu and Chang (1995). Chapter Four: Literature Review on Penetration Mechanisms and Modelling Greaves (1992) observed delamination in 15 mm thick E-glass composites which was as large as 160 mm in diameter on the distal surface of the plate, i.e., about 10 times the plate thickness and 20 times the projectile diameter. Although delamination may not dissipate a major amount of impact energy as shown in Zhu et al. (1992a), it will greatly affect the local deformation and subsequent damage patterns. For example, in dynamic penetration tests of laminated aluminium plates conducted by Woodward et al. (1997), the penetration mechanism changed from transverse plugging to dishing/petalling due to delaminations in the plates. In the penetration tests of T300H/F593 CFRP laminates conducted by Ursenbach et al. (1995), delamination increased the local deformation and greatly reduced the flexural stiffness of the plate. 4.2 Transition of Penetration Mechanisms Woodward (1984) also observed dishing deformation instead of hole expansion in thick plates when the plate material had a very low toughness in the through-thickness direction. It was found that the target material could bulge rearward (similar to dishing of thin plates) instead of being pushed aside when the projectile penetrated to a certain depth in the target. Goldsmith and Finnegan (1986) performed a series of dynamic penetration tests on metallic plates by conical (60°) and blunt (180°) projectiles. The plate thicknesses ranged from 1.78 mm to 25.4 mm. The penetration mechanisms such as dishing, petalling, plugging, and shear banding were identified from experimental observations and metallurgical tests. The transition of these penetration mechanisms were also noted, which meant that both the target 72 Chapter Four: Literature Review on Penetration Mechanisms and Modelling thickness and the projectile cone angle played an important role in the transitions from one penetration mechanism to another. Based upon energy analysis, Woodward (1984) proposed the following transitions of penetration mechanisms for conical projectiles: • When the thickness of the plate still to be penetrated, h, is less than the projectile diameter 2R , h<2R , dishing instead of hole expansion will become the favourable penetration p p mechanism for metallic materials with low toughness in the through thickness direction. • When the thickness of the plate still to be penetrated, h, is less than V3 / 2 times the projectile diameter, h < *j3R , ductile plug formation and ejection instead of hole p expansion will become the favourable penetration mechanism for metallic materials with high strength, low work-hardening rate, and high thermal softening rate. 4.3 Penetration Modelling of Composites In this section, the available penetration models for composites are reviewed. However, before discussing the models it is instructive to highlight the similarities between static and dynamic penetration patterns as obtained from experiments. 73 Chapter Four: Literature Review on Penetration Mechanisms and Modelling 4.3.1 Static vs. Dynamic Tests Experimental study conducted by Lee and Sun (1993b) indicated that in static penetration tests of carbon/epoxy laminates the relationship between the contact force and indenter displacement showed approximately the same trends for various laminate planar sizes. It was inferred from the indenter force-displacement curves that the load at the onset of damage and the ultimate load were about the same. Impact test results also showed that the dynamic failure modes were very similar to those obtained under static loading. This close agreement in damage patterns indicated the possibility of using static penetration tests to characterise the dynamic penetration process. Similar conclusions were also drawn by Zhu et al. (1992a), Lee and Sun (1993a), Sun and Potti (1996), Jackson (1996), and Potti and Sun (1997). Sanders (1997) and Delfosse (1994b) examined the micrographs of cross-sections of IM7/8551-7 CFRP laminates in static penetration and dynamic penetration tests. They are shown in Figure 4.2, where the damage area and pattern are very similar in both tests. Virostek et al. (1987) measured the time history of impact force when projectiles with conical or hemispherical head shapes impacted thin plates at different angles of incidence. Different plate materials were studied, ranging from aluminium, steel, lexan, nylon, ceramic, and Kevlar. It was found that the peak impact force obtained for a given plate material under normal incidence was relatively independent of the initial projectile velocity for impacts where perforation occurred (i.e., above ballistic limit). For experiments at velocities below the ballistic limit, the peak impact force was found to be approximately proportional to the initial impact velocity. 74 Chapter Four: Literature Review on Penetration Mechanisms and Modelling Based upon the above work, it appears that static penetration tests can be used to identify the damage patterns and penetration mechanisms of some laminated composites when strain-rate effect is neglected. The closeness of peak impact force for a certain target which completely perforated by projectiles in different velocities (Virostek et al, 1987) means that their forcedisplacement curves have similar shapes, because for a certain projectile and target the projectile displacement span is almost the same (sum of projectile length plus target thickness and its permanent deflection). This was verified by the measurements in ballistic tests of Sanders (1997) as well. Thus the damage modes and deformation patterns can be assumed to be independent of impact velocity when perforation occurs. 4.3.2 Analytical Models for Projectiles with Spherical Head Shapes Cantwell and Morton (1990) developed a simple analytical model for perforation of CFRP laminates by a spherical projectile. Using balancing of energy, the projectile kinetic energy was dissipated into the laminate elastic flexure, indentation, delamination, and transverse plugging, represented by Ef, E , E& and £ , respectively. Ef, the maximum strain energy c p absorbed by the laminates subjected to transverse loading, was determined by classical laminate theory in conjunction with a maximum strain failure criterion. E , the energy c dissipated in indenting the laminates, was determined by a Hertzian contact law. E&, the energy dissipated in delamination, was determined by measuring the area of delamination after the tests and multiplying this area by the fracture energy of delamination. E , the p energy dissipated in plugging of the plates, was determined by multiplying the surface area of fracture zone by the transverse fracture energy of the composite. This was a very simple 75 Chapter Four: Literature Review on Penetration Mechanisms and Modelling model and the predicted ballistic limits were in good agreement with experimental results for thin laminated plates. However, details of the impact event such as the impact force cannot be obtained using this kind of approach. In most cases, the peak impact force governs the damage mode and progression and is thus another important factor that needs to be considered in addition to the energy absorbed by the target. Cantwell (1988) studied the perforation events in CFRP beams resting on steel supports with different opening sizes and shapes. Test data showed that changing the opening size did not influence the ballistic response of the targets and its energy absorption capability, due to the nature of localised impact loading. However, changing the target thickness could significantly affect the impact energy required to perforate the target. The penetrating impact response of woven E-glass/epoxy composite laminates by a hemispherically-tipped projectile was studied in Wu and Chang (1995). The impact velocity ranged from well below the ballistic limit of the target material to twice its ballistic limit. The relationships between the peak impact force and the extent of damage (delamination area) versus the initial impact energy of the projectile were studied. The details on the experimental set-up and procedures were described in Wu et al. (1994). 4.3.3 Analytical Models for Blunt-Ended Projectiles Lee and Sun (1993a & b) developed an analytical model for the penetration of graphite/epoxy laminates impacted by a blunt projectile. Delaminations and transverse plugging were 76 Chapter Four: Literature Review on Penetration Mechanisms and Modelling identified as the two major mechanisms during penetration. Since the damage patterns in dynamic penetration resembled those observed in static penetration tests, the model used results from the static tests to characterise the dynamic penetration. Finite element analyses were conducted to obtain the detailed distribution of stress components and predict the initiation of delamination in the laminates. The force for plug initiation was a material property determined from the static penetration tests. The validity of the model prediction was restricted to limited laminate thicknesses. In an attempt to improve their previous model, Sun and Potti (1996), and Potti and Sun (1997) developed a ring model to predict damage progression and energy absorption for thicker laminates. A force-displacement curve from quasi-static penetration tests by a blunt indenter also served as the "structural constitutive model" which captured the highly nonlinear behaviour of the laminate during dynamic penetration events. The transverse shear modulus was degraded to model the progressive reduction of plate stiffness appearing in static penetration curves. Four different kinds of initiation criteria for transverse plugging were suggested. They were based on contact force, shear force, plate shear strain, and plate deflection. In the latter, the differential plate deflection between the periphery of the indenter and the boundary of local area was used as an initiation criterion. The local area was defined in the quasi-static penetration tests as a circular area with a diameter three times that of the projectile diameter. In order to model the behaviour of a partially delaminated plate, Ursenbach et al. (1995) assumed an embedded isotropic plate with reduced stiffness to represent the region of delamination. The deformation behaviour of the laminates in the post-delamination stage Chapter Four: Literature Review on Penetration Mechanisms and Modelling was described using the experimental measurements of strains and displacements at the plate centre. The model was capable of estimating the delamination size and the effective bending modulus of the delaminated zone. The predicted delamination sizes for various laminate thicknesses were successfully correlated with those measured using C-scans. For GFRP composite laminates, a simple analytical model based upon one-dimensional stress wave theory was developed by Howlett and Greaves (1995) to predict the ballistic limit of the targets for blunt-ended projectiles. The authors also conducted static penetration tests on GFRP laminates using conical-tipped indenters. 4.3.4 Analytical Models for Projectiles with Conical Head Shapes Zhu et al. (1992b) developed an analytical model for the penetration of Kevlar 29/polyester laminated plates by a conical projectile with 60° included cone angle. The penetration process was divided into three successive and non-interactive stages: indentation, perforation, and projectile exit. The analysis consisted of both global and local deformation, where the former employed linear-elastic laminate plate theory, while the latter consisted of the indention (hole expansion) on the frontal surface of the target, bulging at the distal surface, fibre breakage, and delamination. Friction, neglected in the first two stages, was the only force acting during the exit phase. Resistance to projectile penetration was used in conjunction with Newton's law to determine the projectile and structural response and a maximum strain criterion was applied in the penetration process to reduce the material strength due to the impact damage. A comparison of predicted ballistic limits with test measurements showed good agreement. In their tests, the ballistic limit was found to vary 78 Chapter Four: Literature Review linearly with the laminate thickness. on Penetration Mechanisms and Modelling The lay-up of the laminates and prior partial delaminations in the laminates did not play a significant role in the energy absorption capability of the laminates. For thin, woven, carbon fibre/epoxy laminates impacted by 60° conical projectiles, Goldsmith et al. (1995) developed a penetration model based upon energy balance, taking account of energy absorption due to global plate deflection, fibre breakage, delamination, formation and bending of petals, and friction between the projectile and the plate. The maximum delamination size was found to be as large as four times the projectile diameter. A simplifying model based upon one-dimensional wave propagation theory for impacts on fibre-reinforced polymer with fibre architectures of unidirectional fibres and woven fabric composites was developed by Navarro et al. (1994). In this model, the kinetic energy of the projectile was assumed to be absorbed in the form of elastic and kinetic energy in the fibres directly in contact with the projectile. The interaction effects between warp and filling fibres at crossovers were considered using an approximate solution. The predictions by the model agreed with the measured projectile displacement and velocity drop. However, the predicted impact force was far from the experimental measurements. Pierson et al. (1993) applied the penetration models in Awerbuch and Bodner (1974) and Zhu et al. (1992b) to CFRP laminates and compared the predictions with measurements from ballistic tests. They pointed out that because of the inherent differences between the metallic and composite materials, models initially developed for metals could not capture the 79 Chapter Four: Literature Review on Penetration Mechanisms and Modelling evolution, sequence, and magnitude of the damage growth if such models were directly applied to composite materials. 4.3.5 Analytical Model for FSP (Fragment Simulating Projectiles) In the ballistic studies of S2-glass laminates conducted by Bless and Hartman (1989), the post-test examination of the perforated target cross-sections revealed that there was a region that had experienced compressive failure and expulsion of cut fibres, followed by a region which appears to be dominated by tensile failure and parting of the cut fibres. The former was in the frontal part of the target, considered by the authors as "compressive flow around the penetrator" and yielded the same equation calculating the force as the hole expansion mechanism in present approach (see section 5.2.2). The latter mechanism was in the distal side of the target, defined as "shear failure around the periphery of the penetration cavity", which is the transverse plugging mechanism. The effective compressive strength and effective transverse strength of the laminates were proposed by the authors to measure the penetration resistance of the laminates. Increasing the fracture strength of the resin matrix resulted in increase in the ballistic limit of the laminated composites. The projectiles used in their tests were FSP (fragment simulating projectiles), which have a chisel shape nose. The ballistic limit was found to be proportional to the square root of the transverse compressive strength of the laminates. 80 Chapter Four: Literature Review on Penetration Mechanisms and Modelling 4.3.6 Numerical Modelling Because experimental methods for dynamic perforation problems can become very expensive when numerous variables are involved in the design of laminated composite structures, numerical analysis has been found to be extremely useful in interpreting experimental results and providing a tool for parametric studies. A number of existing computer programs suitable for simulating penetration problems have been carefully reviewed in Vaziri et al. (1989). The most commonly used codes are DYNA2D, DYNA3D, AUTODYN, and PAMCRASH. DYNA3D, is an explicit threedimensional finite element code originally developed at Lawrence Livermore National Laboratory, and subsequently enhanced and marketed by Livermore Software Technology Corporation (LSTC). Many material models are already built in and users can also import their own material models to DYNA3D for penetration analysis. Currently implemented damage models in DYNA for composite materials are the Chang and Chang (1987) model (see Blanas, 1991) and Matzenmiller et al. (1991) model (see Williams and Vaziri, 1995). Lu and Vaziri (1994) provided an extensive review of constitutive and failure models for numerical analysis of impact response of composites using DYNA2D and DYNA3D. Williams (1998) formulated a physically-based damage model based on continuum damage mechanics and implemented it into LS-DYNA3D. This model takes account of stiffness reduction and strain softening in composite laminates. Williams (1998) used his model to simulate the response of thin CFRP laminates impacted by a hemispherical projectile. 81 Chapter Four: Literature Review on Penetration Mechanisms and Modelling 4.4 Summary From the review conducted here, it appears that there has been some limited work on modelling the penetration process in laminated composites. These models provide good estimates of the energy absorption and ballistic limits of composites for certain material systems and certain ranges of geometric sizes of projectiles and targets. However, little or no work has been reported in the literature on prediction of impact force-time history or forceprojectile displacement relationship, which are very important when predicting damage initiation, damage progression, and transient response of laminated structures. Also, the effects of projectile cone angle on ballistic performance of composite materials have not been reported yet. The present work aims at fulfilling these gaps. A penetration model will be developed in Chapter Five and Six. This model, which is driven by experimental observations and engineering mechanisms will be used in conjunction with the finite element model developed earlier to simulate the ballistic response of various laminates in Chapter Seven. 82 Chapter Figure 4.1 Four: Literature Review on Penetration Mechanisms and Modelling Schematic illustration of transition of penetration mechanisms in static penetration tests of thin metallic plates by conical indenters with different cone angles (Johnson et al., 1973). 83 Chapter Four: Literature Review on Penetration Mechanisms and Modelling (a) Static Penetration Test (b) Dynamic Penetration Test Figure 4.2 Micrographs of cross-sections of IM7/8551-7 CFRP laminates in (a) static penetration tests and (b) in dynamic penetration tests (Delfosse, 1994b and Sanders, 1997). 84 Chapter Five: Penetration Mechanisms Chapter Five: Penetration Mechanisms Guided by the literature review in the previous chapter and the supporting experimental work carried out at UBC, several penetration mechanisms are studied in this chapter. These mechanisms include elastic deformation, hole expansion, delamination and dishing/petalling, transverse plugging, and friction. Before examining these mechanisms, the experimental penetration results of Carbon Fibre-Reinforced Polymer (CFRP) and Glass Fibre-Reinforced Polymer (GFRP) are summarised first. 5.1 Experimental Investigations Static penetration tests and indentation tests are two types of experiments used to investigate penetration mechanisms of composite materials (Bless et al. 1989, Lee et al. 1993b, Zhu et al. 1992a, and Howlett et al. 1995). These tests were conducted at UBC by Delfosse (1994b) and Sanders (1997) on CFRP and GFRP laminates. They provide physical insight into penetration mechanisms outlined in this chapter and the penetration model developed in the next chapter. 5.1.1 Static Penetration Tests Static penetration tests shown in Figure 5.1 are performed on an Instron Universal Testing Machine for CFRP and GFRP laminated plates. The indenter size and material properties are 85 Chapter listed in Table 5.1. Five: Penetration Mechanisms The geometric sizes and material properties of CFRP and GFRP laminates are summarised in Table 5.2 and Table 5.3, respectively. The static penetration tests are conducted under displacement control with the penetration force being measured. A video camera and microphone are also used to record the penetration process as well. Two types of opening sizes for rigid (steel) support plates are used in the tests. One is called a large-geometry opening, which is rectangular with dimensions of 127 mm x 76.2 mm. The other is called a small-geometry opening, which is circular with a diameter of 25.4 mm. The tests on the small-geometry opening are treated as material characterisation tests in the present work. This is because the small specimen minimises the global bending of the laminate and hence the response is primarily material rather than structural. The laminated specimens with a rectangular size of 152.4 mm x 101.6 mm are used for the large-geometry testing, while the specimens with a square size of 50.8 mm x 50.8 mm are used in the small-geometry tests. The specimens are categorised in three groups (Delfosse 1994b and Sanders 1997). The first group of specimens is used to get the whole indenter force-displacement curves up to the complete perforation of the laminates. The second group is also tested up to the complete perforation but during the tests it is loaded, unloaded, and reloaded at various points on the force-displacement curve in order to check the visible damages on the frontal and distal surfaces of the specimens. The third group of specimens is loaded up to the check points and unloaded. Subsequently these specimens are cut along the 0° direction. Micrographic Chapter Five: Penetration Mechanisms images of the cross-sections of these specimens and their deformation profiles are then carefully studied. 5.1.2 Static Indentation Tests A schematic illustration of the static indentation tests is shown in Figure 5.2, where a laminated plate is supported by a rigid (steel) flat plate without any openings, and thus the bending of the laminate is inhibited. Such tests are used to characterise the compressive behaviour of the material for penetration problems, because the compressive strength obtained from traditional uniaxial compression tests is not suitable for penetration problems. To characterise the penetration behaviour, Woordward et al. (1994) carried out a set of constrained compressive tests on various materials (Aluminium, GFRP, Kevlar, Nylon, and Polyethylene) and for aluminium compared the derived stress-strain curves with those under simple compressive loading. They found that there was a substantial difference between these two curves. For a given strain, the stresses in the constrained compressive tests were much larger than those in the simple compressive tests. Static indentation tests are considered to be a more realistic measure of the penetration resistance of a material. The first application of the static indentation tests in composite materials was by Zhu et al. (1992a & b), where a mean indentation pressure for woven Kevlar was obtained from the tests. In the present approach, the static indentation tests were conducted using the indenters with two different kinds of head shapes, namely 37° and 120° included cone angles. The results 87 Chapter Five: Penetration Mechanisms of the tests (such as the force-displacement relationship and the penetrated hole shape) are used to calculate the effective compressive resistance of the material to penetration. 5.1.3 Experimental Force-Displacement Relationships The results of static penetration tests on CFRP and GFRP laminates are prescribed and discussed in the following sections. 5.1.3.1 CFRP Laminates The relationship between the measured penetration force and indenter displacement in the static penetration tests on CFRP laminates up to the point when the indenter tip penetrates through the target is shown in Figure 5.3 for indenters with four different included cone angles: 37°, 60°, 120°, and 180° (Delfosse 1994b). The tests on 37° and 180° conical indenters were conducted on the support plate with the large-geometry opening while the tests on 60° and 120° conical indenters were conducted on the plate with the small-geometry opening due to the shortage of material supply. Details of the experimental investigation have been presented in Delfosse (1994b) and Sanders (1997). Some general trends are summarised here: 1. The force-displacement relationship for small cone angle indenters (37° and 60°) are quite different from those corresponding to large cone angle indenters (120° and 180°). Indenters with small cone angles result in force-displacement curves that are more flat at the peak load and extend over a wide range of indenter displacement. However, indenters gg Chapter Five: Penetration Mechanisms with large cone angles result in force-displacement curves that are more steep leading up to the peak load and then drop abruptly. 2. The initial part of the measured load-displacement curves always has a nearly parabolic shape. The rate of increase of the force depends on the included cone angle of the indenter. Larger cone angles lead to higher rates of force increase. The specimen penetrated by a 37° conical indenter was unloaded and cut into two halves (Sanders, 1997). Figure 5.4 shows the cross-sectional micrograph of the specimen, where material around the indenter is seen to be crushed and pushed aside by the indenter. These are the characteristics of hole expansion mechanism reviewed in the previous chapter. 3. There is a load drop in the force-displacement curve when the penetration force reaches its peak value. For penetration by an indenter with a small cone angle, this load drop is caused by a number of fibre breaks as shown in Figure 5.5 (Sanders, 1997); while for an indenter with a large cone angle, it is caused by the initiation and push-out of a transverse plug as shown in Figure 5.6. The former case corresponds to the indenter tip approaching the distal surface of the laminate, while in the latter case the tip stays well inside the laminate. 4. Delaminations are initiated as shown in Figure 5.7 and are well developed by the time the indenter shaft reaches the frontal surface of the laminate as shown in Figure 5.8 (Sanders, 1997). Because of the presence of the delaminations, the whole laminate is separated to several sublaminates in the thickness direction. Each sublaminate behaves like a thin plate and has a significant bending behaviour. 89 Chapter Five: Penetration Mechanisms 5.1.3.2 GFRP Laminates Static penetration tests on the GFRP laminates were conducted by Sanders (1997) for three different laminate thicknesses (6.35 mm, 12.7 mm, and 19.05 mm). The relationship between the measured penetration force and the indenter displacement in these tests is shown in Figure 5.9 for the indenter with an included cone angle of 37° and in Figure 5.10 for the indenter with an angle of 120°. The support plate has a large-geometry opening. Details of these tests have been presented in Sanders (1997). The general characteristics of the forcedisplacement curves are summarised here: 1. Similar to CFRP, the force-displacement relationship for small cone angle indenter (37°) is substantially different from the corresponding curve for large cone angle indenter (120°). The force-displacement curve for the indenter with a small cone angle is triangular in shape, while the force-displacement curve for the indenter with a large cone angle has a plateau regime spanning over a wider displacement range than that corresponding to the small cone angle indenter. 2. The whole penetration curve is quite smooth and continuous. There are no sudden load drops for both the large and small cone angle indenters. 3. The micro-delaminations and matrix cracks start very early during the penetration as shown in Figure 5.11. The delamination pattern in GFRP is quite different from that in CFRP laminates. The massive micro-delaminations are developed around the indenter while in CFRP macro-delaminations are developed only below the indenter. Macro- delamination is observed in the GFRP laminates when penetrated by a 120° conical 90 Chapter Five: Penetration Mechanisms indenter as shown in Figure 5.12. A big gap between the sublaminates stemming from the macro-delamination is clearly visible. 4. The indenter with a 120° cone angle does not actually penetrate through the 6.35 mm thick laminate as shown in Figure 5.13. It simply pushes the laminate through the opening of the support plate. For the 12.70 mm and 19.05 mm thick laminates, the indenter penetrates through to a certain depth in the laminates. After that, a macro delamination initiates, which causes the sub-laminate below the indenter to separate from the original laminate. Then, similar to the 6.35 mm thick laminate, the indenter pushes the sublaminate through the opening of the support plate and never penetrates through it. 5.1.4 Characterised Force-Displacement Relationships Based on the experimental observation summarised in the previous section, the penetration process can be characterised by the schematic curves shown in Figure 5.14 for CFRP and GFRP laminates when penetrated by an indenter with a general cone angle. The critical points are shown as capital letters on these curves. The penetration mechanism that operates between two adjacent critical points are identified. The whole penetration event is divided into four phases, from phase I to phase IV. The start and end points of these phases are labelled by capital letters A-D. It should be mentioned here that these are typical force-displacement curves during penetration of CFRP and GFRP laminates. They capture the essence of the force- Chapter Five: Penetration Mechanisms displacement relationship but could not represent exactly all the cases of different cone angles, different target thicknesses, and different projectile lengths. Phase I: Line OA In this stage, the laminate material around the indenter is crushed and pushed aside by the indenter. Comparing with the severe damage (a penetration hole) on the frontal surface of the laminate, there is just a slight deformation occurring on the laminate distal surface according to the micrographic study in Figure 5.4. The penetration mechanism during this phase is categorised as hole expansion. The local stresses (transverse shear stresses) are built up with increasing indenter penetration depth. When the increasing local shear stresses meet the initiation criterion for delamination or transverse shear plugging, either delaminations are initiated in the laminate below the indenter as shown by curve 1 in Figure 5.14 or transverse plugging is initiated in the laminate below the indenter as shown by curve 2 in the same figure. The initiation point for these penetration mechanisms is labelled as "A," for curve 1 and "A " for curve 2. 2 Phase II, Line AB In this stage, the penetration mechanisms are different for the two characterised forcedisplacement curves. On curve 1, the laminate is divided into two deformation zones as shown in Figure 5.15. In zone I, the frontal part of the laminate, the major penetration mechanism is hole expansion (see Figure 5.5). In zone II, the distal part of the laminate, the bending rigidity is reduced due 92 Chapter Five: Penetration Mechanisms to the delaminations thus leading to severe bending deformations(see Figure 5.7 and Figure 5.8). The penetration mechanism in zone II is flexural deformation of a delaminated plate with a number of delaminations (splits). We will refer to this as "flexural deformation of a split-plate". There is a large amount of local deformation on the distal surface of the laminate. At point B,', the indenter tip penetrates through the target if the indenter cone length L is greater than the target thickness (L > h). Thus the target is defined as "partially c c perforated" at this stage. At point B„ the indenter shaft reaches the frontal surface of the target. In curve 2, the penetration force dropsfrompoint A to point B on the force-displacement 2 2 curve. This drop is caused by transverse shear failure of the laminate (see Figure 5.6). The laminate material below the indenter loses its transverse shear capacity and a transverse plug is initiated as shown in Figure 5.16. At point B ', friction between an indenter-plug system 2 and the adjacent target material initiates. With increasing indenter shaft length in the laminate, the frictional force between the plug and the adjacent laminate material is gradually replaced by the frictional force between the indenter and the laminate material around the indenter. At point B , the indenter tip penetrates through the target and the target is partially 2 perforated. It should be noted that a horizontal curve is used here to represent the penetration stage between points B and B . However, both the experimental measurement and model 2 2 prediction show that the curve between B and B may not be horizontal. 2 Phase III, Line BC 2 Chapter Five: Penetration Mechanisms In this phase, the target has been partially perforated for both curves. The friction becomes the major penetration mechanism when the indenter shaft reaches the laminate frontal surface. The length of the shaft in the laminate becomes larger and larger with the indenter penetration. For curve 1, the penetration force resulting from hole expansion and flexural deformation of the split-plate both decrease with increasing shaft length in the laminate. However, for both curves the friction force between the indenter and the adjacent laminate material increases with the advancing indenter displacement. At point C, for both curves, the indenter shaft reaches the distal surface of the laminate (/'. e., the indenter's conical head exits the laminate). Thus only the indenter shaft stays in the laminate after point C. Phase IV, Line CD In this stage, the force is purely frictional and energy is absorbed through friction only. 5.2 Identification of Penetration Mechanisms Based on the literature review in the previous chapter and experimental studies on penetration behaviour of CFRP and GFRP laminates, one deformation mechanism and four penetration mechanisms have been identified. They are elastic deformation, hole expansion, flexural deflection of a split-plate (delamination and dishing/petalling), transverse shear plugging, and friction. Each mechanism is discussed here in separate sub-sections. Chapter Five: Penetration Mechanisms 5.2.1 Elastic Deformation When the penetration force is very low, the global deformation of the laminate at the impact site is linear-elastic. If the penetration depth is very small, the indenter displacement is almost the same as the laminate deflection at the impact site. Thus the relationship between the penetration force and the indenter displacement is linear until appreciable penetration depth sets in or the penetration force becomes high. When either the penetration depth or the penetration force becomes large, the material around the projectile or indenter undergoes severe non-linear deformations and a large amount of damage occurs under or around the projectile. However, the material far away from the indenter continues to deform elastically, and its deflection can be calculated using the super finite element computer program developed in Chapters Two and Three. The equivalent laminate elastic modulus and Poisson's ratio of the CFRP laminates are calculated using the classical lamination plate theory (e.g. Whitney, 1987), yielding E= 62.5 GPa and v=0.32. The equivalent laminate elastic modulus of the GFRP laminates is back-calculated from three-point flexural tests on 6.35 mm, 12.70 mm, and 19.05 mm thick specimens resulting in a mean elastic modulus, E = 16.2 GPa. This value is close to the one measured by Woodward et al. (1994) as 18.1 GPa for plain woven S2-glass laminates with nominal fibrevolume fraction of 60%. The Poisson's ratio is taken to be v= 0.2 (Greaves, 1993). Chapter Five: Penetration Mechanisms 5.2.2 Hole Expansion In the hole expansion mechanism, the material is cut at the apex of the cone and pushed away from the conical indenter as shown in Figures 5.17 and 5.18. This mechanism initiates when the contact pressure under the cone reaches a certain mean indentation pressure value. From Figure 5.17 and equilibrium of force in the vertical direction, the penetration force F can be written as F = a (l+-^-)A tan/? 0 = a n{a + b f tan B (5.1) 2 p m 0 0 0 where a is the penetration depth defined in Figure 5.18 and cr is the mean indentation m pressure defined as cT =cr (l + - ^ - ) tan/? m (5.2) 0 0 In Equations (5.1) and (5.2), /? is the half included cone angle of the indenter, cr is the hole 0 0 expansion pressure, A is the projected area of the indenter, b is the bluntness of the indenter p 0 as shown in Figure 5.19 which is assumed to be very small, and ju is the coefficient of friction. When Equation (5.1) is used to calculate a penetration force at a certain depth of penetration, CT and ju have to be known first. 0 a is a measure of the compressive resistance of the 0 laminate material, while ju measures its interfacial or frictional resistance. In the present approach, the static indentation tests are conducted using indenters with two different kinds of head shapes, 37° and 120° cone angles. From the measured force- 96 Chapter Five: Penetration Mechanisms displacement curves in these tests, the mean indentation pressure <j in each case can be m calculated by rearranging Equation (5.1): F cr = 2 Q (5.3) r 7r(a + b ) tan 2 0 O The hole expansion pressure CT , which is treated as the compressive strength of the material, 0 and coefficient of friction ju, can then be obtained by using Equation (5.2) and the two known values of mean indentation pressure cr . Thus cr and ju are empirical parameters that best fit m 0 the current experimental measurements. With the knowledge of <x and //, the indentation pressure for an indenter with a general 0 included cone angle can be readily calculated from Equation (5.2). This kind of approach will provide a realistic estimation of the effective compressive resistance of the material to projectile (or indenter) penetration. 5.2.2.1 Material Characterisation Material characterisation tests (static indentation tests) are conducted for CFRP and GFRP laminates. The characterisation for CFRP and GFRP is discussed in the following sections. 5.2.2.1.1 CFRP Laminates The indentation force and indenter displacement in the static indentation tests of CFRP laminates are shown in Figure 5.20 where the indenter has an included cone angle of 37°. The thickness of the specimen is 6.15 mm. The indentation depth in the target is equal to the 97 Chapter Five: Penetration Mechanisms indenter displacement because the distal surface of the laminate is supported by a rigid block. The bluntness of the indenter (see Figure 5.19) is b = 0.5 mm. The relationship between the 0 indentation pressure and indenter displacement is also shown in Figure 5.20 and its mean value is calculated as a = 1585 MPa. m The indentation force versus the indenter displacement of a 120° indenter penetrating the same specimen is shown in Figure 5.21. The calculation of the indentation pressure ends when the indenter shaft reaches the frontal surface of the laminate, i. e., after 2.2 mm of indenter displacement. Careful examination of the specimen cross-section revealed the socalled "sinking-in" phenomenon on the frontal surface around the indenter as shown in Figure 5.22. This was not observed in the indentation tests by the 37° indenter. "Sinking-in" was first identified in the indentation tests on metals by Tabor (1951). When the indenter material is harder than the specimen material, the specimen material is depressed around the indentation and left at a lower level than the material farther away from the indenter. This is only observed close to the rim of the indentation and is referred to as "sinking-in" by Tabor (1951). Because of the "sinking in" phenomenon, the indentation depth will be over-predicted if we simply take the indenter displacement to be the indentation depth. Measuring A and a in Figure 5.22 will provide a relationship between the indentation depth and indenter displacement as a « 0.7A. Based on the above penetration depth, the indentation pressure is calculated using Equation (5.3). The mean pressure turns out to be a = 697 MPa as shown m in Figure 5.21. The indenter is assumed to have no bluntness in this calculation. 98 Chapter Five: Penetration Mechanisms 5.2.2.1.2 GFRP laminates For GFRP (G-glass) laminates, Delfosse (1994b) measured the elastic modulus in the through-thickness direction to be E = 14.0 GPa, a value close to the elastic modulus in the z in-plane direction E =19.0 GPa. Because of its relatively high elastic modulus in the x transverse direction, the deformation of this material around the indentation is in the form of "piling-up" instead of "sinking-in" (Tabor 1951), especially for an indenter with a small included cone angle. This phenomenon was observed by Sanders (1997) in indentation tests on GFRP (S2-glass) laminates as well. In the material "piling-up", there is an upward extrusion of displaced material to form a raised crater in the specimen. Thus if we simply use the indenter displacement A as the indentation depth in the specimen, the actual indentation cc will be under-predicted as shown in Figure 5.24. The under-prediction of the indentation depth will result in under-prediction of contact area and therefore over-prediction of indentation pressure. Following the same approach as that for the CFRP laminate, the relationship between the indentation depth and the indenter displacement is estimated from the examination of the cross-section of GFRP specimen shown in Figure 5.24 a s a * 1.3A. The indentation pressures are calculated from Equation (5.3) as shown in Figures 5.23 and 5.25 and their mean values are a m = 860 MPa for the 37° indenter and a m = 368 MPa for the 120° indenter. 99 Chapter Five: Penetration Mechanisms Based on the measured mean indentation pressure values corresponding to the two included cone angles of 37° and 120°, the hole expansion pressure a and the coefficient of friction // 0 for CFRP and GFRP laminates are calculated using Equation (5.2) as cr = 485 MPa and ju = 0.75 for CFRP 0 a - 250 MPa and ju = 0.81 for GFRP 0 Figure 5.26 shows the comparison between the measured and predicted force-displacement relationships of a 6.15 mm CFRP specimen when statically penetrated by an indenter with an included cone angle of 60°. The predicted curve is calculated using the above values of cr 0 and ju. The good agreement between the measured and predicted curves serves as a verification of the empirical parameters cr and ju. 0 Having the knowledge of the hole expansion pressure cr and the coefficient of friction ju, 0 mean indentation pressure cr for a general conical indenter can be calculated using Equation m (5.2). The mean indentation pressure cT is plotted against the included cone angle of the m indenter 2fi in Figure 5.27. The mean indentation pressure decreases with increasing cone 0 angle. For angles greater than 40°, the mean indentation pressure reaches a relatively constant value for both materials. 5.2.2.2 Calculation of Penetration Force Before the onset of delaminations, the penetration force due to hole expansion is calculated using Equation (5.1). Then, at a certain penetration depth c^, delaminations are initiated in the laminate below the indenter. After the delaminations are initiated, deformations due to 100 Chapter Five: Penetration Mechanisms hole expansion continues to increase, but at a rate less than that before the initiation of the delamination. Figure 5.28 shows the forces that come into play for calculation of the hole expansion force in the delaminated laminates. It needs to be emphasised that the mean indentation pressure cr is obtained from the static m indentation tests, where the specimens are backed-up by a rigid support without any openings. Therefore, there are no lateral deformations on the distal surface of the specimen, only compressive deformation. When the specimens are backed-up by a rigid support with an opening, experimental work conducted by Sanders (1997) indicated that there was very little lateral deformation on the distal surface of the specimens before delaminations were initiated (see Figure 5.4). So <j m from the static indentation tests can still be applied to this case. However, when delaminations occur below or around the indenter (or projectile), the flexural rigidity of the specimen is reduced. The mean indentation pressure o~ is not applicable in m this case anymore. However, for simplicity in modelling, a is still used but multiplied by a m reduction parameter D in the following form D =\ (5.4) max(L ,h)-a c 0 where L is the indenter's cone height and h is the laminate thickness. c Thus, the mean indentation pressure a ' after the onset of delaminations can be written in the following form m cr' = a D =<j (\ m m ) m max(Z ,«)-a c (5.5) 0 101 Chapter Five: Penetration Mechanisms The hole expansion force is then calculated as F = a' A m = & ntan 2 p m B [{a + b f - (a - a ) ] 2 0 0 0 (5.6) instead of using the Equation (5.1). 5.2.3 Flexural Deflection of a Split-Plate As mentioned before, flexural deformation of a split-plate includes the initiation of delamination followed by bending of the delaminated laminate (i.e., dishing/petalling). A simple application of the split-plate theory (presented in Quan et al, 1996) has proven that it is quite an effective tool for modelling the delamination and post-delamination behaviour in laminated composites during penetration process. This gives us confidence in applying the split-plate theory to other cases involving delaminated layers. 5.2.3.1 Delamination Initiation and Progression Hole expansion causes the material around the indenter to crush and form a damage zone. This is denoted as zone I in Figure 5.15, which is the projected area underneath the indenter or the projectile. The laminate material inside zone I loses its shear strength and transfers the indentation pressure to the neighbouring material and the lower part of the laminate. The calculation of the transverse shear force and stress follows the schematic diagram shown in Figure 5.29. On the cylindrical surface of radius (a + b )tanj3, we have the following 0 equilibrium condition 102 Chapter Five: Penetration Mechanisms F Q= (5.7) 2n(a + b )tan /? 0 0 where Q is transverse shear force per unit circumferential length acting on the boundary surface. It is assumed that when the shear stress rat the corner edge as shown in Figure 5.29 reaches a critical value, delamination occurs. This shear stress r is calculated from the Q a multiplying a geometric factor — h average shear stress through the thickness h-a h-a h The critical value of T , T can be obtained from the static penetration tests on the specimen. S Based on the cross-sectional examination of the CFRP specimens penetrated by an indenter (with 37° cone angle) and unloaded at different points along the force-displacement curve, the delamination initiation point can be detected as shown in Figure 5.7. The delamination initiation stress T is then calculated based on the contact force and the penetration depth S measured at that point as shown in Figure 5.7 by applying Equations (5.7) and (5.8). Thus, the calculations lead to r = 100 MP a. s For GFRP laminates, r = 13.7 MPa, taken from the three-point flexural tests on GFRP s specimens as the transverse interlaminar shear strength (Lau, 1997). Delamination initiates when T > T S (5.9) 103 Chapter Five: Penetration Mechanisms When Equation (5.9) is satisfied at a certain depth of indenter penetration, delaminations are assumed to initiate below the indenter. Subsequently, the flexural deformation of the splitplate starts. Experimental study conducted by Sanders (1997) showed that many of the matrix cracks that occured in the CFRP laminate formed an angle of 45° with the fibre direction. This meant that the matrix cracks were likely caused by transverse shear stresses. When matrix cracks grow and finally reach an interface between two plies delamination occurs. Therefore, our assumption that delaminations occur when a critical shear stress is reached is justified. Intensive C-scan and pulse-echo ultrasound (PEUS) studies revealed that delaminations generally form in pie-shaped sectors in a staircase helical pattern through the whole sublaminate thickness, as shown in Figure 5.31 (Delfosse et al., 1994b). It was also found that for quasi-isotropic lay-ups, one full revolution (circle) of the helical delamination pattern will occur in each sublaminate (Delfosse et al., 1994b). Therefore, to simplify the analysis in the present approach, one full circular delamination is assumed to occur at the interface between two adjacent sublaminates. Based on micrographic examinations and to enable theoretical developments, the delamination size is assumed to be bounded by the normal to the cone surface and shown as zone II in Figure 5.15. When the penetration depth increases, the delamination size also expands in a self-similar shape. The inner and outer radii of the delamination zone will increase until the indenter shaft reaches the frontal surface of the specimen. The increase in 104 Chapter Five: Penetration Mechanisms the inner radius is due to the continuous crushing of the material as the indenter advances in the target. It is worth noting that the delamination pattern in CFRP and GFRP are quite different. There are obvious (visible) macro-delaminations that initiate in the CFRP laminates, while in the GFRP laminates, there are individually invisible micro-delaminations that start very early in the penetration process. This explains why the delamination initiation stress in the GFRP laminates is much smaller than that in the CFRP. The micro-delaminations in the GFRP are also approximated as circular macro-delaminations in the present analysis. 5.2.3.2 Calculation of Compliance of a Single Split Because each split is assumed to be a circular plate, compliance of a single split can be calculated from a circular plate or ring using classical flexural theory of plates. Let a- and b y } denote the outer and inner radii of the fth split clamped along its boundary as shown in Figure 5.32. The total number of splits is n . s 5.2.3.2.1 Compliance of a Circular Plate without a Central Hole In this case, bj = 0. From flexural theory of circular plates presented in Timoshenko and Woinowsky-Krieger (1959), the compliance of this plate is given by 105 Chapter Five: Penetration Mecha'nisms where D is the bending rigidity of the y"th split. Let h denote the distance in thickness dj } Eh ~^r> 12(1-v) 3 direction between two adjacent delaminations the y"th and (/+7)'th, then D = dj J where vis the Poisson's ratio of the material. 5.2.3.2.2 Compliance of a Circular Plate with a Central Hole The compliance of the y'th split can be calculated as (Timoshenko and Woinowsky-Krieger, 1959) bj b, 1 , , / , =^—(ln^--1) + — - — ( b + a ) J j %xD dj 2 \6nD y a j Kj j dj ) i „ i 1 a, - + J - 1 1+v 2 %nD di u ^jl-v (5.11) b;i + v (b -a -2a \n^-) 2 2 2 Details of derivations of this equation are given in Appendix E. From Figure 5.33, we can write a and bj as y aj =(a + b )tmfi + 0 [a +(j-Y) hj ] / tan 0 bj =[a + b -a 0 0 0 -C/-1)hj]tm/3 0 fi 0 (5.12) (5.13) where hj is the distance between two delaminated interfaces. For the quasi-isotropic CFRP laminates, hj is equal to a sublaminate thickness. The development pattern of delaminations is shown in Figure 5.34. 106 Chapter Five: Penetration Mechanisms 5.2.3.3 Total Compliance of a Split-Plate Total compliance is calculated for two different cases of interaction between the splits. In the first case, it is assumed that all the splits have the same lateral deflections. In the second case, all the splits are assumed to have the same force acting on them. 5.2.3.3.1 Splits Having the Same Deflections If we assume that the splits satisfy the compatibility condition that all of them have the same deflection shape at any time, then the total force F on the split-plate is the sum of the forces s on the each split, i.e., F =F s i + F 2 + (5.14) +F ns Rewriting Equation (5.14) in terms of deflections xv. w, w 2^ ZL+ 2^_ = f f\ fl + xv„ + (5.15) fn s Because all of the splits have the same transverse deflection, i.e., xv = w =w = s l 2 = xv , n the total compliance / o f this split-plate, can be written in terms of compliance of each split f as If the split-plate deflection is denoted by w , then the penetrating force according to this s deflection is 107 Chapter Five: Penetration Mechanisms 5.2.3.3.2 Splits Subjected to the Same Force If we assume that all the splits are subjected to the same amount of force and do not necessarily have the same deflections as shown in Figure 5.31, then the total deflection of the split-plate can be approximated as the sum of the deflection of each split, i.e., (5.18) +w, "s This approximation will underestimate the total deflection of the split-plate. But if the deflection of each split is small compared with the split thickness, and the planar size of the new generated split is smaller than the splits already there, which is the same situation as in the current case, the error caused by this approximation is small. Rewriting Equation (5.18) in terms of the force on the splits M=f F +f F i 1 2 2+ +f F n (5.19) ni Because the forces acting on the each split are the same, i.e., F = F = x 2 = F„ =F , the total s S compliance from Equation (5.19) is given by (5.20) If the total split-plate deflection is w , then the corresponding penetration force is s (5.21) 108 Chapter Five: Penetration Mechanisms Equations (5.16) and (5.20) are two ways of calculating the total compliance of the splitplate. Which one is used in the analysis depends on the pattern of delamination during the penetration process in that material system. 5.2.3.4 Local Deflection of a Delaminated Plate Based on the experimental observations in Delfosse (1994b) and Sanders (1997), for the CFRP laminates, all delaminations are assumed to occur simultaneously and their shapes are taken to be one full circle for each single sublaminate. For the GFRP laminates as shown in Figure 5.11 (a-e) (Sanders, 1997), delaminations initiate and develop around the projectile. As the projectile penetrates the target deeper, more delaminations occur through the thickness. Thus the circular delaminations in GFRP laminates are assumed to be initiated sequentially, one by one. When delamination occurs in the laminate, the delaminated region of the laminate will behave like a split-plate as shown in Figure 5.32 and deflects more than the remainder of the plate due to its reduced bending rigidity. The plate will be more compliant (flexural) in the region with the delaminations, especially when the laminate has been penetrated through at the centre by the indenter or projectile tip. The local plate deflection w , or the so-called deflection of the split-plate, is calculated at s each increment of the penetration depth. The laminate material in the local deformation region, zone II shown in Figure 5.15, undergoes flexural deformations as shown in Figure 5.36. 109 Chapter Five: Penetration Mechanisms Let's consider a line element OA, which is the outer radius of the current split. It is horizontal before the indenter tip penetrates through it. The line OA bends and moves to a new position, OB, after it is perforated by the indenter tip. Assuming that there is no tensile or compressive deformation in this line, OA = OB. The local deflection is w° = BC. In AOBD, OB = ctj, the angle between the lines OD and BD is equal to ZODB = n - ZCDB . In AAED, the angle between line AD and ED is equal to ZADE = —- ZAED = —-B . 2 2 0 0 Since ZCDB = ZADE, we have ZODB = ^ + B . (5.22) Q Also, the length of OD is calculated by OD = OA-DA = aj - A « t a n / J (5.23) 0 In AOBD, we have sin ZDBO sin ZODB OD a (5.24) } Substituting Equations (5.22) and (5.23) into Equation (5.24), sin ZDBO = s 0 m Z 0 D D B = °J ( 1 _^ t a n yjj c o s p ( 5 25) J a In AOBC, we have no Chapter Five: Penetration ZCOB + ZOBC = - /. e., Mechanisms (5.26) + A> = y ZCOB+ZDBO Thus, ZCOB = — - /? - sin [(1 - — t a n Z?) cos J3 ] 2 a, (5.27) -1 0 0 0 Therefore, the local deflection of the split is equal to w° =O5sinZC6>v3 = a .sin{--/? -sin [(l-—tany? )cos/? ]} 2 cij _1 ; 0 0 0 (5.28) The above derivations imply that point O is a pin and line OA rotates as a rigid body about point O to line OB. This is not true in contradiction to our assumption of a split-plate, where a split is assumed to be clamped along its boundary. Therefore, Equation (5.28) needs to be modified in order to consider the effect of clamped boundary. The displacement function of a simply-supported circular ring with an outer radius a- and y loaded by a uniformly distributed load (of total magnitude FJ) along the perimeter of the hole is given by Timoshenko and Woinowsky (1959) * = UD l ^ w dJ [ ^2(1 ^ + v)J ( a 2 - r 2 ) + r 2 l n -aj ] ( 5 " 2 9 ) The rotation on the support edge is J °j AnD 1 + v F dr (5.30) dJ in Chapter Five: Penetration Mechanisms Therefore, the local deflection when edges are clamped is w =wl+0 sj (5.31) jaj 5.2.3.5 Lateral Force on the Split-Plate At the instant when delamination occurs, since no central hole exists for the delaminations under the indenter, the compliance of the split-plate is calculated using Equation (5.10), where the inner radius of the split-plate bj = 0. As the indenter continues to penetrate into the target, a central hole begins to appear in the splits. Depending on the penetration depth in the laminate, some splits will have a central hole while others won't. Accordingly, the compliance of each split is calculated using either Equations (5.10) or (5.11). The total compliance of the split-plate is then calculated using Equations (5.16) or (5.20). The force is calculated from Equation (5.17) or (5.21). After substituting Equations (5.30) and (5.31) into Equations (5.17) or (5.21), F , = ^ =^ / p _ s 2>, 7=1 n, _ 7=1 / =^ L - I ^ ^ L _ ; / / 4 ^ 1 + v' 1<+1*A >1 °- 1 ^ < =^ - - V - ^ W F for CFRP (5.32a) ^ for GFRP (5.32b) For CFRP laminates, we assume that all the splits have the same deflection, f F = fF , so x x s FF that — = — in Equation (5.32a). Rearranging the terms in Equation (5.32a), then f f\ 112 Chapter Five: Penetration 4xD (\ + v) Mechanisms forCFPvP (5.33a) dl 47rD (l + v)f+a 2 in For GFRP laminates, we assume that all the splits are under the same force, F = F and s p therefore rearrangement of the terms in Equation (5.32b) will lead to 1 for GFRP (5.33b) [1 + Because the outer and inner radii of the circular split-plate change with increasing penetration depth, compliance is a function of the penetration depth and therefore has to be recalculated at each increment of the indenter or projectile displacement. The bending deformation of the split-plate terminates when the indenter shaft reaches the frontal surface of the specimen. 5.2.4 Transverse Plugging Transverse plugging is a penetration mechanism that usually arises in transverse indentation by blunt or obtuse indenters. Static penetration tests with the blunt indenter are therefore used to find the transverse shear strength of laminates. The penetration force and indenter displacement are recorded until the laminate is perforated. A typical force-displacement relationship in a small-geometry test is shown in Figure 5.37 (Delfosse 1994b). The transverse shear strength r is calculated based on the maximum shear force required to form 0 a transverse plug through the laminate thickness. 113 Chapter Five: Penetration Mechanisms For CFRP laminates, the transverse shear strength r is calculated to be 142 MPa, 0 corresponding to a maximum contact force of 20.8 kN (Figure 5.37) for a specimen with 6.15 mm thickness penetrated by a blunt-ended cylinder of diameter 7.62 mm. The friction force F pf between the plug and the adjacent material can be calculated from the measured force- displacement curve. At point/in Figure 5.37, the contact force drops and a transverse plug initiates. At point g, the indenter travels a distance equal to the target thickness, which means that the plug has been pushed out from the target at this time. The trapezoidal area represents this plug push-out stage. By extrapolating the line CD to the point E, the friction force between the plug and the rest of the target material is estimated as F = 11.4 kN. pf The initiation criterion for transverse plugging mechanism is in the following form T > T (5.34) 0 For a given penetration depth a, the transverse shear stress T can be calculated as F T = and compared with the plugging strength r . 0 If the initiation 27r(a + b )htanj3 0 0 condition in Equation (5.34) is met, then a transverse shear plug is deemed to form. In the case of GFRP laminates, the blunt-ended indenter did not perforate the specimens. Instead, for 6.35 mm thick specimen the laminate deflected and was pushed out from the opening of the rigid support plate. For thicker laminates (12.7 mm and 19.05 mm), the indenter yielded. 1 14 Chapter Five: Penetration Mechanisms It can also be inferred that the transverse shear strength for the GFRP laminates considered in this study is large enough such that the transverse plugging mechanism will likely not occur in such laminates. If transverse plugging occurs (i.e., r > r ), the contact force will drop to a value of 0 h—a F = F P f ~ f ( 5 - 3 5 ) 5.2.5 Friction Friction starts when the indenter shaft reaches the frontal surface of the specimen. It is assumed that the penetration force arising from hole expansion and flexural deformation of the delaminated plate (split-plate) or from push-out of a transverse plug decreases linearly in this phase, while the frictional force increases linearly with increasing indenter or projectile shaft length in the laminate. When the projectile shaft penetrates through the laminate, the penetration force reaches a value denoted by F . ( This value is measured from static penetration tests of the same material penetrated by an indenter of the same diameter. It is the magnitude of the force corresponding to post-perforation plateau in the forcedisplacement curves. It should be noted that the coefficient of friction /u discussed in Section 5.2.2 couldn't be used here to calculate the friction force F . ju is merely the empirical parameter derived from static { indentation tests. It may not really reflect the physical nature of friction between the projectile and target. 115 Chapter Five: Penetration Mechanisms For the 6.15 mm thick CFRP laminates considered here, this frictional force is found to be F = 2000— (AO (see Figure 5.37), while for GFRP laminates, it is F, = 4270— (TV), h f is the laminate thickness in millimetre, and 2000 N and 4270 N are the friction force corresponding to the 6.15 mm CFRP and 12.7 mm GFRP laminates, respectively. 5.2.5.1 Effects of Geometric Sizes of Projectiles and Targets In static penetration tests, the indenter shaft is still in the laminate at the end of the penetration process when the indenter cone completely penetrates through the laminate thickness. The effects of the projectile length on the penetration process is only brought up in dynamic penetration tests, where the long projectiles will absorb more impact energy in friction than the short projectiles. The frictional force is assumed to start when the projectile shaft reaches the frontal surface of the laminate. If the projectile cone length is greater than the laminate thickness, the projectile cone will stick out from the distal surface of the laminate when friction starts. Frictional force changes for different relative positions of the projectile and the target. Therefore, calculation of the penetration force depends on the geometric sizes of the projectile and target thickness, and the position of the projectile in the target. There are four different cases depending on the projectile cone length L , projectile shaft c length L -L , p c and target thickness h: 1. Z is smaller than the local thickness of the target which is the sum of the target thickness c h and the local bending deflection w , while the projectile shaft length is greater than the s 116 Chapter Five: Penetration Mechanisms local thickness of the target, i.e., L <h + w and L - L > h + w as shown in Figure c s c s 5.38(1); 2. Z is greater than the local thickness of the target h + w , while the projectile shaft length c s is still greater than the local thickness of the target, i.e., L > h + w and L - L > h + w c s c s as shown in Figure 5.39 (1); 3. Projectile shaft length L -L p is smaller than the local thickness of the target h + w , c s while the projectile cone length L is smaller than the local thickness of the target, i.e., c L -L <h p + w and L < h + w as shown in Figure 5.38 (2); c s c s 4. Projectile shaft length L - L is still smaller than the local thickness of the target h + w , p c s while the projectile cone length L is greater than the local thickness of the target, i.e., c L - L < h + w and L > h + w as shown in Figure 5.39 (2). p c s c s The maximum frictional length in the target as shown in Figures 5.39 and 5.40, can be written as L =mm(h fx + w ,L -L ) s p (5.36) c L defines the maximum length in the target over which friction is operative. n For a thin target (L >h + w ) penetrated by a long projectile (L -L c s p c >h + w ), the total s frictional length at that time is the target thickness plus the local bending deflection, i.e., L =h + w as shown in Figure 5.39(c) and Figure 5.39(bl). For a thick target (L <h + w ) /X s c penetrated by a short projectile (L -L p c s <h + w ), when the back end of the projectile has s 117 Chapter Five: Penetration Mechanisms penetrated into the target, the projectile tip may not have completely penetrated through the target yet, so that the total frictional length at that time is the projectile shaft length, i.e., L fX = L -L p as shown in Figure 5.38 (b2) and Figure 5.39 (b2). c 5.2.5.2 Calculation of Penetration Force for Different Positions of Projectile during the Friction Phase Let us set four typical checking points: a = h + w , a = L +h + w , a = L + L , and a = L + h + w . s c s c f2 p s where L denotes the distance, n L = max(/z + w , L -L ) f2 s p (5.37) c Figures 5.39 and 5.40 show how we can use L and L to check the start of friction and n n perforation of the target. The equations used to calculate the penetration force are shown in this section, which depend on the positions of the projectile within two checking points. 1. L < a < h + w as shown in Figure 5.39(a), c s F = F + F f - ^ , if a - L < L , or h+ w sQ f c fl (5.38) s F=F +F -^,ifa-L >L ; h+ w s0 / c (5.39) /] s where F is the penetration force at a = L and F is the penetration force in static s0 c { penetration tests when only the indenter shaft is penetrating the target. 118 Chapter Five: Penetration Mechanisms 2. h + w < a < L + h + w , in the case when the projectile cone length Z is smaller than the 5 c C s local target thickness h + w , as shown in Figure 5.38(b); s F=F L c + 1 / 1 s0 ~ + F ?—^, if g - I < L , and L <h + w or a } F=F Lc + L / e JX c s ~ + F y ^ , i f ^ - 4 > Z a n d 4</i + w,; 2 a s0 / 4 / 1 (5.40) (5.41) 4z and in the case where the projectile cone length 4 is greater than the local target thickness h + w as shown in Figure 5.39(a), s F=F L c + 1 / 1 s0 F= F £ c + ~ + F ^ — ^ , if a a y ~ L / 2 j0 a +F ^-, 3. L + h + w <a<L c s c +L f2 ifa-L > f 4i -4 <Z c , and 7 1 Z C > / J + W, or L and L >h + w ; fl c s as shown in Figure 5.39(c) and Figure 5.38(b), (5.44) f c f2 (5.43) 4z F=F -^-; 4. Z + L (5.42) < a < L + h + w as shown in Figure 5.39(d) and Figure 5.38(c), s ^ _ Z„ + Z + Z„ - or F= F — f n £ £ h + w. ; (5.45) 5. a > L + h + w as shown in Figure 5.38(e) and Figure 5.39(d), p s F=0. (5.46) 119 Chapter Five: Penetration Mechanisms 5.3 Summary Based on the literature review and experimental work done in our research group, hole expansion, delaminations and associated flexural deformations of a split-plate, transverse plugging, and friction are identified as dominant mechanisms during penetration of laminated composites. For each penetration mechanism, analytical models for their initiation criteria and subsequent development patterns are developed in this chapter. A summary of the equations used to calculate the contact force is provided in Table 5.4. These developments are based on experimental studies and thus have a solid physical background. 120 Chapter Table 5.1 Five: Penetration Mechanisms Geometric sizes and material properties of the indenters used in static penetration and indentation tests. Material hardened steel, Diameter 2R 30 7.62 mm p Included cone angle 26 37°, 60°, 120°, 180° Bluntness b 0.5 mm 0.2 mm 0.0 mm 0.0 mm 0 0 Table 5.2 37° 60° 120° 180° Geometric sizes and material properties of the CFRP laminated plates used in static indentation and penetration tests. Material IM7 carbon fibres with 8551-7 epoxy matrix Lay-up [-45/90/45/0 ] Properties* E n G 1 3 2 4s = 142 GPa, E = 7.9 GPa, G 2 2 =G 2 3 1 2 = 4.1 GPa, = 3.0GPa, v =0.34 12 Equivalent isotropic properties** E = 62.5 GPa, v = 0.32 Thickness h 6.15 mm Delfosse (1994b). Calculated using lamination plate theory (Whitney, 1987). Table 5.3 Geometric sizes and material properties of the GFRP laminated plates used in static indentation and penetration tests. Material S2-glass plain weave rovings with phenolic resin matrix Elastic properties E = 16.2 G P a , v = 0.2** Density p = 1.9 g/cm Thickness h 6.35 mm, 12.7 mm, and 19.05 mm. * ** 3 Lau(1997). Greaves (1992). 121 Chapter Table 5.4 Five: Penetration Mechanisms Summary of the equations used to calculate the contact force during penetration of a laminate. Mechanism hole expansion Equations Equation Number (5.1) no delaminations: F= cr n(a + b ) tan J3 ; 2 m 2 0 0 with delaminations: F = < A P = °>tan B [(a + b f - (a - a f ], 2 0 0 0 (5.5) max(L ,h) - a c flexural deflection of a split-plate (5.6) 0 all the splits have the same deflection: o 4^,(1 + v) (5.33a) aj 4^D (\ + v)f + 2 : cn J\ all the splits have the same force: F ° f (5.33b) ' W 1 " a 4nf(\ + transverse plugging friction F-F F - F k « ' 2 ~ v)j^D 1 dj a (5.35) h see pages 118-120 (5.38)-(5.46) 122 Chapter Indenter Penetration _<i> t Mechanisms 2R 7 T T 7 T 7 Laminate \ (|)2R or axb t Rigid Support Figure 5.1 Five: \ \ \ \ \ \ \ L/ \ Schematic illustration of static penetration tests. The specimen sits on a rigid support with a circular or rectangular opening. R denotes the radius of a circular opening, while a and b denote the length and width of a rectangular opening. The indenter diameter is 2R = 7.62 mm. t P Indenter Z Laminate Rigid Support Figure 5.2 Schematic illustration of static indentation tests. 1 23 Chapter Five: Penetration Mechanisms 24 Indenter Displacement (mm) Figure 5.3 Penetration force versus indenter displacement in static penetration tests on the CFRP laminates defined in Table 5.2 penetrated by indenters with included cone angles of 37°, 60°, 120°, and 180° (Delfosse, 1994b). Figure 5.4 Micrographic illustration of the initial penetration stage in a 6.15 mm thick CFRP laminate penetrated by an indenter with 37° cone angle (Sanders, 1997). The material around the indenter is crushed and is pushed aside by the indenter. 124 Chapter Five: Penetration Mechanisms Figure 5.5 Micrographic illustration of failure in a 6.15 mm thick CFRP specimen penetrated by an indenter with a cone angle of 37° (Sanders, 1997). The material below the indenter is broken and bent away from the indenter. Figure 5.6 Cross-sectional micrograph of a 6.15 mm thick CFRP laminate penetrated by the indenter with an included cone angle of 120°. The plug initiation site can be seen very clearly. 125 Chapter Five: Penetration Mechanisms Figure 5.7 Micrographic illustration of delamination initiation in a 6.15 mm thick CFRP specimen when penetrated by an indenter with 37° cone angle (Sanders, 1997). The delaminations are located below the indenter and matrix cracks nearby. Figure 5.8 Micrographic illustration of delamination development in a 6.15 mm thick CFRP specimen penetrated by an indenter with 37° cone angle (Sanders, 1997). The delaminations are well developed by the time the indenter shaft reaches the frontal surface of the specimen as shown in the figure. 126 Chapter Five: Penetration Mechanisms 24 0 5 10 15 20 25 30 Indenter Displacement (mm) Figure 5.9 Experimental force versus indenter displacement curves for 6.35 mm, 12.70 mm, and 19.05 mm thick GFRP laminates penetrated by an indenter with 37° included cone angle (Sanders, 1997). 28 Indenter Displacement (mm) Figure 5.10 Experimental force versus indenter displacement curves for 6.35 mm, 12.70 mm, and 19.05 mm thick GFRP laminates penetrated by an indenter with 120° included cone angle (Sanders, 1997). 127 Chapter Five: Penetration Mechanisms (a) projectile striking energy E = 154 J, V = 153 m I s s s (b) projectile striking energy E = 284 J, V = 207 m i s s s '""^^HfiflfeBMlK- J ••HMMi^lSliMMBI (c) projectile striking energy £ = 365 J , V. = 235 m i s v s (d) projectile striking energy E = 469 3\V = 267 m / s s S (e) projectile striking energy E = 763 / , V = 340 m / 5 s Figure 5.11 s Cross-sectional micrographs of the penetrated and the perforated specimens and corresponding sections with dye penetrant for a 19.05 mm thick GFRP laminate impacted by a projectile with 37° cone angle at different striking energies (Sanders, 1997). The micro-delamination sites are shown as the black areas in the sections with dye penetrant. 1 28 Chapter Five: Penetration Mechanisms Figure 5.12 Micrographic illustration of macro-delamination for a 12.7 mm thick GFRP specimen impacted by an indenter with 120° cone angle (Sanders, 1997). The development of macro-delamination is clearly visible. 129 Chapter Five: Penetration Mechanisms Figure 5.13 Top and side view of a 6.35 mm thick GFRP specimen penetrated by a 120° indenter (Sanders, 1997). The indenter simply pushes the laminate through the opening of the support plate and does not actually penetrate it. I 30 Chapter Penetration Aj • delamination initiation on curve 1 ^ ^ transverse shear plugging initiation on curve 2 2 Bi, B ' Mechanisms • friction initiation 2 B/, B • projectile tip penetrates through the target J-J JJ • friction plateau 2 Figure 5.14 Five: Schematic illustration of the force-displacement curves of CFRP and GFRP laminates when penetrated by an indenter with a general cone angle. The penetration mechanisms follow the initial hole expansion mechanism (OA or OA ) are hole expansion and flexural deformation of a delaminated plate (splitplate) in curve 1 (A B ) and transverse plugging in curve 2 (A B ). Friction force initiates and gradually dominates the penetration process after 5, in curve 1 and B in curve 2. x 2 X X 2 2 2 131 Chapter Po /' , ' Five: Penetration Mechanisms y 1 • •. l l Figure 5.15 Local deformation zones: zone I, hole expansion; zone II, flexural deformation of the delaminated plate (split-plate). 7 A transverse shear plugging Figure 5.16 Schematic illustration of transverse shear plugging initiation. 132 Chapter Five: Penetration Mechanisms Pc /J A. Figure 5.17 Schematic illustration of hole expansion mechanism. A is the projected area of a projectile or indenter, rj is the hole expansion pressure of the laminate, and u is the coefficient of friction of the laminate material. p 0 i Po /t1 Figure 5.18 a h Schematic illustration of indentation pressure cr in static penetration tests. The relationship between cr and cr in Figure 5.17 is shown in Equation (5.2). m m 0 133 Chapter Figure 5.19 Schematic illustration of bluntness b indenter. 0 Five: Penetration Mechanisms on the tip of a conical projectile or Figure 5.20 Calculation of mean indentation pressure based on measured forcedisplacement curve for the CFRP laminate. The conical indenter has an included cone angle of 37°. The specimen thickness is 6.15 mm. 134 Chapter Figure 5.21 Five: Penetration Mechanisms Calculation of mean indentation pressure based on the measured forcedisplacement curve for a CFRP laminate. The conical indenter has an included cone angle of 120°. The specimen thickness is 6.15 mm. The calculation of indentation pressure ends when the indenter shaft reaches the frontal surface of the laminate, /. e., 2.2 mm of indenter displacement. Figure 5.22 Micrograph examination of cross-section of the CFRP specimen in static indentation tests by a 120° indenter. "Sinking-in" (a<A) is clearly observed. The measured relationship between a and A is a~ 0.7'A. 135 Chapter 2 3 4 5 6 7 8 Five: 9 Penetration 10 Mechanisms 11 Indenter Displacement (mm) Figure 5.23 Calculation of mean indentation pressure based on the measured forcedisplacement curve for a S2-glass laminate where the conical indenter has an included cone angle of 37° and a diameter of 7.62 mm. The specimen thickness is 19.0 mm. Figure 5.24 Cross-sectional microgragh showing "piling-up" in the 19.05 mm thick GFRP specimen when penetrated by an indenter with an included cone angle of 37° and a diameter of 7.62 mm, a ~ 1.3A. 136 Chapter 0.6 1.2 1.8 Five: 2.4 Penetration 3 Mechanisms 3.6 Indenter Displacement (mm) Figure 5.25 Calculation of mean indentation pressure based on the measured forcedisplacement curve for a S2-glass laminate penetrated by a conical indenter with an included cone angle of 120° and a diameter of 7.62 mm. The specimen thickness is 19.05 mm. 15 Indenter Displacement (mm) Figure 5.26 Comparison between the predicted and the measured contact force-indenter displacement curve in static indentation tests on a 6.15 mm thick CFRP specimen by an indenter with an included cone angle of 60°. 137 Chapter Five: Penetration Mechanisms Figure 5.27 Relationship between the mean indentation pressure and the included cone angle of an indenter for CFRP and GFRP laminates. F Figure 5.28 Schematic illustration of hole expansion mechanism after delamination is initiated in the laminate. A is the projected area undergoing hole expansion. D is a reduction factor accounting for lateral deformation on the distal surface of the laminate. p 138 Chapter Five: Penetration Mechanisms Po a delamination initiates here delamination initiates here (a+b )tan(3 0 0 JJJ.LLLLLL Q ! t Q Figure 5.29 Free body diagrams used in calculation of shear stresses for delamination criterion. 10 Indenter Displacement (mm) Figure 5.30 The experimental force-displacement curve for a 6.15 mm thick C F R P laminate statically penetrated by a 37° indenter. The support plate has a small-geometry opening. 1 39 Chapter Five: Penetration Mechanisms Figure 5.31 Schematic illustration of delamination patterns in a T300H/F593 CFRP laminate with a stacking sequence of [45/0/-45/90] under static loading by a flat indenter (Delfosse et al, 1994b). ns Figure 5.32 Schematic illustration of the geometry and loading condition of a typical split in the multi-split plate. 1 40 Chapter Five: Penetration Mechanisms a \ i I 7 ; <—— Figure 5.33 Schematic illustration for calculation of the inner and outer radii of each split in the split-plate. Ab Aa ^ w \ / / : N i / \ A Arv ••• \ V/ . / / / I Figure 5.34 Schematic illustration of growth of the outer and inner radii of delaminations. 141 Chapter Figure 5.35 Five: Penetration Mechanisms Schematic illustration of flexural deformation of a split-plate in section 5.2.3.3.2. 142 Chapter Five: Penetration Mechanisms Figure 5.36 Nomenclature for the split-plate deflection after the projectile tip penetrates a certain layer of the laminate. (XQ is the penetration depth just when delaminations have initiated. 143 Chapter Five: Penetration Mechanisms Figure 5.37 Estimation of the transverse shear strength and friction force between the plug and adjacent material for CFRP laminates. 144 Chapter Five: Penetration (2) (1) (a) a = L, h+w a *H 4 Mechanisms y (b) a a = h + w, « — • (c) a = h+ w + L s a c < —> K H > (d) « = < + L n a L (e) a = h + w + L s a a p > Figure 5.38 Calculation of penetration force at different positions of the projectile in the target when the projectile cone length is smaller than the local target thickness for (1) long projectiles ( L - L >h+w ) and (2) short projectiles ( L - L <h+w ). p c s p c s 145 Chapter c (d) Penetration Mechanisms (2) (1) (a) a = L Five: a a = h+ w + L s r Figure 5.39 Calculation of penetration force at different positions of the projectile in the target when the projectile cone length is greater than the local target thickness for (1) long projectiles (L - L >h+w ) and (2) short projectiles (L - L <h+w ). p c s p c s 146 Chapter Six: Static Penetration Model and Applications Chapter Six: Static Penetration Model and Applications Based on the penetration mechanisms identified in the previous chapter, a penetration model is developed here. This model predicts the resistive force in addition to the energy absorbed by the composite target during projectile penetration. The model accounts for the penetration mechanisms and their transitions for different cone angles of projectiles. Two different material systems, CFRP and GFRP which are considered before (see Table 5.2 and Table 5.3), will be studied for the purposes of verifying the present model. 6.1 Model Development The penetration depth is the difference between the projectile (or indenter) displacement and target deflection at the impact point. This can be written as a = A - w - A-(w + w ) 0 (6.1) s where A is the projectile (or indenter) displacement, w, w , and w are the total, global, and 0 s local target deflections at the impact site, respectively. The assumptions used in developing the penetration model are: 1. The projectile is nondeformable (rigid) during penetration. Because a certain part of the kinetic energy of the deformable projectile is removed by elastic and/or permanent deformations, the deformable projectile needs more initial kinetic energy to perforate the 147 Chapter Six: Static Penetration Model and Applications target than rigid projectiles. So this assumption will give a conservative estimate of the perforation energy. 2. The energy dissipated in heat, sound, and other sources not identified in the previous chapter is ignored in the penetration model. 3. Dynamic material characterisation tests (Delfosse, 1994b) showed that the CFRP is a strain rate insensitive material. GFRP is more sensitive to strain rate than CFRP. However, the force-displacement curves obtained from static penetration tests and dynamic tests on GFRP have been shown by Sanders (1997) to be very similar. Thus for simplifying the problems, material strain rate effects are ignored here. Based on the penetration mechanisms discussed in the previous section, a flow chart of the proposed penetration model and its description will be presented first. 6.1.1 Description and Flow Chart of the Penetration Model A flow chart of the proposed penetration model is illustrated in Figure 6.1. The initial penetration mechanism is hole expansion. The material around the projectile is pushed aside from the impact point and the deformation on the distal surface of the specimen is assumed to be negligible. In this phase, delamination has not initiated in the laminate and thus the local plate deflection w = 0. s The penetration force is calculated using F = a 7r(a m + b ) tzn 2 0 2 6 0 (5.1) 148 Chapter Six: Static Penetration Model and Applications At each increment of penetration depth, initiation of delamination and transverse plugging are checked using the following conditions: for delamination initiation r. T > for transverse plugging initiation T>T N (5.9) (5.34) If delaminations occur, the flexural deflection of the split-plate combines with hole expansion to affect the penetration response of the laminate. At this time, we need to calculate the local plate deflection w° in the laminate due to the split-plate bending deformation v w = l w + ° j (5.31) j a The total resistive force is calculated as F= F F h+ (6.2) s where F » = a ^ 1 7TT? ] 7 rt a n ' Po[{cc + max(L ,h)-a c K) -(a-a f 2 0 (5.6) 1 0 F= 4*D (l + v) rfl < s f 2/ for CFRP (5.33a) for GFRP (5.33b) 1 V J 4^(l + v ) ^ Z V a [1 + If transverse plugging occurs, the force drops to a value of pf h (5.35) 149 Chapter Six: Static Penetration Model and Applications When the projectile (indenter) shaft reaches the frontal surface of the target, friction starts. The equations to calculate the penetration force after initiation of the friction phase presented in the previous chapter [Equations (5.38) - (5.46)]. The penetration model can also be illustrated using the spring system shown in Figure 6.2, where k , k , k denote the spring stiffnesses due to hole expansion, flexural deformation of h s p the split-plate, and transverse plugging, respectively. The spring corresponding to the hole expansion is continually active from the beginning of the penetration event to the time when the projectile (or indenter) shaft reaches the frontal surface of the target, while the other two springs k and k do not trigger until a certain h p criterion has been met. Experimental observations have shown that in the GFRP laminates delaminations have a "thickening" effect. Part of the "thickening" is due to the local bending of the laminates and can be modelled as split-plates as in CFRP laminates. The other reason for "thickening" is the gap between the micro-delaminations in the laminates (see Figure 5.12). Therefore, when the split-plate model is applied, the total compliance of the split-plate is calculated using Equation (5.20) because in this case the splits do not have the same deflection. Furthermore, the increasing gap between delaminations will cause changes in applying the hole expansion mechanism. The penetration force F in hole expansion deformation before any delamination initiate is calculated from Equiation (5.1). When T > r , new delamination s initiates in the target material around the projectile tip. Let (XQ denote the penetration depth when a new delamination initiates. Then in subsequent penetration the projectile penetrates a 150 Chapter Six: Static Penetration Model and Applications new surface from very beginning until new delamination initiates, which is schematically shown in Figure 6.3. The calculation of hole expansion force becomes F=F h + b -a + G K{a 0 m f tan 2 0 fi ; Q for GFRP (6.3) where F is calculated from Equation (5.6). h For CFRP laminates, all the delaminations initiate at the same time while for GFRP laminates, delaminations initiate sequentially, one by one. Therefore, there is a penetration stage in GFRP laminates which CFRP laminates do not have, i.e., the stage from the initiation of the first delamination to the last delamination. Equation (6.3) is used to calculate the penetration force during this stage. The new delamination interface is then generated using the criterion of T= Q h-a A ~ A ° > T h-a S ; for GFRP (6.4) Q where Q is calculated using Equation (5.7). Then CCQ are updated until next delamination initiates. When all the delamiantions have been initiated in the laminate, the combined flexural deformation of the split-plate and hole expansion acts as the second penetration mechanism. The penetration force is still calculated in the same way as CFRP except that the calculation of the compliance of the split-plate follows Equation (5.20) instead of Equation (5.16). In SENACS, a flag is used in the input file load definition to direct the computational procedure for a CFRP and GFRP laminate. 151 Chapter 6.1.2 Six: Static Penetration Model and Applications Sensitivity Analysis In this section, a sensitivity analysis is conducted on the present penetration model for CFRP laminates. The objective is to better understand the initiation criteria for delamination and transverse plugging, and to know how definitive these material constants obtained from static penetration and indentation tests are. 6.1.2.1 Initiation of Delamination From Equations (5.7), (5.8), and (5.9), we have F a (6.5a) 2n(a + b )tan/? h(h-a) Q 0 Because b is very small, a + b « a. While the projectile (indenter) shaft has not reached the 0 Q target frontal surface [(a + 6 )tan/? < L ], Equation (6.5a) can be written as 0 0 c a =h F (6.6a) Irzhtan fl t 0 s When the projectile (indenter) shaft reaches the target frontal surface [(a + Z> )tan/? > Z J , 0 0 Equations (6.5a) and (6.6a) become F a (6.5b) 2nR h(h-a) p or after rearranging terms F = 2nR hT ( p s a \) (6.6b) 152 Chapter Six: Static Penetration Model and Applications Thus, a series of curves of delamination initiation corresponding to different included cone angles are shown in Figure 6.4 for CFRP laminates. When a pair of contact force and penetration depth locates above the curve corresponding to that indenter cone angle, delamination initiates. As the cone angle increases, higher forces are required to initiate delamination. In other words, according to the present model, penetration by an indenter with a small cone angle is more likely to cause delamination. 6.1.2.2 Initiation of Transverse Shear Plugging From Equations (5.34), while the projectile (indenter) shaft has not reached the target frontal surface [(a + b )tanJ3 < L ], we have 0 0 c F r (6.7a) 0 2n(a + 6 )/ztan B 0 0 Thus F = 2a(a + b )hr tmB 0 0 (6.8a) 0 When the projectile (indenter) shaft reaches the target frontal surface [(a + 6 )tan/5 > Z J , 0 0 Equations (6.7a) and (6.8a) turn to be ^ = ^ o 2nR (6.7b) p and F = 2TTR T P 0 (6.8b) 153 Chapter Six: Static Penetration Model and Applications In Figure 6.5, a series of curves of transverse shear plugging initiation corresponding to an indenter with different included cone angles are shown for CFRP. The plateau on the curves for large cone angles (90°, 120°, and 150°) means that the indenter shaft has penetrated into the laminate and there is no more increase in the indenter projected area A . The indenter p bluntness causes the different start points of the curves of 15°, 37°, and 60°. When a point falls above the curve corresponding to a given indenter cone angle, plugging mechanism dominates. Each curve is the locus of forces and penetration depths that correspond to initiation of plugging. For a given penetration depth, an indenter with a large cone angle needs higher force to initiate transverse plugging than an indenter with a small cone angle. For a given contact force, an indenter with a large cone angle is more easily to initiate the plugging than that with a small cone angle. 6.1.2.3 Sensitivity of Hole Expansion Pressure a on Initiation of Penetration Mechanism 0 The relationship between hole expansion force and penetration depth, delamination initiation curve, and transverse plugging initiation curve are shown in Figure 6.6 for the CFRP laminate penetrated by an indenter with 37° cone angle. It can be seen that the delamination initiation curve meets the force curve earlier than the plugging initiation curve. Therefore, delamination initiates instead of plugging. If the value of current hole expansion pressure o 0 is increased by 35%, plugging initiation curve would meet the force curve first instead of delamination initiation curve. Then, transverse plugging rather than delamination would initiate. 154 Chapter Six: Static Penetration Model and Applications The same curves are also plotted for an indenter with an included cone angle of 120° as shown in Figure 6.7. It can be seen that the plugging initiation curve meets the force curve first and thus causes transverse shear plugging. If the value of the current hole expansion pressure c> is decreased by 32%, delamination rather than transverse plugging would initiate. 0 6.2 Results and Discussion We apply the penetration model developed in the previous sections to predict the CFRP and GFRP penetration responses and compare the model predictions with experimental measurements. 6.2.1 CFRP Laminates The foregoing model is used to simulate the static penetration responses of large-geometry or small-geometry specimens of CFRP laminates. The material properties and geometric sizes are listed in Table 5.2. Specimens with dimensions of 152 mm x 102 mm or 50.4 mm x 50.4 mm are placed on a steel support with a 127.0 mm x 75.4 mm rectangular opening or circular opening of diameter 25.4 mm. The predicted force-displacement curves and energy-displacement curves for indenters with included angles of 37°, 60°, 120°, and 180° are compared with the corresponding experimental results in Figures 6.8 to 6.15, respectively. The static penetration tests on 60° and 120° conical indenters were conducted on the small-geometry specimen due to the shortage of material supply. The labels A , B or B , C , A^ B or B ', and C on the curves x x x x 2 2 2 155 Chapter Six: Static Penetration Model and Applications correspond to the critical points shown on the characterised force-displacement curves shown in Figure 5.14. 6.2.1.1 Penetration by 37° and 60° Conical Indenters A comparison between the predicted and measured results are shown in Figures 6.8 and 6.9 for 37° conical indenter, and Figures 6.10 and 6.11 for 60° conical indenter. It can be seen that the current model compares fairly well with the experimental forcedisplacement and energy-displacement curves for both the 37° and 60° conical indenters. The parabolic increase in the force at the early stage of penetration, delamination initiation, and friction plateau that are characteristics of the experimental curves are reasonably well predicted by the model. The predicted potential energy (i.e., integration of the force- displacement curve) versus indenter displacement compares well with the experimental results for the 37° and 60° conical indenters. 6.2.1.2 Penetration by 120 °and 180° Conical Indenters A comparison between the predicted and measured results are shown in Figures 6.12 and 6.13 for the 120° conical indenter, and Figures 6.14 and 6.15 for the 180° conical indenter. The model is seen to capture very well the essence rather than details of the forcedisplacement curves, such as the load drop and friction plateau. The model predicts that a transverse plug forms in the laminate and this is supported by examining the cross-sectional micrograph of the specimen as shown in Figure 5.6. The peak force in a 180° conical 156 Chapter Six: Static indenter is very close to that in the experiment. Penetration Model and Applications However, the comparison between the predicted and measured peak force is rather poor for the 120° indenter. One reason for this discrepancy is that the model does not predict the delaminations that have been observed in the static penetration tests with 120° and 180° conical indenters. Delaminations will reduce the flexural rigidity of the laminate and cause larger bending deflection and hence lead to an increase in the indenter displacement. A major deficiency of the current model is that it does not account for both delamination and transverse plugging. As shown in the flow chart (see Figure 6.1), either delamination or plugging can occur but not both. 6.2.1.3 Penetration by an Indenter with a General Cone Angle Figure 6.16 shows the comparison among the predicted force-displacement curves for a range of included cone angles of indenters. These curves have all been generated for large- geometry specimens. With increasing cone angle, the peak penetration force increases while the span of the indenter displacement decreases. This trend is similar to the experimental force-displacement curves for Kevlar penetrated statically by a range of cone angles (60°, 90° , and 120°) presented in Zhu et al. (1992). For small indenter cone angles, such as 15°, 37°, and 60°, the material around the indenter undergoes hole expansion as well as flexural deformation of delaminated plate (split-plate). For large cone angles, such as 90°, 120°, and 150°, the plate material undergoes hole expansion and then formation and push-out of a transverse plug. A similar trend was also observed by Woodward (1984) for metallic materials. 157 Chapter Six: Static Penetration Model and Applications Johnson et al. (1973) found that plug formation occurred when the included cone angle of the indenter was greater than 90° in the plates when there were no friction between the indenter and the plate. If the coefficient of friction were to increase, plug formation would occur at included cone angles less than 90°. The predicted initiation force of delamination and transverse plugging, and peak penetration force are also shown in Figure 6.17 and Figure 6.18, respectively. 6.2.2 GFRP Laminates The present model is also applied to static penetration tests of GFRP laminates defined in Table 5.3. The indenters are specified in Table 5.1, where only those with 37° and 120° included cone angles are used in the tests. 6.2.2.1 Penetration by 37"ConicalIndenter The calculated force-displacement curves are compared with experimental ones in Figure 6.19 to Figure 6.24. The results of penetration tests on 6.35 mm laminate are shown in Figures 6.19 and 6.20. The label A ' represents the initiation of the first delamination in the laminate. With the { advancement of the indenter, more delaminations occur. After all the delaminations have been initiated, the hole expansion and flexural deformation of the split-plate commences. This is marked as A v At point y5„ the indenter shaft reaches the frontal surface of the laminate and the friction phase starts. Finally, at point C the indenter shaft reaches the l 3 158 Chapter Six: Static Penetration Model and Applications distal surface of the laminate. The comparison between the predicted force-displacement and energy-displacement curve is reasonably good. Points, does not appear on the predicted force-displacement and energy-displacement curves corresponding to penetration tests on 12.70 mm and 19.05 mm thick laminates. According to the model, before all the delaminations are initiated, the indenter shaft has already reached the frontal surface of the laminate and the friction phase starts. For the penetration tests on all the three laminate thicknesses, the predicted maximum indenter displacements are very close to the measurements. 6.2.2.2 Penetration by 120 "ConicalIndenter The predicted and measured force-displacement and energy-displacement curves are compared in Figure 6.25 to Figure 6.30. In the experiments, the indenter did not penetrate through the laminate. It merely pushed the laminate through the opening on the support plate. The predicted contact force is higher than the measured one while the predicted maximum indenter displacement is smaller than the measurement value as shown in Figures 6.25 and 6.26 for penetration test on the 6.35 mm thick laminate. In the cases of 12.70 mm thick and 19.05 mm thick laminates, both the peak contact forces and the maximum indenter displacements are smaller than the measured values. It is difficult to assess the performance of the model in this case since penetration was not achieved in the experiments. 159 Chapter Six: Static Penetration Model and Applications 6.3 Summary Based on the mechanisms identified in the previous chapter, analytical models for penetration of CFRP and GFRP laminates have been developed in this chapter. The predicted and measured penetration force and energy absorption versus indenter displacement curves agree well in most cases for different indenter included cone angles (37° , 60°, 120°, and 180°), different target thicknesses (6.35 mm, 12.7 mm, and 19.05 mm), and different material systems (IM7/8551-7 CFRP, S2-glass/phenolic resin GFRP). The penetration model developed here has been implemented into the super finite element code, SENACS, to enable analysis of dynamic penetration problems. The results of application of the code to ballistic impact of laminated composite structures are presented and discussed in the next chapter. 160 Chapter Six: Static Q Hole Expansion Penetration S t a r Model and Applications t Equation (5.1) Transverse Plugging Hole Expansion Split-Plate Bending Equation (5.35) Equations (5.6), (5.33), (6.2) Projectile shaft reaching frontal surface of target Friction Equations (5.38) - (5.46) Projectile tail reaching distal surface of target End Figure 6.1 Flow chart of the penetration model. 161 Chapter Figure 6.2 Six: Static Penetration Model and Applications Illustration of the penetration model as a spring system. F denotes the penetration force, k\, k , k denote the spring stiffnesses due to hole expansion, flexural deformation of a split-plate, and transverse plugging, respectively. s p 162 Chapter Figure 6.3 Static Penetration Model and Applications Schematic illustration of the penetration behaviour in GFRP laminates when one delamination has already been initiated and developed. Total force is the sum of F and F . h Figure 6.4 Six: { Curves of delamination initiation force versus penetration depth for a CFRP laminate. 163 Chapter Six: Static Penetration Model and Applications Figure 6.5 Curves of transverse shear plugging initiation force for CFRP laminates versus the penetration depth. When a pair of contact force and penetration depth falls above the curve corresponding to a given indenter cone angle, plugging mechanism dominates. Figure 6.6 Hole expansion force, delamination initiation, and plugging initiation curves of CFRP laminates when penetrated by an indenter with a cone angle of 37°. 164 Chapter Figure 6.7 Static Penetration Model and Applications Hole expansion force, delamination initiation curve, and plugging initiation curve of CFRP laminates when penetrated by an indenter with a cone angle of 120°. 0 Figure 6.8 Six: 4 8 12 16 Indenter Displacement (mm) 20 24 Comparison between the predicted and measured static force-displacement curves for the CFRP laminate defined in Table 5.2 penetrated by an indenter with an included cone angle of 37°. 165 Chapter Figure 6.9 Six: Static Penetration Model and Applications Comparison between the predicted and measured energy-displacement curves for the CFRP laminate defined in Table 5.2 penetrated by an indenter with an included cone angle of 37°. 16 . , Indenter Displacement (mm) Figure 6.10 Comparison between the predicted and measured static force-displacement curves for the CFRP laminate defined in Table 5.2 penetrated by an indenter with an included cone angle of 60°. 166 Chapter Figure 6.11 Six: Static Penetration Model and Applications Comparison between the predicted and measured energy-displacement curves for the CFRP laminate defined in Table 5.2 penetrated by an indenter with an included cone angle of 60°. 24 21 - /'I 18 i A 15 / 7 / 9 0) Experiment / 12 Q. , 4r / / / / / ** / / / / / / / / I ' // // 6 3 y ^ > M - £ - — Predicted ^ • • || B 2 ^\-^--^ /s 0 S i i i i i i i 1 2 3 4 5 6 7 ^2 i 10 Indenter Displacement (mm) Figure 6.12 Comparison between the predicted and measured static force-displacement curves for the CFRP laminate defined in Table 5.2 penetrated by an indenter with an included cone angle of 120°. 167 Chapter Figure 6.13 Six: Static Penetration Model and Applications Comparison between the predicted and measured energy-displacement curves for the CFRP laminate defined in Table 5.2 penetrated by an indenter with an included cone angle of 120°. 0 2 4 6 8 10 12 14 16 Indenter Displacement (mm) Figure 6.14 Comparison between the predicted and measured static force-displacement curves for the CFRP laminate defined in Table 5.2 penetrated by an indenter with an included cone angle of 180°. 168 Chapter Six: Static Penetration Model and Applications 120 Indenter Displacement (mm) Figure 6.15 Comparison between the predicted and measured energy-displacement curves for the CFRP laminate defined in Table 5.2 penetrated by an indenter with an included cone angle of 180°. 25 0 3 6 9 12 15 Indenter Displacement (mm) Figure 6.16 Comparison between the predicted and measured static force-displacement curves for the CFRP laminate defined in Table 5.2 penetrated by indenters with included cone angles of 15°, 37°, 60°, 90°, 120°, 150°, and 180°. 169 Chapter Six: Static Penetration Model and Applications 25 Predicted delamination 2 20 L aT o L_ 4 Predicted transition point between delamination and plugging • • Predicted transverse plug o c 15 o .2 c T3 a> +-> o ^ 10 o • <; _l 20 L_ 40 60 80 100 120 140 160 180 Indenter Included Cone Angle (degree) Figure 6.17 Predicted forces at the onset of delamination and transverse plugging for IM7/8551 -7 CFRP laminates. 25 Predicted o Experimental ^20 u c 15 o S 1 0 0) g 10 Q. I • ra 20 40 60 80 100 120 140 160 180 Indenter Included Cone Angle (degree) Figure 6.18 Comparison between the predicted and measured maximum penetration force as a function of the indenter included cone angle. 170 Chapter Six: Static Penetration Model and Applications 12 I Indenter Displacement (mm) Figure 6.19 Comparison between the predicted and measured static force-displacement curves for 6.35 mm thick GFRP laminate penetrated by an indenter with cone angle of 37°. Figure 6.20 Comparison between the predicted and measured energy-displacement curves for 6.35 mm thick GFRP laminate penetrated by an indenter with cone angle of 37°. 171 Chapter Six: Static Penetration Model and Applications 18 Indenter Displacement (mm) Figure 6.21 Comparison between the predicted and measured static force-displacement curves for 12.7 mm thick GFRP laminate penetrated by an indenter with cone angle of 37°. 300 Indenter Displacement (mm) Figure 6.22 Comparison between the predicted and measured energy-displacement curves for 12.7 mm thick GFRP laminate penetrated by an indenter with cone angle of 37°. 172 Chapter Figure 6.23 Six: Static Penetration Model and Applications Comparison between the predicted and measured static force-displacement curves for 19.05 mm thick GFRP laminate penetrated by an indenter with cone angle of 37°. Figure 6.24 Comparison between the predicted and measured energy-displacement curves for 19.05 mm thick GFRP laminate penetrated by an indenter with cone angle of37°. 173 Chapter Figure 6.25 Six: Static Penetration Model and Applications Comparison between the predicted and measured static force-displacement curves for a 6.35 mm thick GFRP laminate penetrated by an indenter with a cone angle of 120°. Figure 6.26 Comparison between the predicted and measured energy-displacement curves for a 6.35 mm thick GFRP laminate penetrated by an indenter with a cone angle of 120°. 174 Chapter Six: Static Penetration Model and Applications 28 Indenter Displacement (mm) Figure 6.27 Comparison between the predicted and measured static force-displacement curves for a 12.7 mm thick GFRP laminate penetrated by an indenter with a cone angle of 120°. Indenter Displacement (mm) Figure 6.28 Comparison between the predicted and measured energy-displacement curves for a 12.7 mm thick GFRP laminate penetrated by an indenter with a cone angle of 120°. 175 Chapter 0 4 8 Six: 12 Static 16 Penetration 20 24 Model 28 and Applications 32 Indenter Displacement (mm) Figure 6.29 Comparison between the predicted and measured static force-displacement curves for a 19.05 thick GFRP laminate penetrated by an indenter with a cone angle of 120°. Indenter Displacement (mm) Figure 6.30 Comparison between the predicted and measured energy-displacement curves for a 19.05 mm thick GFRP laminate penetrated by an indenter with a cone angle of 120°. 176 Chapter Seven: Ballistic Applications Chapter Seven: Ballistic Applications In this chapter, the penetration model developed in the previous chapter is adapted to dynamic analysis and implemented in the super finite element code SENACS described in Chapters Two and Three. Using this code, the transient response of a laminated composite structure under projectile impact causing penetration and perforation can be predicted. The effects of projectile cone angle, projectile velocity, target thickness, target planar size, and target boundary condition on the penetration event are investigated in this chapter. In the global (structural) analysis of the laminates, the material behaviour is assumed to be linear elastic with no damage. The target inertia consists of two parts. The first part originates from the global deformation and has been considered in the global (structural) analysis. The second part originates from the local deformation. The inertial force resulting from this is ignored in the local analysis due to its relatively small value compared with the other resisting forces. 7.1 Solution Scheme The dynamic solution algorithm for the penetration model is illustrated in Figure 7.1. The time integration method follows the Newmark-P method with B = 0.25 and y = 0.5 as used in Chapter Two. The finite element formulation is the same as that in Chapter Three for nonpenetrating impact problems, except that the impact force is now calculated using the 177 Chapter Seven: Ballistic Applications Equations developed in Chapter Six instead of the Hertzian contact law given by Equation (3-1). Let n represent the time step number, n = 1,2, ,N , where N is the total number of time 0 0 steps and At is the time increment. Also, let the solution at time station t be known. Then n sequence of calculation for a dynamic penetration problem is as follows: 1. Increase the time t by one time step: n h+x (7.1) =t„ + At 2. Calculate the increment of the projectile displacement using Newton's 2nd Law of Motion: A(A) n+1 = F„Ar-i^(A0 2m (7.2) 2 where V and F are the projectile velocity and impact force at the «'th time step, n n respectively, and m is the projectile mass. 3. Calculate the increment of penetration depth: A «„ i = A(A) + n + 1 - O „ - w „_,) - (w - M V , ) 0 0 (7.3) m where w is the global elastic bending deflection of the target at the impact site and w is the 0 s local bending deflection at the impact site due to damage mechanisms induced by projectile penetration. This equation comes from the subtraction of the following two displacement compatibility conditions: «+i a = » i - ( o„ + > 0 A w + (7.3a) 178 Chapter Seven: Ballistic Applications a»=A„-(Wo„_,+w„_,) (7.3b) The reason for using Equation (7.3) instead of Equations (7.3a) or (7.3b) is that the former can be easily used to check the sign of Aa . n+l 4. If Aor n+1 > 0, update the penetration depth a as « i n + = « „ + and then calculate the penetration force F A « „ + (7-4) i using the equations developed in the previous n + 1 Chapter, which can generally be expressed in the following form F =f (a,Q n+l (7.5) F In Equation (7.5), a represents the penetration depth, and ^represents the other geometric and material parameters. Depending on which penetration mechanism is initiated, a can be either the penetration depth at the current time step, penetration depth at that time step, Ac\+ V or the increment of The explicit form of the parameter £ in Equation (7.5) is provided in Table 5.4. 5. If Aor n+1 <0, the increment of projectile displacement is not sufficiently large to cause further penetration in the target. Thus skip updating of F and a, because in the present work decreasing penetration depth (rebounding) which has been considered in nonpenetrating impact problems, is not considered for penetrating type of problems. In nonpenetrating impact events, decreasing indentation depth and impact force are caused by projectile rebounding from the target. However, in penetrating impact problems, the penetration force decreases due to perforation (i.e., when the projectile completely 179 Chapter Seven: Ballistic penetrates through the target), and not necessary due to rebounding. Applications Therefore, the penetration depth always increases until the projectile completely loses its kinetic energy and causes to a halt. It is conceivable that when the projectile comes to a halt the stored energy in the target results in oscillations which could induce relative motion between the projectile and the target. However, this effect is ignored here. 6. Calculate the global deformation and update the global bending deflection at the impact site, w 0n+1 , using global structural analysis described in Chapter Two. This deflection can be generally expressed as (7.6) On+1 7. If the projectile displacement is greater than the sum of the projectile length, the target thickness, and the global and local deflections of the target, then the projectile has completely penetrated through the target, or "completely perforated" the target. If the projectile velocity is equal to or less than zero while the projectile has not caused complete perforation of the target, the projectile stops within the target. The target is referred to as "partially perforated" if the projectile tip penetrates through the target as the projectile stops. Program will stop if any of the above two cases occurs. Otherwise, go back to step 1 and repeat steps 1 to 6. This is summarised as the following expression of either A n+1 > L +h + w + w , completely perforated; p 0n sn f A„ > h, partially perforated; or K , < 0, { I A < h, embedded. +1 (7.7) n+1 180 Chapter Seven: Ballistic Applications 7.2 Comparison with Experiments The predicted impact force and impact energy absorption are compared with the experimental measurements on CFRP and GFRP laminates to verify the present penetration model and its implementation in SENACS. There are three types of instrumented dynamic impact experiments that have been conducted at UBC: drop weight (DW), gas gun (GG), and power gun (PG) tests. In the DW tests, the projectile has a mass of 6.14 kg and the impact velocity is of the order of 10 m/s depending on the drop height. In GG tests, a force transducer is attached to the projectile tail which records the impact force as a function of time during impact. The projectile velocity and displacement are then obtained by integrating the force-time curve. The projectile mass is 0.32 kg and depending on the gas pressure, the impact velocity can easily reach 50 m/s (Delfosse et al, 1993a). The PG tests are ballistic type of tests where the projectile mass is low (4.2 g or 13.2 g) and depending on the amount of powder used the velocity can reach 1000 m/s. The projectiles are fired from a .308 calibre standard rifle barrel. The projectile motion (displacement) in PG tests is recorded using the ELVS (Enhanced Laser Velocity Sensor). The projectile velocity and acceleration are then calculated by appropriate time differentiations. The impact force in turn is calculated by multiplying the projectile acceleration by the projectile mass. The impact energy absorption at a certain time is calculated as the difference between the initial and current projectile kinetic energy. The 181 Chapter Seven: Ballistic Applications details of the experimental set-up and calibration of ELVS are provided in Starratt et al. (1998). It should be noted that once the projectile passes through the entire width of the laser sheet (/'. e., projectile leaves the measurement window), the ELVS can no longer detect the projectile motion. The measured results from ELVS will therefore not cover the whole range of penetration process as the penetration model does. Perforation energy in this chapter is the projectile impact energy that is consumed in completely perforating the target. In dynamic penetration tests and numerical calculations, it is the difference between the measured (or calculated) initial and residual kinetic energies of the projectile. In static penetration tests, it is the sum of the area under the force- displacement curve when the indenter cone completely perforates the target and the friction energy resulting from the motion of the indenter shaft. 7.2.1 CFRP Laminates Specimens of the CFRP laminate shown in Table 5.2 were penetrated by instrumented projectiles with either a conical tip (37° cone angle) or a blunt tip (180° cone angle), fired from a gas gun (Delfosse 1994b). The projectile mass and length are given in Table 7.1. The tests were conducted on a large-geometry opening, where a 101.6 x 152.4 mm ( 4 x 6 inch) specimen was placed on a steel support with a 76.2 x 127 mm (3x5 inch) opening. 182 Chapter Seven: Ballistic Applications 7.2.1.1 37°Cone Angle The predicted and measured impact force versus projectile displacement curves are shown in Figure 7.2. The predicted maximum projectile displacement agrees very well with the measurements. The initial slope of the predicted force-displacement curve is in good agreement with the measured slope. However, the peak impact force is over-predicted. It is also interesting to see that the peak contact force in the static penetration test is higher than that in the dynamic penetration test. The physical reason for this is not known at present. For clarity, the predicted position of the projectile in the target is also shown schematically in the graph. The labels A, B, and C denote the initiation of delamination and subsequent flexural deformation of the delaminated plate (split-plate), the initiation of friction phase, and the completion of projectile cone penetration, respectively. Subscript 1 on the labels in the figure indicates that delamination rather than transverse plugging initiates in the penetrated target and thus the penetration response is characterised by Curve 1 in Figure 5.14. Due to the over-predicted impact force, the predicted impact energy absorption (/. e., the difference between the current and initial projectile kinetic energy) will be greater than the measured loss of projectile kinetic energy, as shown in Figure 7.3. The time histories of impact force, energy absorption, projectile velocity, projectile displacement, and target deflection (at the impact site) during the penetration event are shown in Figure 7.4 to Figure 7.6. When the projectile cone has penetrated through the target, the impact force reaches a friction plateau and the target starts to oscillate flexurally with an almost constant frequency (see point C, in Figure 7.6). The time period of the oscillation is 183 Chapter Seven: Ballistic Applications 253 /JS as shown in Figure 7.6, which is very close to the natural period of the laminate, 248 /JS (corresponding to a natural frequency of 4.032 kHz). 7.2.1.2 180° Cone Angle (Blunt) The GG test by a blunt-ended projectile (180° cone angle) with a velocity of 30.2 m/s was conducted on the same specimen and support condition as the 37° projectile (Delfosse, 1994b). Due to the singularity of tan/? when /? = 90°, the cone angle used in the analysis 0 is 2/? =179°. 0 0 The comparison between the model prediction and experimental measurements for the impact force and energy absorption versus the projectile displacement are shown in Figures 7.7 and 7.8. The predicted peak impact force and peak projectile displacement agree very well with the experimental results. The labels A^B^, and C denote the initiation of transverse plugging, the initiation of friction 2 phase, and the completion of projectile cone penetration. Before transverse plugging initiates at point A , penetration force increases with the projectile displacement very rapidly due to 2 the large cone angle of the projectile. Point B represents the instant when the projectile head 2 reaches the distal surface of the laminate. For a projectile with a finite cone length (cone angle 2/? <180°), the impact force decreases gradually from B to C ; while in Figure 7.7, 0 2 2 the force drops from B to C abruptly due to a very small length of projectile cone when the 2 2 cone angle 2/J = 179°. The predicted results show that the projectile completely penetrates 0 through and exits the target with a residual velocity of 18.9 m/s. 184 Chapter Seven: Ballistic Applications In the experimental impact force-displacement curve shown in Figure 7.7, the projectile has perforated the target when the impact force reaches a zero value. Because of onboard instrumentation that is attached to the tail end of the projectile in GG tests, the projectile remains in the target and vibrates with it. That is why we see the oscillations in the measured impact force and energy absorption shown in Figures 7.7 and 7.8. The penetration force and energy absorption obtained from the static test of the same specimen under the same support condition are also superposed on the dynamic penetration results in Figures 7.7 and 7.8. It can be seen that the peak force in the static test is higher than that in the dynamic test. A similar trend was observed for the 37° projectile penetration tests. 7.2.2 GFRP Laminates Instrumented ballistic tests were performed by Sanders (1997) on GFRP laminates defined in Table 5.3 for three different laminate thicknesses (6.35 mm, 12.7 mm, and 19.05 mm). The steel projectiles used which were hardened to HRc30 are defined in Table 7.2. In all these tests, the GFRP specimens were supported by a steel plate with a large-geometry opening (a rectangular opening of 76.2 x 127 mm, or 3 x 5 inch). 7.2.2.1 6.35 mm Thick Plate Figure 7.9 (a-e) and Figure 7.10 (a-e) show the comparisons between the predicted and experimental time histories of projectile displacement, projectile velocity, and impact force when the projectile impacts a 6.35 mm thick GFRP specimen. The impact energy absorption 185 Chapter Seven: Ballistic Applications in the target and impact force as a function of the projectile displacement are also shown in the figures. The 37° projectile has an initial impact velocity of 104 m/s, while the initial impact velocity of the 120° projectile is 211 m/s. For the 37° projectile test, the predicted and measured impact force agree very well from the start of penetration to the point where the projectile displacement reaches 3.0 mm [Figure 7.9 (e)]. After that point, the measured force starts to oscillate while the predicted force continues to increase until the critical point A which is the point at which the projectile tip x reaches the distal surface of the target (partial perforation). between the points A\ and A . x Four delaminations occur The delaminated target bends like a split-plate in the subsequent penetration. The model predicts that the projectile stops inside the target, with a 0.3 mm length sticking out from the distal surface of the target, i.e., partially perforated. This agrees quite well with the experiment. In the test, the projectile did not completely perforate the target (Sanders, 1997) and there was a small hole observed on the distal surface of the target after the projectile was removed, which means that the projectile tip had indeed pierced from the target distal surface. The projectile displacement and velocity are under-predicted, and impact energy absorption is over-predicted, relative to the experimental results. force-displacement The oscillations on the measured curve which cause the discrepancy between the prediction and measurement need to be further studied. 186 Chapter Seven: Ballistic Applications The same comparison is conducted for 120° projectile penetrating the same GFRP specimen which is shown in Figure 7.10 (a-e). The experiment showed that the projectile had penetrated through the target with an exit velocity less than 85 m/s (Sanders, 1997). The model predicts that the projectile completely perforates the target with a residual velocity of 123 m/s. Both the model prediction and experimental measurements conclude that the projectile completely perforates the target in this case. As shown in Figure 7.10 (e) and Figure 7.10 (c), the predicted initial slopes of the curves agree with the experiment very well. In the force-displacement curve, A x denotes the initiation of the first delamination in the target, while B denotes that the projectile shaft x reaches the target frontal surface and the friction phase starts. The 2nd and 3rd delaminations are initiated between points A and B . At the point B , the projectile tip exits from the distal x x x surface of the target. The target at this stage is partially perforated. After this point, the experimental force starts to oscillate while the predicted force decreases until the point C,, where the whole projectile cone tip penetrates through the target leaving the shaft still inside the target. The predicted time histories of the projectile displacement and velocity, and the impact force as well as the energy absorption as a function of the projectile displacement, are in good agreement with the experiment until the point B . x The reasons for the increase and subsequent oscillations of the measured impact force (when the projectile displacement is larger than 8.0 mm) is not clear at this moment and needs to be further studied. Chapter Seven: Ballistic Applications 7.2.2.2 12.7 mm Thick Plate The 12.7 mm GFRP specimens were impacted and penetrated by a 37° projectile with impact velocities of 304 m/s and 170 m/s, and a 120° projectile with impact velocity of 278 m/s. The comparisons between the predicted and experimental results for these cases are shown in Figure 7.11 (a-e), Figure 7.12 (a-e), and Figure 7.13 (a-e), respectively. For the GFRP specimen penetrated by a 37° projectile with a striking velocity of 304 m/s, the experiment showed that the projectile had perforated the target with an exit velocity of 200 m/s (Sanders, 1997). The model predicts that the projectile completely perforates the target with an exit velocity of 196 m/s. The agreement between the predicted and measured time histories of projectile displacement and velocity is good. The predicted impact force and energy absorption are close to the measured ones with the predicted value falling somewhat below the measurements. The labels A , B , B ', and C in Figure 7.11 (e) have the same x x x x meaning as those in Figure 7.10 (e) discussed before. Figure 7.12 (a-e) shows the comparisons between the predicted and experimental results when the same target is penetrated by a projectile with the same cone angle but with a lower incident velocity of 170 m/s. According to the experiment the projectile did partially perforated the target but not perforate it completely. The cross-sectional examination of the specimen indicated that there was about 2.0 mm of the target thickness which was not penetrated through by the projectile. The model predicts that there is a 5.8 mm length of the projectile sticking out from the target distal surface. Chapter Seven: Ballistic Applications The predicted and experimental peak impact force shown in Figure 7.12 (a-e) agree quite well. Again, the labels on the prediction curve in Figure 7.12 (e) have the same meaning as those defined before. The target is being partially perforated. For the GFRP target penetrated by a 120° projectile with a striking velocity of 278 m/s, the experiment showed that the projectile completely perforated and exited the target. Unfortunately, the exit velocity of the projectile could not be measured by the ELVS system. The model predicts that the projectile completely perforates the target with a residual velocity of 134 m/s. Therefore, the model prediction and experimental observation both conclude that the projectile perforates the target, which is encouraging from the modelling viewpoint. The overall agreement between the predicted and measured results is reasonable as shown in Figure 7.13 (a-e). The peak impact force and energy absorption are under-predicted. 7.2.2.3 19.05 mm Thick Plate The 19.05 mm GFRP specimens were impacted by projectiles with a 37° cone angle and striking velocities of 318 m/s and 211 m/s, and by projectile with a 120° cone angle and striking velocity of 384 m/s. The comparisons between the predicted and experimental results for these cases are shown in Figure 7.14 (a-e), Figure 7.15 (a-e), and Figure 7.16 (a-e), respectively. For the 37° projectile with striking velocity of 318 m/s, the experiment showed that the projectile did not completely perforate the target and finally came to rest in it (Sanders, 1997). Cross-sectional examination of the target revealed that the penetration hole was close 189 Chapter Seven: Ballistic Applications to the target distal surface and had a conical shape, suggesting that the projectile cone had not perforated the target. According to the model, the projectile also fails to completely perforate the target in this case. For the same projectile but with a striking velocity of 211 m/s, the model seems to capture the essence of the ballistic response. Cross-sectional observation of the target revealed that there was about 2.0 mm of the target thickness that had not been penetrated, while the model predicts that there is a 2.8 mm of un-penetrated target thickness. Again, the model and experiment are in agreement. The impact force as well as impact energy absorption are over-predicted by the model as shown in Figure 7.15 (c) and (e) while the projectile displacement and velocity are underpredicted as shown in Figure 7.15 (a) and (b). For the GFRP target penetrated by a 120° projectile with a striking velocity of 384 m/s, experimental observation showed that the projectile completely perforated the specimen and exited from it with a velocity of 160 m/s. The model predicts that the projectile completely perforates the target with an exit velocity of 243 m/s. Therefore, both the model and experiment arrive at the same conclusion. The overall agreement between the predicted and measured results is reasonable as shown in Figure 7.16 (a-e). The model under-predicts the impact force and energy absorption and over-predicts the projectile displacement and velocity. 190 Chapter Seven: Ballistic Applications 7.3 Effect of Projectile Cone Angle Experimental results (Figure 5.3, Figure 5.9, and Figure 5.10) showed that penetration response of the target is highly dependent on the projectile cone angle. In this section we conduct a series of numerical experiments to gain insight into the effect of cone angle on the ballistic response of composite materials. While keeping all other parameters constant, we carry out the analysis by systematically changing the cone angle. We will consider both the CFRP and GFRP systems. Where available, experimental results conducted by Delfosse (1994b) and Sanders (1997) will be shown on the graphs for references. 7.3.1 CFRP Laminates The ballistic tests on CFRP laminates were conducted on a small-geometry opening, where a 50.8 x 50.8 mm (2x2 inch) specimen was placed on a steel support with a circular opening with a diameter of 25.4 mm (1 inch). The projectile mass and length used in the analysis are given in Table 7.1. The projectile shaft length is always assumed to be a constant (L — L = 9.0 mm) for all projectile cone angles. p c Figure 7.17 shows the relationship between the predicted impact force and the projectile displacement for a range of projectile cone angles. It can be seen that the force-displacement curves become shallower with decreasing projectile cone angle. With increasing cone angle, on the other hand, the force-displacement curves exhibit a sharper peak force followed by a sudden drop. When the included cone angle reaches 90°, the deformation mechanism following the initial hole expansion phase changes from delamination initiation and 191 Chapter Seven: Ballistic Applications progression to plug initiation and ejection. When the projectile cone angle is greater than 90°, there is always a distinct load drop in the force-displacement curves. Figure 7.18 shows the prediction of the impact energy absorbed by the target for dynamic penetration of projectiles with various cone angles. Impact energy absorption is calculated as the difference between the projectile incident and residual kinetic energy. The relationship between the peak impact force and the projectile cone angle is shown in Figure 7.19. Projectiles with larger cone angles will generate higher peak impact forces, while the target absorbs more impact energy when it is penetrated by projectiles with smaller cone angles. Thus based upon energy absorption, a projectile with a small cone angle will encounter a higher penetration resistance than a projectile with a large cone angle. When the projectile cone angle approaches 90°, the penetration resistance reaches its minimum. Then it will increase again at a small rate when the cone angle continues to increase. Therefore, energywise projectiles with obtuse cone angles will penetrate through the CFRP targets more easily than the projectiles with acute cone angles. 7.3.2 GFRP Laminates A comparison of predicted force-displacement relationships for 37°, 45°, 55°, 60°, 75°, 120°, and 180° projectile is shown in Figure 7.20. The 6.35 mm thick GFRP specimen rests on a rigid support with the large-geometry opening (127 mm x 76.2 mm, or 5" x 3"). The projectile mass is 4.2 g and its length is given in Table 7.1 for various cone angles. For all of the cases studied, the projectile striking velocity is 600 m/s. The difference among the 192 Chapter Seven: Ballistic Applications character of these force-displacement curves is not as severe as that seen in the CFRP laminates. The 60° projectile has a smaller peak impact force than the other projectiles, similar to the 90° projectile for CFRP laminates. The relationship between the perforation energy (impact energy absorption) and the projectile cone angle is shown in Figure 7.21. The perforation energy for the projectile with a very small cone angle (15°) is the largest. The perforation energy decreases with increasing projectile cone angle until the cone angle reaches 60°. The perforation energy of 60° projectile is the smallest. When the projectile angle continues to increase, the perforation energy increases slightly and reaches a constant for projectile angles greater than 120°. The relationship between the peak force and the projectile cone angle is shown in Figure 7.22. 7.4 Effect of Impact Velocity The study on the effect of the projectile impact velocity on the ballistic response of the target is very important, because it results in the "ballistic limit" of the target material. It is characterised by V in the present thesis. The ballistic limit or V is considered to be the ct a impact velocity at which the projectile just completely perforates the target with a zero residual velocity. 193 Chapter Seven: Ballistic Applications 7.4.1 CFRP Laminates Figures 7.23 to 7.27 show the relationships between the projectile initial striking velocity (V ) s and the residual (exit) velocity (V ) after penetrating through a 6.15 mm thick CFRP laminate T at various striking velocities. The V -V s r curves have the classical initial parabolic shape followed by a linear rise. The projectile in Figure 7.23 has a 37° cone angle with a mass of 0.32 kg; while the projectile in Figure 7.24 has a mass of 4.2 g with 37° and 180° cone angles. The ballistic limits for 37° projectiles are 29.1 m/s for a 0.32 kg projectile and 261.6 m/s for a 4.2 g projectile. The ballistic limit for a 180° projectile with a mass of 4.2 g is 206.3 m/s. The laminate is not completely perforated until the projectile velocity reaches these critical values. After that, the projectile exit velocity increases sharply with increasing striking velocity. After the sharp increase, the projectile exit velocity increases at almost the same rate as the striking velocity. In Figure 7.25, two more V -V s r curves from 15° and 120° projectiles with a mass of 4.2 g have been added to the figure. The curves on the right absorb more impact energy than those on the left. The relationship between the projectile residual kinetic energy E and initial kinetic energy E r for 15°, 37°, 120°, and 180° projectiles is shown in Figures 7.26 and 7.27. s The model prediction is very close to the test measurements for the 180° projectile. The perforation energy is E perf = 261 J for the 15° projectile, E for the 120° projectile, and E perf peif = 144 J for the 37° projectile, E perf = 11 J =89 J for the 180° projectile. 194 Chapter Seven: Ballistic Applications Figure 7.23 to Figure 7.27 show the capability of a 6.15 mm thick CFRP laminate to absorb the impact energy when perforated by projectiles with various cone angles, which is very useful for design engineers. Another important information is the knowledge of how these absorbed impact energies distribute among the different penetration phases. The impact energy absorption consists of three parts on the force-displacement curves as shown in Figure 7.28 (a) and (b). For both curves (a) and (b), a major energy absorption mechanism in phase I is hole expansion. In phase II, major energy absorption mechanisms are either hole expansion and flexural deformation of a split-plate [curve (a)], or transverse plug initiation and push-out [curve (b)]. The friction is a major energy absorption mechanism in phase III for both curves. Energy absorption by the target global deformation and motion coexists with all these three phases. The global energy absorption is calculated as the integration of the impact force over the target central deflection w , while the local energy absorption is the integration of the force 0 over the sum of the penetration depth a and local target deflection w . s The partitioning of the impact energy absorption is shown in Figure 7.29 (a-d) for projectiles with 15°, 37°, 120°, and 180° cone angles where the CFRP laminate is placed on a rigid support with a small-geometry opening. The impact energy absorbed in the form of global deformation and motion is very small compared with the other three types of energy absorption mechanisms and thus is not shown in the figures. The projectile residual kinetic energy after perforating the target is also shown in the figures. The energy absorbed in phase II contributes the most to the total energy absorption for all four cases (about 247 J for 15°, 195 Chapter Seven: Ballistic Applications 128 J for 37°, 52 J for 120°, and 76 J for 180°). Because of the same shaft length L -L and p c target thickness h, and small local deflection w for these projectiles, the energy dissipated in s Phase III is very similar for all four cone angle (15°, 37°, 120°, and 180°), about 12 J. The energy absorbed in phase I is very small for very sharp (15°) projectile and non-existent for blunt-ended projectile (180°). Therefore, phase II plays a very important role in energy absorption and ballistic resistance of the CFRP laminates. 7.4.2 GFRP Laminates The effect of impact velocity on energy absorption capability of GFRP laminates with three thicknesses (6.15 mm, 12.7 mm, and 19.05 mm) is studied in this section. 7.4.2.1 6.35 mm Thick The predicted relationship between the exit velocity V and the initial striking velocity V for T s 15°, 37°, 120°, and 180° projectiles is shown in Figure 7.30. The projectile mass is 4.2 g and the length is given in Table 7.1 for various cone angles. The 6.35 mm thick GFRP specimen rests on a rigid support with a large-geometry opening (127 mm x 76.2 mm, or 5" x 3"). The right-most V -V curves correspond to more impact energy absorption than those to the s r left. So the target absorbs the most impact energy when it is impacted by a 15° projectile and the least impact energy when impacted by a 180° projectile. The ballistic limit V for these a different projectiles is tabulated in Table 7.3. 196 Chapter Seven: Ballistic Applications The relationship between the projectile residual kinetic energy E and its initial kinetic energy r E for different conical projectiles is shown in Figure 7.31. s The perforation energies are listed in Table 7.3. The composition of impact energy absorption is shown in Figure 7.32 (a-d) for projectiles with 15°, 37°, 120°, and 180° cone angles. The energy absorbed in phase II constitutes the most to the total energy absorption for all four cases (see Table 7.3). The energy absorbed in phase I is small compared with the energy absorption in the other phases. The impact energy dissipated in the global deformation and motion of the target increases with the projectile striking energy up to the complete perforation of the target, and then decreases. This confirms the widely accepted notion that the global energy absorption plays more important role in low velocity impact events than in high velocity impact events. The high velocity impact can be treated as a local event. Due to the decreasing global energy absorption, the target perforation energy also decreases with the striking energy when the impact velocity is above the ballistic limit. 7.4.2.2 12.7 mm Thick Consider the GFRP laminate with a thickness of 12.7 mm, supported on a rigid plate with the large-geometry opening (127 mm x 76.2 mm) and impacted by projectiles with 37° and 120° cone angles. The projectile mass is 13.2 g and length are listed in Table 7.2. The predicted relationship between the projectile exit velocity V and its initial striking velocity V is shown x s in Figure 7.33. The solid points are the experimental data. 197 Chapter Seven: Ballistic Applications The 120° projectile absorbs more impact energy than the 37° projectile does. Further ballistic tests are needed to obtain the perforation energy in both cases and to verify this prediction. The predicted ballistic limit V for both projectiles are V = 234 mf s (37° a cr projectile), and V = 245 mIs (120° projectile). cr The relationship between the projectile residual kinetic energy E and initial kinetic energy E r is shown in Figure 7.34. The perforation energy is E peif E perf s = 362 J for the 37° projectile and = 397 J for the 120° projectile. The composition of impact energy absorption is shown in Figure 7.35 (a-b), where as in the previous case, the energy absorbed in phase II contributes the most to the total energy absorption (see Table 7.4). The energy dissipated in phase III is also large because the target thickness in this case is longer than in the previous examples. The energy absorbed in phase I is also small compared with the energy absorption in the other phases. 7.4.2.3 19.05 mm Thick Finally, consider the GFRP laminate with a thickness of 19.05 mm on a rigid support with a large-geometry opening (127 mm x 76.2 mm), impacted by the same projectile as for the 12.7 mm thick laminate. The predicted relationship between the projectile exit velocity V t and initial striking velocity V is shown in Figure 7.36. s The 37° projectile dissipates more impact energy than the 120° projectile. The ballistic limit V for both projectiles are given in Table 7.5. a 198 Chapter Seven: Ballistic Applications The E -E curves for the two conical projectiles are shown in Figure 7.37. The perforation s r energy and the composition of impact energy absorption are shown in Figure 7.38 (a-b) and Table 7.5. Once again the energy absorbed in phase II still accounts for the most part of the total energy absorption. The energy dissipated in phase III is also large because the target is thicker than in the previous examples. In general, the model prediction suggest that phase II always plays a very important role in energy absorption and ballistic resistance of the GFRP laminates. 7.4.3 Comparison Between the Ballistic Resistance of CFRP and GFRP Laminates A comparison between the predicted ballistic resistance (perforation energy) of a CFRP laminate (6.15 mm thick) and a GFRP laminate (6.35 mm thick) is shown in Figure 7.39. It is seen that GFRP laminates have higher ballistic resistance than CFRP laminates for all four projectile cone angles (15°, 37°, 120°, and 180°). For large cone angles, the perforation energy of the GFRP and CFRP laminates is about the same. However, for small cone angles the difference between the perforation energy of two laminates is very large. 7.5 Effect of Target Thickness The predicted relationship between the impact force and projectile displacement for the three different thicknesses (6.35mm, 12.7mm, 19.05mm) of GFRP specimen is shown in Figure 7.40 for a 37° projectile and Figure 7.41 for a 120° projectile. The specimen in each case is supported on a rigid plate with a large-geometry opening and the projectile mass is 13.2 g. 199 Chapter Seven: Ballistic Applications The 37° and 120° projectiles have a velocity of 384 m/s for all three GFRP laminates (6.35 mm, 12.7 mm, and 19.05 mm thick). The greater the thickness, generally the higher the peak impact force and projectile displacement. 7.6 Effect of Boundary Conditions In all the calculations so far, the laminates were taken to be simply-supported along their edges. In this section, the laminates are assumed to be clamped on all edges in order to demonstrate the effect of boundary conditions on their ballistic response. A comparison between the predicted impact force versus projectile displacement curve for simply-supported and clamped targets is shown in Figure 7.42, where the projectile mass is 0.32 kg and its cone angle is 37° or 120°. The projectile velocity is 30.2 m/s. 7.7 Effect of Planar Size The effect of laminate planar size on its ballistic response is shown in Figure 7.43 for a CFRP laminate with a thickness of 6.15 mm. The projectile has a mass of 0.32 kg, a cone angle of 37°, and a striking velocity of 30.2 m/s. The planar size of the laminate (i.e., opening size of the rigid support plate) is seen to affect the maximum projectile displacement. A large opening size will increase the maximum projectile displacement. The perforation energy will increase as well due to more energy being absorbed in the form of global deformation and 200 Chapter Seven: Ballistic Applications motion energy. However, the effect on the peak impact force remains fairly constant with increasing size. When the projectile velocity increases, the global deformation and motion energy decreases rapidly for the target sitting on a support plate with a large opening and increases slightly for the target sitting on a support plate with a small opening as shown in Figure 7.44. Thus for a large target, penetrating impact acts like a local event when the projectile velocity is very high. 7.8 Effect of Projectile Mass The predicted time history of the impact force for the 6.35 mm thick GFRP laminate impacted and penetrated by a 37° projectile with three different masses is shown in Figure 7.45. The projectiles all have the same impact energy of 343 J. The projectile with high mass causes a larger impact duration. However, the peak impact force is almost the same for all cases. The relationship between the target deflection at the impact site and the projectile displacement is shown in Figure 7.46. For a given projectile displacement the target deflection is larger for the high mass projectile. Oscillations in the target deflection appear in the curves for the high mass (6.14 kg) and intermediate mass (0.42 kg) projectiles. There are no oscillations in the curve for the low mass projectile. The relationship between the energy absorption and projectile displacement is shown in Figure 7.47. 201 Chapter Seven: Ballistic Applications 7.9 Summary In this chapter, the penetrating impact model in SENACS have been used to perform numerical analyses of the ballistic response of various laminates. Based on comparisons with experiments conducted by Delfosse (1994b) and Sanders (1997), the present model proves to be a useful tool for predicting the nature and essence of penetration events in composite structures. The accuracy of ballistic limit V prediction is A very good as shown in Table 7.6 with an error generally less than 10%. Observation of the impacted specimen cross-sections showed that there are often big gaps between delaminated sections in GFRP laminates which the current model does not consider. After extensive parametric studies, it has been found that the projectile cone angle and target thickness play a very important role in the penetration response of laminates. The projectile cone angle determines which penetration mechanism will initiate in the target. Generally, a large cone angle projectile causes higher peak impact force, while a small cone angle projectile leads to longer projectile displacement. Although the impact force in penetration by a small cone angle is lower, the projectile impact energy used to perforate the target is higher. Target thickness will affect the development of the penetration mechanisms initiated by a projectile cone in the target. For a given projectile, the impact force is higher in thicker targets. A thick target also increases the distance that a projectile travels through to perforate. Therefore, the projectile consumes more energy to perforate thicker targets. 202 Chapter Seven: Ballistic Applications The target boundary condition and planar size will also affect the penetration response only when the target flexural deformation is significant, e.g., when the target is thin or the planar size is large. Flexible target will absorb more projectile impact energy in the form of strain energy. The ballistic limit, V , varies significantly with the target thickness and projectile cone angle. a V is generally higher for sharp projectiles and thick targets. a For a given level of impact energy, the mass of the projectile launched at different velocities will affect the time history of the target deflection, and thus affect the perforation energy and V. a The target deflection does not affect the penetration response very much for a high impact velocity. It should be noted that the present penetration model which describes the local target behaviour, can be used separately from the global finite element analysis program. The local model is simple and can be run on a spreadsheet. So for a simple structure, this simplifies and improves the efficiency of the analysis. The present penetration model includes the penetration mechanism of flexural deflection of a split-plate. The relationship between force and deflection in this mechanism is derived from lateral bending theory of a thin plate. So the present model may cause some error when predicting ballistic resistance of a very thick plate as shown in Table 7.6 (the error of prediction of V„ for a 19.05 mm thick GFRP laminate is larger than that in the other cases). 203 Chapter Table 7.1 Seven: Ballistic Applications Mass and length of projectiles with different cone angles in gas gun and powder gun tests on CFRP laminates. The projectile length is chosen in such a way that all the projectiles have the same shaft length (44) regardless of their cone angles. _ Length 15° 4 4( ) 28.93 (mm) mm 37.90 b (mm) 0.5 0 37° 45° 60° 90° 120° 150° 180° 11.38 9.19 6.60 3.81 2.20 1.02 0.0 20.35 18.16 15.57 12.78 11.17 9.99 9.0 0.5 0.5 0.2 0.0 0.0 0.0 0.0 Diameter: 7.62 mm Mass: 0.32 kg in gas gun tests 4.2 g in powder gun tests Table 7.2 Mass and length of projectiles with two different cone angles used in powder gun tests on GFRP laminates. Cone Angle 37° 120° Diameter (mm) 7.62 7.62 Mass (g) 13.2 13.2 46.1 40.0 Length, L (mm) p Table 7.3 Partition of the predicted perforation energy for a 6.35 mm thick GFRP laminate impacted by 4.2 g projectiles with various cone angles. Projectile Cone Angle Ballistic Limit V (m/s) Perforation Energy E (J) Eglobal 15° 37° 120° 180° 452 302.5 249 229.5 676 320 131 110 52 34 19 17 cr perf (J) F Ephase n •'-'phase I (J) 0.5 1 2.5 0 (J) 586 267 96 80 F ^phase ffl (J) 37.5 18 13.5 13 204 Chapter Table 7.4 Ballistic Limit V (m/s) Perforation Energy E (J) Eglobal 37° 120° 234 245 362 397 8 12 Table 7.5 cr Perforation Energy E (J) Eglobal 37° 120° 343 299 778 591 9 5 GFRP Applications Ephase II ^phase I (J) (J) 5 14 213 223 Ephase I I I 136 148 Partition of the predicted perforation energy for a 19.05 mm thick GFRP laminate impacted by 13.2 g projectiles with 37° and 120° cone angles. Ballistic Limit V (m/s) CFRP F perf Projectile Cone Angle Target Ballistic Partition of the predicted perforation energy for a 12.7 mm thick GFRP laminate impacted by 13.2 g projectiles with 37° and 120° cone angles. Projectile Cone Angle Table 7.6 Seven: cr perf (J) F •^phase I •"^phase II (J) (J) 15 14 563 367 F •'-'phase I E (J) 191 205 Comparison of the predicted and measured ballistic limit V for CFRP and GFRP laminates. The projectile mass is 4.2 g for CFRP and 13.2 g for GFRP. cr thickness(mm) /cone angle (m/s) 6.15/37° 311 203 168 236 261 10% 6.15/180° 293 180 208 231 206 10% 6.15/180° 261 143 159 218 205 6% 12.7/37° 304 200 196 229 232 1% 19.05/120° 384 160 243 349 297 15% V (m/s) measured predicted r V /(m/s) measured predicted c error (V ) cr * Calculated as V = ^V? - V? . cr 205 Chapter Seven: Ballistic Applications START Time Integration Method No Newton's Law of Motion Penetration Stages Check A+l / ( n^n) ) = V D Equ. (7.2) J Global Structural Model Local Penetration Model Displacement Compatibility Condition Equs. (7.3-4) J Figure 7.1 Flow chart of the solution scheme for the dynamic penetration model in SENACS. 206 Chapter 0 3 6 9 12 15 Seven: 18 21 Ballistic 24 Applications 27 Projectile Displacement (mm) Figure 7.2 Comparison of the predicted and experimental impact force versus projectile displacement when a 0.32 kg steel projectile with a 37° cone angle penetrates through a 6.15 mm thick CFRP laminate with a striking velocity of 30.2 m/s. Projectile Displacement (mm) Figure 7.3 Comparison of the predicted and experimental impact energy absorption of the target versus projectile displacement curves when a 0.32 kg steel projectile with a 37° cone angle penetrates through a 6.15 mm thick CFRP laminate with a striking velocity of 30.2 m/s. 207 Chapter Figure 7.2 Seven: Ballistic Applications Comparison of the predicted and experimental impact force versus projectile displacement when a 0.32 kg steel projectile with a 37° cone angle penetrates through a 6.15 mm thick CFRP laminate with a striking velocity of 30.2 m/s. 0 4 8 12 16 20 24 28 Projectile Displacement (mm) Figure 7.3 Comparison of the predicted and experimental impact energy absorption of the target versus projectile displacement curves when a 0.32 kg steel projectile with a 37° cone angle penetrates through a 6.15 mm thick CFRP laminate with a striking velocity of 30.2 m/s. 208 Chapter Seven: Ballistic Applications Impact Time (ns) Figure 7.6 Time histories of the predicted target deflection during penetration of a 6.15 mm thick CFRP laminate by a 0.32 kg steel projectile with a 37° cone angle and a striking velocity of 30.2 m/s. The impact force is also plotted here as a reference. 25 . Dynamic Experiment 20 ,, 15 % Q. E 1 ° Static Experiment 2 7B77 \ i ; i : i ; i .' i ; i \ :> ;i •! 1 •" 1 • ' ' / , i Predicted \ ;/ s \ 1 8 12 16 1 20 24 Projectile Displacement (mm) Figure 7.7 Comparison of the predicted and experimental impact force versus projectile displacement curves when a 0.32 kg steel projectile with a blunt-ended head penetrates through a 6.15 mm thick CFRP laminate with a striking velocity of 30.2 m/s. 209 Chapter Seven: Ballistic Applications 105 Projectile Displacement (mm) Figure 7.8 Comparison of the predicted and experimental impact energy absorption versus projectile displacement curves when a 0.32 kg steel projectile with a bluntended head penetrates through a 6.15 mm thick CFRP laminate with a striking velocity of 30.2 m/s. 210 Chapter Seven: Ballistic Applications 2P =37° Experiment 0 2P =37° 0 ~ 12 Experiment Predicted o k. 0. 0.04 0.08 0.12 0.16 0.2 0.24 0.04 0.08 Impact Time (ms) 0.12 0.16 0.2 0.24 Impact Time (ms) Predicted . 4 8 12 Projectile Displacement (mm) (d) (C) 97-D-2005-3 3 6 9 12 15 Projectile Displacement (mm) (e) Figure 7.9 Comparison of the predicted and experimental results when a steel projectile with a 37° included cone angle impacts a 6.35mm thick GFRP laminate with a striking velocity of 104 m/s. Both the predicted data and experimental observation show that the projectile stops in the target. 211 Chapter Applications 2P„=37° Experiment — f E. 1 Ballistic 120 16 c Seven: 2P„=37° E 90 2 Experiment £ a. Q — Predicted 8 o a 0.08 0 . 1 2 0.16 0.2 0.24 0.04 0.08 Impact Time (ms) 0.12 0.16 0.2 0.24 Impact Time (ms) (b) 2P„=37° 2P„=37° Predicted -60 c o v - Q. O «40 < ^ - i 0.04 0.08 0.12 0.16 Impact Time (ms) 0.2 i 4 8 12 Projectile Displacement (mm) 0.24 12 Predicted I f 16 (d) (c) 10 Experiment A, *~ v 97-b-2005-3 8 Experiment \J jr O l£ 6 o \ ro4 2 - / // / ' 1 Ii K C _ 0 1 1 3 6 9 12 15 Projectile Displacement (mm) (e) Figure 7.9 Comparison of the predicted and experimental results when a steel projectile with a 37° included cone angle impacts a 6.35mm thick GFRP laminate with a striking velocity of 104 m/s. Both the predicted data and experimental observation show that the projectile stops in the target. 212 Chapter Seven: Ballistic Applications 240 48 ? Predicted 2p\,=120° 2p>120° E. Predicted £ 36 a> E ai • Experiment a 12 Experiment a a 0.15 0.1 0.05 0.2 0.05 0.3 0.25 0.15 0.1 0.3 0.25 0.2 Impact Time (ms) Impact Time (ms) (b) (a) 24 280 2p\,=120 0 JI A Experiment j 2p\,=120° — Experiment 2 1 8 o a. a; o ,? / Predicted — 12 » 140 < a i 2! — Predicted 6 [ t i 0.05 0.1 .1 0.15 . . 0.2 0.25 70 UJ / \ 0.3 i i 16 24 32 40 48 Projectile Displacement (mm) Impact Time (ms) (d) (C) 25 97-b-0306-2 Experiment 20 0, 15 p 8 ° a. 1 i 0 W 1 •ii'' i—. 8 Predicted 1 16 v v i 1 1 1 24 32 40 \ 48 Projectile Displacement (mm) (e) Figure 7.10 Comparison of the predicted and experimental results when a steel projectile with a 120° included cone angle penetrates through a 6.35mm thick GFRP laminate with a striking velocity of 211 m/s. Both the predicted data and experimental measurement show the non-zero projectile exit velocity. 213 Chapter Seven: Impact Time (ms) Ballistic Impact Time (ms) (a) 0 Applications (b) 10 20 30 40 50 60 Projectile Displacement (mm) (e) Figure 7.11 Comparison of the predicted and experimental results when a steel projectile with a 37° included cone angle completely perforates a 12.7 mm thick GFRP laminate with a striking velocity of 304 m/s. Predicted projectile exit velocity is very close to the measured one. 214 Chapter 2p>37° 2p>37° §1,6 E, 135 >. o 5. » 90 Predicted 5 8 o Experiment o O 12 CO .2, - s o m g" 45 4 - a 0.08 0.12 0.16 Impact Time (ms) Predicted 0.04 0.24 0.08 0.12 0.16 Impact Time (ms) 15 / \ j 2P,=37 Experiment Predicted — 150 — 12 z c o s S a < 4 0.24 200 2P„=37° ^ 0.2 (b) (a) o P Applications Experiment if o Ballistic 180 20 S Seven: Predicted • // 0.04 100 ' Experiment 0.08 0.12 0.16 Impact Time (ms) 0.2 4 0.24 8 12 16 Projectile Displacement (mm) 20 (d) (c) 97-D-1205-2 8 12 Projectile Displacement (mm) (e) Figure 7.12 Comparison of the predicted and experimental results when a steel projectile with a 37° included cone angle impacts a 12.7 mm thick GFRP laminate with a striking velocity of 170 m/s. Both the predicted data and experimental observation show that the projectile finally rests in the target. 215 Chapter 60 Seven: Ballistic Applications 300 2p„=120° 2p\,=120° E 225 • Predicted o » 150 75 - Experiment 0.06 0.12 0.18 0.24 0.06 0.3 0.12 0.18 0.24 0.3 Impact Time (ms) Impact Time (ms) (b) (a) 18 27 36 Projectile Displacement (mm) (e) Figure 7.13 Comparison of the predicted and experimental results when a 120° steel projectile penetrates through a 12.7 mm thick GFRP laminate with a velocity of 278 m/s. The predicted projectile residual velocity is 134 m/s. Experimental observation shows the projectile has penetrated through the target but fails to catch its exit velocity. 216 Chapter 0 6 12 18 24 Seven: 30 Ballistic 36 Applications 42 Projectile Displacement (mm) (e) Figure 7.14 Comparison of the predicted and experimental results when a steel projectile with a 37° included cone angle impacts a 19.05 mm thick GFRP laminate with a striking velocity of 318 m/s. Both the predicted data and experimental observation show the partially perforation in the target. 217 Chapter 21 r E 18 c 15 § Experiment Applications 2P„=37° v — Experiment E 180 > 0) 12 oc c b Ballistic 240 2p\,=37° E a. Seven: • Predicted 9- a •— 6 a> 0 3 a g" / a. Predicted — a. 0 0 - 60 0.02 i i i 0.04 0.06 0.08 0.1 0.12 0.14 0.02 1 1 0.04 0.06 0.08 0.1 0.12 0.14 Impact Time (ms) Impact Time (ms) (b) (a) 300 2p\,=37° Predicted ~225 c 1 / - - o s. o a 150 Experiment < 1 c 75 LLI 0.02 0.04 0.06 0.08 0.1 0.12 4 0.14 8 12 16 2C Projectile Displacement (mm) Impact Time (ms) (d) (C) 30 97-D-2403-1 Predicted Experiment 6 9 12 15 Projectile Displacement (mm) (e) Figure 7.15 Comparison of the predicted and experimental results when a steel projectile with a 37° included cone angle impacts a 19.05 mm thick GFRP laminate with a striking velocity of 211 m/s. Both the predicted data and experimental observation indicate that the projectile finally rests in the target. 218 Chapter 50 Seven: Ballistic Applications 400 2P„=120° 2p =120° 0 E 300 0 E • Predicted OJ o S b 01 1 Predicted 30 Experiment o> • Experiment 15 o 100 a <li 'o a 0.05 0.1 ^ ' 0.15 0 0.2 0.05 0.1 0.15 0.2 Impact Time (ms) Impact Time (ms) (b) (a) 600 40 2p„=120° Experiment Experiment —^ 3, co / 450 • - Predicted s o (0 JO < 300 - EX 0> c 150 • LU 2p„=120° 0 • 0.05 0.1 0.15 1 1 L_ 15 30 45 60 Projectile Displacement (mm) Impact Time (ms) (d) (C) 35 30 8 — \7 \ y Bi / 2 5 £ 97-D-1505-1 V 20 \ i o Experiment i LL CO Q. I 10 I •i 0 | ^ / 10 20 30 Predicted J . 40 P . 50 60 Projectile Displacement (mm) (e) Figure 7.16 Comparison of the predicted and experimental results when a 120° steel projectile penetrates through a 19.05 mm thick GFRP laminate with a velocity of 384 m/s. The predicted projectile residual velocity is 243 m/s, comparing with the measured residual velocity of 160 m/s. 219 Chapter Seven: Ballistic Applications Figure 7.19 Predicted peak impact force when 0.32 kg steel projectiles with various included cone angles penetrate through a 6.15 mm thick CFRP laminate with a striking velocity of 30.2 m/s. Projectile Displacement (ms) Figure 7.20 Predicted family of impact force versus projectile displacement curves when 4.2 g steel projectiles with various included cone angles penetrate through a 6.35 mm thick GFRP laminate with a striking velocity of 600 m/s. 220 Chapter Seven: Ballistic Applications 700 0 I i i i 0 20 40 60 i i i i i I 80 100 120 140 160 180 Cone Angle (degree) Figure 7.21 Predicted impact energy absorption (perforation energy) when 4.2 g steel projectiles with various included cone angles penetrate through a 6.35 mm thick GFRP laminate with a striking velocity of 600 m/s. 20 „ z 16 LL O CO Q. E 8 I Predicted peak impact force ro d> Q. o Peak contact force in static tests 20 40 60 80 100 120 140 160 180 Cone Angle (degree) Figure 7.22 Predicted peak impact force when 4.2 g steel projectiles with various included cone angles penetrate through a 6.35 mm thick GFRP laminate with a striking velocity of 600 m/s. 221 Chapter Seven: Ballistic Applications 80 0 16 32 48 64 80 Projectile Striking Velocity V (m/s) s Figure 7.23 Predicted projectile residual versus striking velocity curve for a 0.32 kg projectile with a 37° cone angle penetrating through a 6.15 mm thick CFRP laminate. The experiment data from a gas gun test is also shown here. Figure 7.24 Predicted projectile residual versus striking velocity curves for 4.2 g projectiles with 37° and 180° cone angles penetrating through a 6.15mm thick CFRP laminate. The experiment data from gas gun tests are also shown here. 222 Chapter Figure 7.25 Seven: Ballistic Applications Predicted projectile residual versus striking velocity curves for 4.2 g projectiles with 15°, 37°, 120°, and 180° cone angles penetrating through a 6.15mm thick CFRP laminate. Impact Energy (J) Figure 7.26 Predicted projectile residual versus striking energy for 4.2 g projectiles with 37° and 180° cone angles penetrating through a 6.15 mm thick CFRP laminate. The experimental data are also shown in the graph. 223 Chapter Seven: Ballistic Applications 550 Impact Energy (J) Figure 7.27 Predicted projectile residual versus striking energy curves for 4.2 g projectiles with 15°, 37°, 120°, and 180° cone angles penetrating through a 6.15mm thick CFRP laminate. 224 Chapter Figure 7.28 Seven: Ballistic Applications Schematic illustration of three parts of impact energy absorption on forcedisplacement curves: in phase I, energy dissipates in the form of hole expansion; in phase II, energy dissipates in the form of: (a) hole expansion and flexural deformation of a split-plate or (b) transverse plug initiation an pushout; in phase III, energy dissipates in the form of friction. 225 Chapter 320 Projectile Residual Kinetic Energy 15° Projectile 280 *-i Phase 111: Friction . 150 c o o 200 e-120 o 160 o (A •Q < |> 120 o UJ Applications Projectile Residual Kinetic Energy Phase III: Friction C .2 Ballistic 3 7 ° Projectile — 240 'fL Seven: < Phase II: Flexural Deflection of Split-Plate and Hole Expansion 80 HI 90 60 \ Phase I: Hole Expansion \. > ^ y< Phase II: Flexural Deflection of Split-Plate and Hole Expansion / 40 0 120 160 200 240 60 Projectile Striking Energy (J) 90 120 Perforation Energy 150 Projectile Striking Energy (J) (a) (b) 150 . 1 2 0 ° Projectile l| 120 90 60 30 Projectile Residual Kinetic Energy Phase III: Friction Projectile Residual Kinetic Energy Phase I: Hole Expansion X/Perforation Energy Phase II: Plug Initiation and Push-Out Phase II: Plug Initiation and Push-Out \ 0 60 90 120 Projectile Striking Energy (J) (c) 60 90 120 150 Projectile Striking Energy (J) (d) Figure 7.29 Composition of impact energy absorption of a 6.15 mm thick CFRP laminate perforated by a 4.2 g projectile with various cone angles: (a) 15°, (b) 37°, (c) 120°, (d) 180°. 226 Ch apter Seven: Ballistic Applications {mis 450 - o o cu > 300 - Resiidual 600 150 - 100 200 300 400 500 600 Striking Velocity (m/s) Figure 7.30 Predicted projectile residual versus striking velocity curves for 4.2 g projectiles with 15°, 37°, 120°, and 180° cone angles penetrating through a 6.35 mm thick GFRP laminate. 900 750 L 3, >. 600 - O) (5 c LU 450 - « 3 T3 '55 CD or 300 - 150 100 200 300 400 500 600 700 800 900 Striking Energy (J) Figure 7.31 Predicted projectile residual versus striking energy for 4.2 g projectiles with 15°, 37°, 120°, and 180° cone angles penetrating through a 6.35 mm thick GFRP laminate. The results of powder gun tests are also shown in the figure. 227 Chapter Seven: Ballistic Applications y 3 7 ° Projectile 450 Phase III: Friction 375 \ Perforation Energy - 300 225 ^^Projectile Residual y<T Kinetic Energy > Global Deformation Energy Phase II: Flexural Deflection of Split-Plate and Hole Expansion 150 75 0 200 300 400 500 600 Projectile Striking Energy (J) 700 150 225 300 375 Projectile Striking Energy (J) (a) (b) (c) (d) Figure 7.32 Composition of impact energy absorption of a 6.35 mm thick GFRP laminate perforated by a projectile with various cone angles: (a) 15°, (b) 37°, (c) 120°, (d) 180°. 228 Chapter 0 50 100 150 200 250 Seven: 300 Ballistic 350 Applications 400 Striking Velocity (m/s) Figure 7.33 Predicted projectile residual versus striking velocity curves for 13.2 g projectiles with 37° and 120° cone angles penetrating through a 12.7 mm thick GFRP laminate. The experimental result in the powder gun test is also shown in the graph. Striking Energy (J) Figure 7.34 Predicted projectile residual versus striking energy for 13.2 g projectiles with 37° and 120° cone angles penetrating through a 12.7 mm thick GFRP laminate. The experimental results in powder gun test are also shown in the figure. 229 Chapter Figure 7.35 Seven: Ballistic Applications Composition of impact energy absorption of a 12.7 mm thick GFRP laminate perforated by a 13.2 g projectile with various cone angles: (a) 37°; (b) 120°. 230 Chapter Seven: Ballistic Applications Striking Velocity V (m/s) s Figure 7.36 Predicted projectile residual versus striking velocity curves for 13.2 g projectiles with 37° and 120° cone angles penetrating through a 19.05 mm thick GFRP laminate. The experimental results in the powder gun test are also shown in the graph. 7000 . . Striking Kinetic Energy E (J) s Figure 7.37 Predicted projectile residual versus striking energy for 13.2 g projectiles with 37° and 120° cone angles penetrating through a 19.05 mm thick GFRP laminate. The experimental results in powder gun test are also shown in the figure. 231 Chapter Seven: Ballistic Applications Projectile Striking Energy (J) (a) Figure 7.38 Composition of impact energy absorption of a 19.05 mm thick GFRP laminate perforated by a 13.2 g projectile with various cone angles: (a) 37°; (b) 120°. 232 233 Chapter Seven: Ballistic Applications 700 C F R P Laminate G F R P Laminate 37 180 120 Projectile Cone Angle (degree) Figure 7.39 Comparison between the predicted ballistic resistance (perforation energy) of a CFRP laminate (6.15 mm thick) and a GFRP laminate (6.35 mm thick). The projectiles used are defined in Table 7.1. 0 10 20 30 40 50 60 70 Projectile Displacement (mm) Figure 7.40 Predicted force-displacement curves for a 13.2 g projectile with a 37° cone angle penetrating through 6.35 mm, 12.7 mm, and 19.05 mm thick GFRP laminates with a velocity of 384 m/s. 234 Chapter Seven: Ballistic Applications Figure 7.43 Predicted force-displacement curves for a 0.32 kg projectile with a 37° cone angle penetrating a 6.15 mm thick CFRP laminate with different planar dimensions. The projectile velocity is 30.2 m/s. Figure 7.44 Predicted ratio of global deformation and motion energy to target perforation energy versus the projectile striking velocity for a 0.32 kg projectile with a 37° cone angle penetrating a 6.15 mm thick CFRP laminate with different planar dimensions. 235 Chapter 1.5 Seven: Ballistic Applications 2 Time (ms) Figure 7.45 Predicted force-time histories for projectiles with three different masses penetrating a 6.35 mm thick GFRP laminate. All projectiles have the same cone angle of 37° and the same striking energy of 343 J. 5 -2 Projectile Displacement (mm) Figure 7.46 Predicted family of target deflection versus projectile displacement at the impact site for projectiles with different masses penetrating a 6.35 mm thick GFRP laminate. All projectiles have the same cone angle of 37° and the same striking energy of 343 J. 236 Chapter Seven: Ballistic Applications 250 Projectile Displacement (mm) Figure 7.47 Predicted family of impact energy absorption versus projectile displacement at the impact site for projectiles with different masses penetrating a 6.35 mm thick GFRP laminate. All projectiles have the same cone angle of 37° and the same striking energy of 343 J. 237 Chapter Eight: Summary, Conclusions, and Future Work Chapter Eight: Summary, Conclusions, and Future Work 8.1. Summary A numerical model has been developed in the present thesis for non-linear structural and impact analyses of plain and stiffened laminated composite structures. A super finite element program, SENACS, that was previously developed for transient nonlinear analysis of isotropic structures, has been modified to handle the response of layered (laminated) composite materials. Material non-linearity is accounted for through implementation of an existing anisotropic yield condition and associated hardening rule. The von-Karman large deflection assumptions have been incorporated to model the geometric non-linearities. The structural analysis capabilities of the code have been demonstrated by successfully comparing the predictions with other experimental, analytical, and numerical results in the literature. Impact problems have been subdivided into two groups: nonpenetrating and penetrating. In each case, appropriate contact laws are introduced to evaluate the local impact load on the structure. In nonpenetrating impact events, where there is only indentation in the targets, the Hertzian contact law has been employed to establish the impact force as a function of the local indentation. Predictions of the nonpenetrating impact response of plain and stiffened laminated composite plate and shell structures have been found to be in good agreement with 238 Chapter Eight: Summary, Conclusions, and Future Work the experimental measurements previously reported from the literature. No other damage modes except indentation are considered in the nonpenetrating impact problems. For penetrating impact events, typical of low mass projectile with high impact velocities, a contact law based on an engineering model of the local penetration mechanism has been developed and implemented in SENACS. The target deflection consists of two components. One is contributed by the global response and is calculated using the structural analysis part of the code. The other is contributed by the local response of the target and is calculated using the engineering-based penetration model developed in the thesis. The penetration model considers the following static penetration mechanisms: hole expansion, flexural deformation of a delaminated plate (split-plate), transverse plugging, and friction. It also accounts for the transitions between these mechanisms for varying geometric quantities such as size of the target and the included cone angle of projectiles with conical nose shapes. Two different material systems, IM7/8551-7 Carbon Fibre-Reinforced Polymer (CFRP) and S2glass/phenolic resin Glass Fibre-Reinforced Polymer (GFRP) laminates provide the physical background and experimental verifications for the present model. For both GFRP and CFRP, the predicted impact energy absorption agrees fairly well with the experimental measurements. The discrepancies in the penetration force between the model prediction and experimental measurement in some cases results from two sources. • First, the present model does not account for coexistence of transverse plugging and delaminations (split-plate). Experimental observations show that projectiles (indenters) 239 Chapter Eight: Summary, Conclusions, and Future Work with large cone angles will trigger delaminations in the laminates prior to transverse plugging. The present model predicts the delamination initiation in the laminates for small cone angle projectiles but fails to predict delaminations for large cone angle (120° and 180°) projectiles (see Figure 6.12, Figure 6.14, and Figure 7.7). • Second, the "measured penetration force" in the ballistic tests on GFRP laminates (Figures 7.9 to 7.16) is not measured directly from the experiments. They are calculated after the experiments by double-differentiating the ELVS measured projectile displacement time history. Successive differentiation deteriorates the accuracy of the results. Therefore, the measured penetration force from these tests has more inherent scatter that has to be known in mind when they are compared with the predictions. 8.2. Conclusions The present model provides a useful and computationally efficient tool for analyses of penetrating/non-penetrating impact and transient nonlinear structural responses of laminated composites. Through extensive numerical parametric studies and verifications on the structural and impact response of composite structures, the following conclusions can be drawn: • The present non-linear finite element formulation based upon super elements is efficient in accurately predicting the non-linear (contact/impact load, elasto-plastic material, and large deflections) structural response of plain and stiffened composite plates and shells. 240 Chapter Eight: Summary, Conclusions, and Future Work • The hole expansion mechanism, where the target material is crushed and pushed aside by a conical projectile (indenter); flexural deformation of a delaminated target (split-plate model); and transverse plugging (shear failure) are the three major penetration mechanisms that have been identified from experimental studies. These mechanisms capture the nature and essence of local deformation state and penetration damage development of composite materials. Thus they are very useful in building the present penetration model. • The ballistic limit of composite targets predicted by the present penetration model compares very well with the experimental measurements, where the error is generally less than 10%. Whether the target is "perforated" or "not perforated" as predicted by the present model is shown to be consistent with the experimental observations. Although the same accuracy in the ballistic limit prediction has already been attained by some authors (Cantwell et al. 1990, Zhu et al. 1992b, and Sun et al. 1996), the ability to predict the ballistic limit for various geometric conditions of the projectile and the target (e. g., nose shape of a conical projectile, target thickness, etc.) and type of composite material (e. g., CFRP and GFRP laminate) has not been demonstrated by these models. The model presented here aims to fill this gap. • The cone angle of a conical projectile plays a major role in whether or not it will penetrate a given target. For CFRP and GFRP laminates studied in the present thesis, a projectile with an obtuse (flat) cone angle generally dissipates less impact energy to penetrate through the laminates than that with an acute angle. 241 Chapter Eight: Summary, Conclusions, and Future Work • The global response has a minimal effect on the ballistic response when a target is impacted by a low mass projectile with a very high impact velocity. The impact energy absorbed in global deformation and motion is also small in the case where a target has a small planar size. 8.3. Future Work Although the present penetration model can represent the local impact behaviour of composite structures fairly accurately, the penetration model is still semi-empirically and based upon the experimental observations of penetration behaviour under static loading. In order to capture the dynamic characteristics of the local penetration response, further experimental investigations have to be carried out leading to model improvements. The following work is suggested for future research: • Investigate dynamic characteristics of the present penetration mechanisms (hole expansion and flexural deflection of a split-plate) and the strain-rate effects on the derived empirical parameters (hole expansion pressure, coefficient of friction, delamination initiation stress, and transverse shear strength). • Develop proper models to account for the local permanent deformation due to the gap (displacement discontinuity) between delaminated interfaces. 242 Chapter Eight: Summary, Conclusions, and Future Work • Conduct penetration tests statically and dynamically using projectiles (or indenters) with a wide range of included cone angles (15°, 60°, 90°, and 150°) and targets with different thicknesses to verify the conclusions derived from the present penetration model. • In the present model (see Figure 6.1), either delamination or transverse plugging initiates when a certain criterion is met. 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B., and Curren, D.R., Impact Dynamics, John Wiley & Sons, 1982. 254 Appendix A: Shape Functions for Shell and Beam Elements Appendix A: Shape Functions for Super Shell and Beam Elements A.l Super Shell Element The matrix of shape functions for a super shell element, which is denoted as [N] in Equation (2.1), is given by 0 0 0 N; 0 0 0 N? 0 0 N; 0 0 0 N; 0 0 N W iV N W • 16 0 0 iV 13 0 0 N V 0 N: N; ZV JV 0 0 JV 10 15 0 0 0 0 0 0 N; N; N 0 0 0 0 0 0 W 0 0 2 0 0 0 0 JV 0 N" 12 N 0 0 W 2 0 0 N; 0 0 K 0 N; 0 0 0 0 N; 0 N W 0 0 N; o N; N; 0 0 0 0 0 K N; 0 0 0 0 0 0 0 0 0 0 N W (A.1) 0 0 0 N" JV 10 16 0 0 0 0 0 0 0 JV 0 0 0 0 0 iv 0 0 0 0 N N JV JV W N V 0 N" JV ] 3 0 16 0 N; 0 0 0 0 0 0 0 0 0 N" 0 N" 14 0 2 5 0 0 JV 14 0 0 JV 0 0 0 N W 0 0 N W - 24 /V 22 0 0 iv 15 0 0 N" iV W 21 N" ] 2 17 0 AT iN" v 0" Nn J v V 0 255 Appendix A: Shape Functions for Shell and Beam Elements The element nodal variable vector {S } is defined as e V w si ,y2 U 2 V ,y4 9 V 8 w «S V 12 v w 4 V ,ys V ,yi w «6 7 w V V, ".0 9 U , 0 3 V 1 3 V 2 w 3 U 3 ",>4 4 W 5 V "2 W 4 W W ,*y2 ^ ,y2 W ,*2 u V ,y\ W l V 5 ,x4 W 6 W W ,y3 ,*4 (A.2) 6 8 } The shape functions N " , NJ, and NJ in Equation (A.l) are explicitly given by /V," = 1,(^(17) N; = L (&H {ri) N; = L\(Z)H {rj) 1 1 3 N = A(^)// (r7) Nl = L (^)H (n) N" = Nl = L.i&H^n) Nl = L (&L (T ) u 2 2 L (£)H (T ) 2 5 I ] K N = Z (£)Z (/ ) K = Z (£Z (/ ) N" = sin(2«i*)Lj(7) N" = sin(2^)Z3(77) u 3 = 2 7 3 = H (%)L (n) N = A(0A(»7) X v /V 2 = v H^L^rf) ^ v K = 4(0/3(17) x x H (t)L(n) = H^L^ri) 3 AT = Z (0sin(2^) 1 v NJ = H ({)H (n) = 2 N = 4(£>sin(2;z77) = ^(^)sin(4^-77) I = I (04(7) iV N 1 = sin(4^)Z,(7) 2 v 3 N" = sin(2^)^(/7) = H (^)L\(ri) H^)L\(n) N; 7 x = sin(4^)Z3(/7) s i n ^ ^ T ? ) = ^,(^(7) ^; = 3 2 14 iV 15 = Z,(0sin(4^7) N = 4(0sin(4^) v iV = w 2 H ($)H (ri) 2 x tf w 3 = H ({)H (n) x 2 256 Appendix N; 16= 19= V I V J Shell and Beam = H (Z)H {rj) H.(QH (T,) = H ($H {n) Ng = H^)H (r ) -M ' I= H.(&H (tj) N = H (QH {rj) = = for N: N? = w Functions H^)H (r ) H (QH (TJ) JV Shape = = N 10 A: 2 2 N; 2 3 1 H^)H (rj) 3 H (?>H (n) 2 4 N w x 4 w iy22 H.{QH.{fj) A N w 2 X = N 4 H^H^TJ) w 1 2 17= H (QftTJ) = H^MTJ) N 2?-$ 4(^ = 4^-^) V J H.(?Mn) x 12 15= <KZ)H {rj) 18 21= K&H {rj) H {QH.{TJ) = </>(Z)H (T ) w A V I = N 2 4 14 V I 1 N w N w Elements V J 4 23 3 w V J N w JV 25 The quadratic Lagrange polynomials are A(£) = 2<f-3cf+l L (£) = 2 (A.6) The cubic Hermitian polynomials are Htf) = 1 -3<f +2? H {£) = + 2 (A.7) H {$ = 3 3?-2f The <j> function is given by «(sinh fit; - s i n + ( c o s h - cos/?£) (A.8) where sin—) +(cosh—2' 2 v a = and ^ = 4.7300407448 257 Appendix A: Shape Functions for Shell and Beam Elements A.2 Super Beam Element The matrix of shape functions for a super shell element, which is denoted as [/V] in Equation (2.2), is given by 4(£) 0 0 0 0 0 0 eH \t) 0 0 0 H (4) 0 0 0 0 0 H ($ x H (g) H {$) 0 H {$ 0 eH ($ eH ({) L\{g) 0 eH \t) 3 0 eHl{$) 0 0 0 0 H ($ 3 0 H ({) 0 0 0 0 0 0 0 H ($ Hj£) 0 tf (£) 0 0 efl^) eff (5) _ 0 Zg(^) eH \§ x x 3 e<p'{%) 0 2 2 2 4 H ($ 4 sin2^ 3 2 4 sin4^ " 0 fa) 0 0 0 0 0 </{£) 0 0 0 0 0 0 e<f{%) x 2 The element nodal variable vector \8 \ is defined as e {S } ={w, e ,*2 V w v xX w 2 &,x2 "3 v, W*2 x e xX 6 X W e (A.10) 3 258 Appendix B: Elastic and Plastic Constitutive Matrices Appendix B: Elastic and Plastic Constitutive Matrices The elastic constitutive matrix is written in the following form (Vaziri et al., 1992) 0 0 [Qe] 0 0 (B.l) G where E and E are the elastic moduli of the lamina in the longitudinal and lateral directions; v and v are the major and minor Poisson's ratios of the lamina, G is the elastic in-plane shear modulus. ] 2 n 21 The constitutive matrices for anisotropic yielding and hardening is written in the following form (Vaziri etal, 1992) (B.2) Where {a} is the plastic flow vector defined by A aj, i,j = 1,2,6; tt (B.3) Li, is an anisotropy parameter (ju = 1 for isotropic yielding and hardening) defined as y (B.4) 259 Appendix C: Numerical Verifications and Structural Applications of SENACS Appendix C: Numerical Verifications and Structural Applications of SENACS In this Appendix, the super finite element method and the associated computer program, SENACS, will be verified by comparing its results with available experimental, analytical, and other numerical results in the literature. At the same time, some numerical applications will be presented for future reference. These examples cover static, eigenvalue, and transient problems involving linear and non-linear geometries as well as elastic and elasto-plastic analysis of laminated composite structures. Cl. Static Problems C . l . l Small Deflection, Linear-Elastic Analysis C. 1.1.1 Laminated square plate under sinusoidally-distributedpressure load This is a typical example for elastic laminated composite plates subjected to lateral loading, which has been used extensively to verify a theoretical formulation and computer program. Consider a simply-supported graphite/epoxy laminated plate with a lay-up of [0/90/0] under transverse sinusoidal pressure load Q(x,y) = q sin(—) sin(—) a b 0 The material constants are E =25E , n 22 G = 0.5 E , 1 2 22 v = 0.25, 12 £ = 10 psi. 6 2 2 The ratio of side length to panel thickness is — = 100. The comparison of present results h with finite element and analytical results in Ochoa et al. (1992), and exact solutions in Pagano (1970) for a square plate (a = b) and a rectangular plate (b = 3a) are shown in Table C l and Table C.2, respectively, where the non-dimensional central deflection and stress components are defined as 260 Appendix C: Numerical Verifications and Structural Applications of SEN ACS _ l00E wh w= -j— qa 3 7 2 0 qa 0 In the present formulation, one shell super element is used to model the whole plate. The element mesh used in Ochoa et al. (1992) was 2x2 and 6x6 QHD40 elements (an eight-node quadrilateral plate element with 40 degrees of freedom) for a quarter plate. The analytical solutions were obtained by using the elasticity theory [Pagano (1970)] and classical lamination plate theory (CLPT) [Ochoa et al. (1992)]. The results in Table C l and Table C.2 show that the super element converges more quickly than QHD40. One super shell element solution is in good agreement with analytical solutions, and numerical results by Ochoa et al. using 4x4 or 12x12 QHD40 elements. C.l.1.2 Laminated stiffened cylindrical shell panel under a point load This example concerns the static analysis of a glass/epoxy stiffened cylindrical shell, which was first presented by Venkatesh et al. (1983) and then verified by Venkatesh et al. (1985) and Goswani et al. (1994). A layered cantilevered cylindrical shell panel reinforced with eccentrically mounted layered stiffeners is loaded by a single concentrated force at one of its free corners as shown in Figure C l . The material properties, geometric sizes, and lay-ups are given in Table C.3. Table C.4 shows a comparison between the present results and those obtained by Venkatesh et al. (1983), Venkatesh et al. (1985), and Goswani et al. (1994). The present mesh consists of 2x3 (axialxcircumferential) elements for the whole shell panel. C.1.2 Large Deflection, Linear-Elastic Analysis C. 1.2.1 Laminated square plate under uniformly-distributed pressure load Present results are also compared with experimental data and numerical results for laminated square plates with lay-ups of [0] and [0/90/90/0] in linear-elastic and large deflection analysis. The geometric sizes and material properties are shown in Table C.5. Comparing the load-deflection curves shown in Figure C.2 for a simply-supported square plate, it is evident that present analysis with moveable boundary (u *0 along the edge y perpendicular to x-axis and v, 0 along the edge perpendicular to .y-axis) is closer to the experimental data much more than that with immoveable boundary (u = 0 and y = 0 on those edges). This is because the experimental set-up generally cannot produce fully immoveable boundary as assumed in the finite element analysis. 261 Appendix C: Numerical Verifications and Structural Applications of SENACS From the comparison of load-deflection relationship for a fully-clamped plate as shown in Figure C.3, the present results, although not in good agreement with the experimental data, are very close to the finite element predictions by Ochoa et al. (1992). This is also attributed to the improper finite element representation of the actual boundary conditions in the experiment. As we expected, present results are on the stiff side of Ochoa et al. (1992)'s which take into account the transverse shear deformations. The same phenomenon occurred in Matsuhashi et al. (1993) when comparing the impact response of a clamped laminated composite plate using Rayleigh-Ritz method with experimental results. A geometrical non-linear factor fi was applied to the large deflection term due to in-plane displacements in the system of equations of motion, where fi = 0 represents a clamped-perfectly sliding boundary condition and fi = l is equivalent to a clamped-fixed boundary condition. It was found from Matsuhashi et al. (1993)'s study that fi = 0.05 would give closer results to the experiments. Putting fi = 1, which is equivalent to the boundary condition used in the present analysis, over-predicts the peak impact force and under-predicts the impact duration. C. 1.2.2 Laminated composite cylindrical shell panel under a point load A composite cylindrical shell panel with a [0 /90 /0 ] lay-up is considered here. The geometric conditions and material constants are shown in Table C.6. This shell panel is subjected to a radially-directed point load P at the centre of the panel. The super element mesh is 2x4 for the whole panel, where 2 elements are arranged along the axial direction and 4 elements along the circumferential direction. 4 4 4 The present results are successfully compared in Table C.5 with finite element solutions by Lee et al.'s (1993) which account for the transverse shear deformation and Saigal et al.' (1986) which are based on CLPT. Because the shell panel is hinged along two straight edges and free on two curved edges, compressive deformation will occur around the panel centre if a point force is applied there. The panel will buckle when the force is increased to its critical buckling load. In the present case, when the load level approaches 1.1 kN, the number of iterations in the analysis increases significantly first and then the solution diverges due to the buckling of the shell. C.1.3 Small Deflection, Elasto-Plastic Analysis C. 1.3.1 Boron/Aluminium plate under uniformly-distributed pressure load This example is concerned with a simply-supported orthotropic square plate which was analysed by Arenburg et al. (1989) and Rahman et al. (1992) using micro-mechanics based models. The material constants are shown in Table C.7. The plate dimensions are a = 0.254 m, and h = 2.54 mm. The mesh used in SENACS consists of 2x2 elements for a quarter plate 262 Appendix C: Numerical Verifications and Structural Applications of SENACS and the results agree very well with micro-mechanics models as shown in Figure C.6. In the micro-mechanics analysis by Rahman et al. (1992), 6x6x2 elements were used for a quarter plate, where 2 elements are arranged through the plate thickness. C.l.3.2 Laminated plate under a uniformly-distributed pressure load, elasto-plastic analysis This example is adopted from Owen et al. (1983) where elasto-plastic analysis of an orthotropic plate was undertaken by using the finite element method accounting for transverse shear deformation. The material constants and geometric conditions are shown in Table C.8. A comparison between the present results and numerical results of Owen et al. is shown in Figure C.7 on a plot of the non-dimensional central deflection of the plate, yw , versus 0 Ma p the non-dimensional external load, Q» f ° both isotropic and anisotropic cases, where r M denotes the plastic bending moment on the cross-section per unit length (MN-rn/m), D denotes the plate's bending rigidity, and q is the transverse pressure load (MPa). For the isotropic material considered by Owen et al. (1983), the present analysis using only one element for a quarter plate can almost represent the same results obtained by Owen et al. in 3x3 element mesh for a quarter plate. p For the anisotropic case, the load-deflection relationship for the present analysis using 3x3 elements for a quarter plate are in good agreement with that obtained by Owen et al. as shown in Figure C.7. The values used by Owen et al. for the yield stresses are given in Table C.8. C.l.3.3 Laminated plate under a uniformly-distributed pressure load, elasto-plastic analysis with either anisotropic or isotropic hardening This is the same example except that the plastic tangent modulus or initial yield stress are raised in the 1-direction while they are kept the same in the 2-direction. Then different hardening rules (isotropic and anisotropic) are used to investigate their effects on the loaddeflection relationship. As shown in Figure C.8, for high anisotropy in plastic tangent modulus, there is a big difference on the load-deflection relationship between the results using anisotropic hardening rule and those obtained using isotropic hardening rule. However, if high anisotropy comes from initial yielding, it will not cause any visible difference in the load-deflection relationship as shown in Figure C.9. C.l.3.4 Laminated cylindrical shell panel under a uniformly-distributed vertical load This example is also adopted from Owen et al. (1983) and is a classical example of the cylindrical shell roof analysed in the literature. The isotropic case was compared with the four anisotropic cases in which one material constant is varied at a time to get material anisotropy. 263 Appendix C: Numerical Verifications and Structural Applications of SENACS The cylindrical shell which is subjected to a self-weight loading and the corresponding mesh as shown in Figure C.10 (a). In Owen et al.'s results, one quarter shell was discretized by 6 elements and 6 equal layers were taken through the thickness. The material constants and geometric dimensions are shown in Table C.9. Different finite element meshes and Gauss integration points are used to compute the isotropic case. There is a small difference between the results for the same mesh but different arrangements of Gauss integration points through the thickness, such as 6x1, or 6x2. Therefore, one Gauss integration point is used through the thickness of each layer instead of two to improve the computational efficiency. Then, the different finite element meshes nxm are used, where the first number "«" is the number of finite elements in the direction of the shell axis, and the second number "m" is the number of finite elements in the circumferential direction. The results of the 2x4 mesh for the whole shell and the 2x2 mesh for a quarter shell are found to be close to the results of Owen et al. Therefore, the first mesh is chosen for the analysis. The relationships between the load and the vertical displacement at point A on the free edge are shown in Figure C.10 (b)-(e). Figure C.10 (b) shows this relationship for the isotropic case, while Figure C.10 (c)-(f) show the relationships for the four following anisotropic cases: E = 2 E ; E = 2E -^ cr = 2cr ; cr =2cr . x 0 2 0 01 0 02 0 The present results agree with Owen et al. (1983)'s very well. In the case of cr = 2cr in Figure C.10 (f), the mesh 2x2 for a quarter shell is used to approach the problem. 02 0 C.1.4 Large Deflection and Elasto-Plastic Analysis C. 1.4.1 Laminated cylindrical shell panel under a uniform vertical load This example is again adopted from Owen et al. (1983). For large deflection and elasticperfectly plastic case, the 2x2 mesh for a quarter plate is chosen in the analysis. In Figure C.10 (b) and Figure C . l l (a), isotropic and anisotropic results for the case E = 2E are compared with Owen et al.'s and they agree very well. In Figure C l 1 (b), a 2x3 mesh for a quarter shell is used to obtain load-deflection relationship for the anisotropic case cr = 2 cr . The present results agree very well with Owen et al.'s. l 0 01 0 C.2 Eigenvalue Problems C.2.1 Fundamental Frequencies of Square Orthotropic Plywood Plates 264 Appendix C: Numerical Verifications and Structural Applications of SENACS The fundamental frequencies of an orthotropic plate [0] made of plywood for different boundary conditions are shown in Table C l 1, where/is the fundamental frequency and D is the bending rigidity in the 2-direction. It is found that these results, which are calculated by using only one element for the whole plate are in good agreement with available analytical results given by Hearmon (1959) and Huffington (1959). Material constants and geometric sizes are given in Table C.10. 2 C.2.2 Natural Frequencies of Boron/Epoxy Plates The natural frequencies of [0] and [45] boron/epoxy square plates with fully-clamped boundary conditions are calculated and compared with the results by Whitney (1987). In order to take account of high-frequency and anti-symmetric vibration modes, three meshes, namely, l x l , 3x3 and 3x6 for the whole plate are considered in the present analysis, where in "nxm", the first number "«" denotes the number of elements along the 0° direction of the laminate, while the second number "w" denotes the number of elements along the 90° direction. More elements are taken along the 90° direction because the plate is more flexible in that direction. The material constants and geometric sizes are given in Table C.12 and a comparison of natural frequencies with other analytical results and experimental data are provided in Table C.13. It is expected that the results from SENACS will be a little stiffer than the analytical results. Naturalfrequenciesobtained from experiments are always lower than the analytical and numerical ones, because it is difficult to produce fully "clamped" boundary conditions in an experimental set-up. C.2.3 Simply-Supported Laminated Plate The natural frequencies for a rectangular [0/90/0] laminated plate are compared with the analytical solution by Ashton et al. (1970) and other available finite element solution by Brockman et al. (1989) in Table C.14. The plate size is a = 30 cm, 6 = 1 0 cm, h = 0.03 cm. E G The material constants are —^ = 25, —— = 0.5, v = 0.25, p = 0.0001 kg/cm , where E , 3 LT L E , G , v denote the elastic modulus in the lamina longitudinal direction and transverse direction, and in-plane shear modulus and Poisson's ratio of the lamina. All four edges are simply-supported. T LT LT First, a 1 x 1 mesh for the whole plate is used to obtain the first two naturalfrequencies.As expected, present results agree well only for the first mode of naturalfrequencyand a large error occurs in computing the second and higher modes of naturalfrequenciesbecause the one element representation is not sufficient to capture the higher modes of vibration. A more refined 4x2 mesh was used for the whole plate. The corresponding results are close to Ashton et al.'s and Brockman et al.'s. The discrepancies are due to the fact that unlike the 265 Appendix C: Numerical Verifications and Structural Applications of SENACS other two analyses transverse shear deformations were not considered in the present formulation. The normalised natural frequencies in Table C. 14 are calculated as _ b [p~h 2 OJ = CO—ir 71 —— \ D 22 C.3 Transient Problems C.3.1. Isotropic cylindrical shell roof under time-dependent uniformly-distributed vertical load Because there are not many examples on transient analysis of composite structures available in the literature for comparison with the present analysis, some examples of isotropic materials are used to verify the present approach. The transient response of an isotropic cylindrical shell roof subjected to a vertically uniformly-distributed half-sinusoidal impulsive loading are analysed using SENACS and compared with the results of Tsai et al. (1991). The geometric sizes and material constants are shown in Table C.15. First, the response of the shell roof without the inertia force for both small and large deflection cases are calculated and compared with Tsai et al. (1991). The present static results agree very well with Tsai et al.'s. The finite element mesh used in the present analysis is the same as in Figure CIO (a). Then transient analysis is performed for the same shell panel using the same mesh. The first natural frequency of the shell is 1.53 Hz. The time increment in each load step is taken to be 0.02 sec. A comparison of transient responses is shown in Figure C.12. It seems that they both exhibit the same trends and vibration periods. The small difference in the amplitudes of free vibration can be attributed to the fact that the transverse shear deformations are ignored in the present analysis. C.3.2 Isotropic cylindrical stiffened shell panel under time-dependent uniformly distributed vertical load This example is an isotropic stiffened cylindrical shell roof, which is supported by a rigid shear diaphragm at the curved ends and is loaded by a step vertical uniform pressure due to its own dead weight as shown in Figure C.13 (a-b). Its geometric size and material properties are shown in Table C.15. The 3x3 super finite element mesh was used to discretize one 266 Appendix C: Numerical Verifications and Structural Applications of SENACS quadrant of the stiffened panel, which involves nine shell and three beam elements. The fundamental frequency of the stiffened shell is 1.95 Hz. A time step of 7 ms taken from Jiang et al. (1991) is used in the linear and non-linear transient analysis. The results obtained are compared with the finite strip solution by Jiang et al. (1991) and are shown in Figure C.13 (c-d). Figure C.13 (c) shows the linear transient response for vertical displacements at the central point of the shell. In Figure C.13 (d), the time histories of the same vertical displacement using large deflection analysis option are compared. It can be seen that the present results are very close to the finite strip predictions in both cases. 267 Appendix Tabled C: Numerical Verifications and Structural Applications of SENACS Comparison of normalised stress components and transverse deflection with Table 3.7-6 in Ochoa et al. (1992) for a simply-supported square graphite/epoxy plate with a lay-up of [0/90/0] subjected to transverse sinusoidal pressure load. Approach - , a a \ 2 2 ,a a h _ a a h ^(0,0,±f) Table C.2 SENACS (lxl) Ochoa (4x4) Ochoa (12x12) Pagano Elasticity CLPT 0.432 — — — 0.435 0.545 0.566 0.542 0.539 0.539 0.181 0.174 0.167 0.181 0.180 0.0230 0.0224 0.0215 0.0213 0.0213 Comparison of normalised stress components and transverse deflection with Table 3.7-8 in Ochoa et al. (1992) for a simply-supported rectangular graphite/epoxy plate with a lay-up of [0/90/0] subjected to transverse sinusoidal pressure load. Approach —,a a w(—,—) 2 2 _ a a h ^(0A±|) SENACS (lxl) Ochoa (4x4) Ochoa (12x12) Pagano Elasticity CLPT 0.504 0.507 0.509 0.508 0.503 0.630 0.657 0.628 0.624 0.623 0.0253 0.0259 0.0231 0.0253 0.0252 0.0089 0.0086 0.0084 0.0083 0.0083 268 Appendix Table C.3 C: Numerical Verifications and Structural Applications of SENACS Constants of a laminated stiffened cylindrical shell panel subjected to a point load as shown in Figure C l (Goswami et ah, 1994). Material properties skin: E = 51.6 GPa, E = 13.7 GPa, G = 8.61 GPa, v =0.25 stiffener: E= 25.85 GPa, v= 0.36 (equivalent isotropic properties) h= 10.0 mm skin [45/-45/-45/45] straight stiffener [45/30/-45/-30/45/30] curved stiffener [-45/45/-45] . shape of cross-section: rectangular web thickness: 10.0 mm web height: 60.0 mm No. of stiffeners: 2 straight, one above and the other below; 2 curved, both below the skin. x Skin thickness Lay-up 2 l2 12 x 2T 2s Stiffener size Table C.4 Static deflection of a laminated stiffened cylindrical shell panel at the loading point as shown in Figure C. 1. Source Venkatesh et al. (1983) Venkatesh et al. (1985) Goswami et al. (1994) SENACS Radial displacement (mm) 2.1100 2.0254 2.2269 2.3704 Tangential displacement (mm) — 0.4231 0.4160 0.3819 269 Appendix Table C.5 C: Numerical Verifications and Structural Applications of SENACS Constants of laminated square plates with lay-ups of [0] and [0/90/90/0] under uniform pressure load in large deflection analysis (Ochoa et al., 1992). Material Properties: E = 3xl0 psi, £ = 1.28xl0 psi, G = 0.37xl0 psi, v =0.32 Plate Dimensions: a = b= 12 in, A = 0.138 in [0/90/90/0] Material Properties: E = 1.8282xl0 psi, E = 1.9315xl0 psi, G = 0.3125xl0 psi, v =0.2395 Plate Dimensions: a = b= 12 in, h = 0.096 in [0] 6 6 x 2 6 12 n 6 6 x 2 6 12 Table C.6 Constants of a laminated cylindrical shell panel with a lay-up of [0 /90 /0 ] subjected to a point load as shown in Figure C.4 (Lee et al., 1993). 4 Material constants (MPa) Dimensions (m) Table C.7 4 4 E = E = 1100, v =0.25, G„= 660.0 X 2 12 L = 0.254, R = 2.54, h = 0.0126, </>= 0.1 rad Material constants for [0] boron/aluminum plate subjected to uniformlydistributed pressure load (Rahman et al., 1992). Units: GPa. Elastic Plastic £ , , = 210, £ = 107, v = 0.2, G = 32 E = 203, E^ = 24, G = 1.5 X = 1.21, Y =0.089, S =0.045 2 2 X2 12 TXX 0 Table C.8 12 T12 0 Q Constants of an orthotropic plate [0] pressure load (Owen et al., 1983). Isotropic (MPa) 8 subjected to uniformly-distributed Elastic #!= £ = 30000.0, v = 0.3, G = 11540.0 12 2 l2 Yield stresses <j = a = cr = 30.0, T = 17.32 Plastic £ = 300.0, G = 100.0 01 02 045 p Anisotropic (MPa) Dimensions (m) QX2 p cr = 30.0, cr = 40.0, o- = 35.0, T = 20.0 the remaining values are the same as for isotropic case a = 6.0, h = 0.2 01 02 045 0 1 2 270 Appendix Table C.9 C: Numerical Verifications and Structural Applications of SENACS Constants of an orthotropic cylindrical shell panel [0] subjected to uniform pressure load as shown in Figure C.10 (a) (Owen et al., 1983). 6 Isotropic (MPa) Anisotropic Size (m) =10500 Elastic E = E = E = 21000, v = 0.0, G Yield Stresses cr = cr = cr = 4.1, r = 2.367 0 x 2 0 n 01 02 1 2 0 1 2 Plastic £ = 0.0, G =0.0 p E =2E ; l 0 p £ =2£ ; 2 0 CX = 0 1 2<T ; cr = 2fT 0 02 0 L = R = 7.60, h = 0.076, ^ = 4 0 ° Table C.10 Properties of [0] plywood plate (Hearmon, 1959). Material constants 1870.0 ksi, E = 600.0 ksi, v =0.12 G = 159.0 ksi, p = 2.52X10- lb s /in . a = 100.0, b = 50.0, h = l.O 12 2 4 2 4 12 Geometric sizes (in) Table C. 11 Fundamental frequencies for [0] plywood plate under various boundary conditions. Boundary Condition iTtfa 1 Normalised Frequency 4D I ph 2 SCSC SSSC SSSS SFSC SSFS SFSF SENACS 94.55 68.56 48.65 26.06 20.65 17.39 Hearmon (1959) 94.57 68.53 48.65 26.22 20.70 17.42 Huffington(1959) 94.56 68.52 48.65 26.06 20.65 17.39 271 Appendix Table C. 12 C: Numerical Verifications and Structural Applications of SENACS Constants of a [0] boron/epoxy plate (Whitney, 1987). Material constants £ , = 31000.0 ksi, E = 2700.0 ksi, G = 750.0 ksi Geometric sizes (in) v = 0.28, p = 0.000192 lb-s /in a = b= 12.0, h = 0.0424 2 n 2 4 n Table C.13 Natural frequencies of [0] and [45] boron/epoxy plates. Plate [0] Mode No. M N 1 2 3 1 2 4 3 4 5 1 2 [45] Tabled4 SENACS 3x3 130 165 241 343 364 362 413 503 523 665 689 110 184 256 270 373 1 1 1 2 2 1 2 2 1 3 3 1 2 3 4 5 Natural Frequencies (Hz) SENACS Analytical Experiment 6x3 130 125 107 165 159 123 241 232 204 342 329 233 363 350 301 363 343 343 413 397 360 505 482 451 525 490 475 664 639 555 681 655 545 110 106 92 186 178 156 254 244 215 276 264 239 377 359 329 Normalised natural frequencies for a rectangular simply-supported laminated plate with a lay-up of [0/90/0]. Mode M N Exact 1 2 3 4 5 1 2 1 3 2 1 1 2 1 2 1.415 2.626 4.420 4.622 5.659 Brockman (15x5) 1.473 2.733 4.864 5.056 6.358 SENACS (4x2) 1.569 2.774 5.623 5.269 6.249 272 Appendix Table C.15 C: Numerical Verifications and Structural Applications of SENACS Constants for the isotropic cylindrical shell panel shown in Figure C.4. Shell Weight Elastic Constants Geometry Size Mass Density 90 lb/sq.ft. £ = 3.0xl0 psi, v=0.0 600 in, R = 300 in, A = 3.0 in, (/>-- 20° W 6 p = — = imibs ft-* 2 hg 273 Appendix C: Numerical Verifications and Structural Applications of SENACS 10 Kg Figure C l A cantilever laminated composite cylindrical shell with eccentrically mounted composite stiffeners in axial and circumferential directions (Venkatesh et al, 1983). Central Deflection of the Plate (in) Figure C.2 Comparison of the present relationship of load and deflection with experimental data in Zaghloul et al. (1975) for large deflection analysis of a simply-supported plate [0] with in-plane moveable or immoveable boundary conditions. 274 Appendix C: Numerical Verifications and Structural Applications of SENACS Central Deflection of the Plate (in) Figure C.3 Comparison of the present relationship of load and deflection with experimental data in Zaghloul et al. (1975) and finite element results in Ochoa et al. (1992) for large deflection analysis of an fully-clamped plate [0/90/90/0]. Figure C.4 Schematic illustration of laminated composite cylindrical shell panel with a layup of [0 /90 /0 ] subjected to a point load (Lee et al., 1993). 4 4 4 275 Appendix C: Numerical Verifications and Structural Applications of SENACS Central Deflection (mm) Figure C.5 Comparison of the present results with other finite element results by Lee et al.'s (1993) and Saigal et al.'s (1986) for linear-elastic material with large deflection analysis of laminated composite cylindrical shell with lay-up of [0 /90 /0 ]. 4 4 4 Central Deflection (in) Figure C.6 Comparison of the present results with predictions of micro-mechanics models in Arenburg et al. (1989) and Rahman et al. (1992) for elasto-plastic material in small deflection analysis. 276 Appendix C: Numerical Verifications and Structural Applications of SENACS 7 Nondimensional Central Deflection of the Plate Figure C.7 Comparison of relationship of non-dimensional load vs. non-dimensional deflection between the present analysis and Owen et al. (1983) for a fullyclamped laminated square plate [0 ] in isotropic or anisotropic elasto-plastic material and small deflection analysis. 8 277 Appendix C: Numerical Verifications and Structural Applications of SENACS 8 0 10 20 30 40 50 Nondimensional Central Deflection of the Plate Figure C.8 Comparison of the predicted load-deflection relationships using isotropic and anisotropic hardening rule in the case of high anisotropy in plastic tangent modulus. 8 , 0 . 2 4 6 8 10 12 14 Nondimensional Central Deflection of the Plate Figure C.9 Comparison of the predicted load-deflection relationships using isotropic and anisotropic hardening rule in the case of high anisotropy in initial yielding stress. 278 Appendix C: Numerical Verifications and Structural Applications of SENACS m Isotropic Case • —-—* SENACS, 2x2fora quarter shell, Large deflection • Owen et al 1983, Large deflection SENACS, 2x4 for a whole shell, Small deflection • Owen et al 1983, Small deflection 0.03 0.06 0.09 0.12 0.15 0.18 Vertical Displacement at Midside of Free Edge (m) (a) (b) 4 o o 3 e 3 Anisotropic Case E = 2Eo m 2 3 2.5 2.5 <A t 3.5 Anisotropic Case E ^ E o 5 2 sr Q. < i i 0 u 2 SENACS, 2x4 for a whole shell. Small deflection Z Z 2 p 1 . 5 1 1 § Owen et al 1983, Small deflection v 1.5 SENACS, 2x4 for a whole shell, Small deflection • Owen et al 1983, Small deflection / 1 / 0.5 0.5 0 0.02 0 0.04 0.06 0.08 0.1 0.12 0.02 Vertical Displacement at Midside of Free Edge (m) 0.04 0.06 0.08 0.1 , (c) (d) 3.5 Anisotropic Case 3 0.12 Vertical Displacement at Midside of Free Edge (m) •• Anisotropic Case 2.8 <TQ2 = 2 CTQ - r • 2 £ ST 2.1 N -A S2 32 SENACS, 2x4 for a whole shell, Small deflection • Owen etal 1983, Small deflection 0.02 0.04 0.06 0.08 £ 0.1 — SENACS, 2x2 for a quarter shell. Small deflection •£ * 1.4 > • Owen etal 1983, Small deflection 0.7 0.12 0.02 Vertical Displacement at Midside of Free Edge (m) 0.04 0.06 0.08 0.1 0.12 Vertical Displacement at Midside of Free Edge (cm) (e) CD Figure C. 10 Comparison of the present results with Owen et al. (1983) for laminated cylindrical shell panel [0 ] in elasto-plastic and large deflection analysis, (a). Schematic illustration of cylindrical shell panel, (b-f). Relationship of load vs. downward deflection at midside of free edge for one isotropic and four anisotropic cases of E =2E , E =2E , a =2a , a =2a . 6 X 0 2 0 } 0 2 0 279 Appendix C: 0 Numerical Verifications 2 4 and Structural 6 8 Applications 10 of SENACS 12 Vertical Displacement at Midside of Free Edge (cm) (b) Figure C. 11 Comparison of the present results with Owen et al. (1983) for laminated cylindrical shell panel [0 ] in elasto-plastic and large deflection analysis: (a) £,=2£ , (b) ai=2o- . 6 0 0 280 Appendix C: Numerical Verifications and Structural Applications of SENACS 0.8 -SENACS, 3x3 for a quarter shell - Tsai et al 1991 Clough etal 1971 1.5 1.8 Time (s) (a) 0.6 SENACS, 3x3 for a quarter shell — Tsai etal 1991 -0.4 Time (s) (b) Figure C.12 Comparison of the present deflection-time history with Tsai et al. (1991) for transient response of an isotropic cylindrical shell roof subjected to a uniformlydistributed vertical half-sinusoidal impulsive load in linear-elastic and (a) small deflection and (b) large deflection analyses. 281 Appendix C: Numerical (c) Verifications and Structural Applications of SENACS (d) Figure C.13 Comparison of the present results with finite strip analysis by Jiang et al. (1991) for transient response of an isotropic stiffened cylindrical shell panel, (a-b) Schematic illustration of the stiffened shell panel, (c) Results in small deflection and linear-elastic analysis, (d) Results in large deflection and linearelastic analysis. 282 Appendix D: High Frequency Oscillations in Contact Force-Time Histories Appendix D: High Frequency Oscillations in Contact Force-Time Histories ^ A two degrees of freedom, spring-mass model was proposed by Bucinell et al. (1991) to investigate the response of composite plates subjected to transverse impact. The model consists of a projectile and a target plate. The projectile mass is m and the effective mass of the target plate is m . The representation of the model is illustrated in Figure D. 1. x 2 The first type of displacement was the indentation a, which was associated with the contact spring stiffness k and the natural circular frequency of the contact spring &>,. The second one is the transverse displacement of the plate, associated with the structural stiffness of the plate &2 and fundamental circular frequency of the plate <»>. x By solving the equations of motion of the two-degree-of-freedom system, the natural frequencies of the system can be written as (k +k )m +k m +^J[(k x 2 i l 2 l + k )m + k^] 2 -4k k m m { l 2 l 2 (D.l) 2/w7n 1 2 where o\ corresponds to the vibration mode of indentation a, and <% corresponds to the transverse vibration mode of the plate. The non-penetrating impact responses of T800H/3900-2 CFRP plates with [45/90/-45/0] lay-up using present approach with and without artificial damping are shown in Figure D.2 Figure D.5. The same results with artificial damping are also shown in the Figures 3.4-3.7. 3s The periods of high frequency oscillations in the force time history are shown in the graphs. These periods are compared with the periods calculated from in Equation (D.l) and shown in the Table D.l. The fairly agreement between the results indicate that these superposed secondary oscillations on the force-time curves are a reflection of the plate vibration. Any other efforts, such as refining the finite element mesh, using a patch load instead of a point load, or decreasing the time step, does not diminish the high-frequency oscillations. Only introducing artificial damping can decrease their magnitudes or make them disappear. The energy absorption of the target during the projectile impact can also be estimated from the spring-mass model of Bucinell et al. (1991). The solution to the 2DOF spring-mass system is also given in Bucinell et al. 's work, x (t) = V (Aj sin ojjt + A sin co t) x s n u x (t) = V (XjAj sin oj,t + X A 2 s n n sin co t) (D.2) n 283 Appendix D: High Frequency Oscillations in Contact Force-Time Histories where V is the projectile impact velocity, A and A are the amplitudes of the vibration, and X and X are the mode shapes of the system. They can be written as follows s x x n n Aj= ± ( l A x ) ( 0 < - J = n (D.3-1) a (D.3-2) n X =(\- ^)co r (D.4-1) 2 1 1 Xn^d-^col (D.4-2) The energy absorbed in the global deformation and motion of the target can be written as E where x 2max global =^ 2^l Lax m (D-5) X is the maximum deflection of the target. Ignoring the energy absorbed by the second mode of vibration, the maximum target deflection can be derived from Equation (D.2), as * 2max = K*,A, (D.6) 1 Let £ impact denote the kinetic energy of the projectile, E jmpact , = —m V . The relationship x s between the ratio of the global energy absorption to the total impact energy, s l o b a l , and the impact frequency ratio of target to contact springs, — - , can be calculated using the above equations. The curves shown in Figure D.6 are plotted for different ratios of the projectile to target mass, M = — . In the calculation, m and k are kept constant. It can be seen from the graph x x CO-) that the stiffer structures (—- » 1 ) absorb less impact energy as expected. Also high mass 284 Appendix D: High Frequency Oscillations in Contact Force-Time Histories impact events ( M = —- » 1 ) lead to more impact energy being absorbed by the target than 1TI2 low mass impact. 285 Appendix Table D.l D: High Frequency Oscillations in Contact Force-Time Histories Comparison of periods of high frequency oscillations in impact force time history of T800H/3900-2 CFRP plates. Projectile Mass (kG) Projectile Velocity (m/s) Predicted Oscillation Period (ms) 6.14 6.14 0.314 0.314 1.76 2.68 7.70 11.85 0.183 0.178 0.191 0.175 Bucinell's Oscillation Period (ms) 0.149 0.139 0.247 0.137 Error % 22 28 30 28 286 Appendix Figure D.l D: High Frequency Oscillations in Contact Force-Time Histories Two degree of freedom, spring-mass system used in Bucinell et al. (1991) to investigate the nonpenetrating impact responses of composite plates. 287 Appendix D: High Frequency Oscillations in Contact Force-Time Histories 10 Time (ms) Figure D.2 Contact force history for 6.14 kg, 1.76 m/s impact on the laminate defined in Chapter Three; the predicted results with and without artificial damping are compared with drop-weight test results (Delfosse et al, 1993). 16 Time (ms) Figure D.3 Contact force history for 6.14 kg, 2.68 m/s impact on the laminate defined in Chapter Three; the predicted results with and without artificial damping are compared with drop-weight test results (Delfosse et al, 1993). 288 Appendix D: High Frequency Oscillations in Contact Force-Time Histories 12 Time (ms) Figure D.4 Contact force history for 0.314 kg, 7.70 m/s impact on the laminate defined in Chapter Three; the predicted results with and without artificial damping are compared with drop-weight test results (Delfosse et al, 1993). Figure D.5 Contact force history for 0.314 kg, 11.85 m/s impact on the laminate defined in Chapter Three; the predicted results with and without artificial damping are compared with drop-weight test results (Delfosse et al., 1993). 289 Appendix Figure D.6 D: High Frequency Oscillations in Contact Force-Time Histories Predicted relationship between energy absorbed in the deformation and motion of the target and the non-dimensional frequency ratio. 290 Appendix E: Derivation of Bending Compliance of a Clamped Circular Ring Appendix E: Derivation of Bending Compliance of a Clamped Circular Ring Consider a circular ring, fully clamped on its boundary, with an inner radius b and outer radius a as shown in Figure E.l. Suppose that there is a transverse uniformly distributed shearing force Q acting along ring's inner edge. 0 Let Q denote the transverse shearing force at radial distance r = a . Then fi = ^r = fZTtr (E.D where P = 27zbQ is the magnitude of the transversal load. 0 The differential equation of equilibrium of the circular ring according to Timoshenko and Woinowsky-Krieger ( 1 9 5 9 ) is d , dw^ v —) dr r dr dr Q 1 (E.2) 2wD D where D is the bending rigidity of the circular ring. Integrating Equation (E.2), dw P r — = —-r(21n dr Q l)--~r a STTD r 2 %nD P ... dW 2 ^dr 2 a = ^%TCD r T ( 2 1 A a r n _ a C, + 1 ) .„ _ . (E.3a) 2 2 2 w = —r (\n--\)-^r -C ln2 C - (E.3b) C " ^2 ^ r" + +Q /y~y o ( R 3 c \ ) Applying the boundary conditions of the ring yields w =0: C =— 3 + ±a 2 (E.4a) 291 Appendix dw ~dr~, r r=b E: Derivation of Bending =0 C C 2 a v dw. Circular Ring (E.4b) =0 r dr r of a Clamped %izD 2 ^,d w vdw^i d w -D(— + ) = 0: ( dr dr r dr r=b z.e., Compliance (2ln—+ l) + —f+v (2ln—l) UD a l b SnD a ^- = 0 1 L l 2 b 2 Rearranging C, — C 1-v P . b P f = ln —+ 1-v 2 2 6 4TTD 1+ v 2 a (E.4c) UD 1+ v Subtracting Equation (E.4c) from Equation (E.4b), we obtain ™ c 2 b 1 ln- + l+v a 4aD, ( a l-v 6 l+ v 2 E 5 ) ' v 2 Substituting Equation (E.5) into (E.4b), and Equation (E.6) into (E.4a), ln —+ q =- — + — — 4^D ltd), a 1-v 1 + -T6 1+v a 1 + (E.6) v / 2 1 2 2 i b 1 Q = — ,2+ -r>2 In—+ \6rcD 8aD, a 1-v 1+6 1+v 3 (E.7) ' 2 v 2 Substituting Equations (E.5-7) into (E.3b), ln- + — w= — r (m--l) + — ( r + a ) " . + (r -a ) UD a 16xD SzD, a 1-v 1 +b- 1 + v 2 2 2 P a 1 v 2 2 2 2 (E.8) 292 Appendix E: Derivation of Bending Compliance 1 2i i n b n P a of a Clamped Circular Ring 1 i In —+ a 1 + v , „>" "a 1 - v a 1+Z> 1+v 4TTZ) , 2 2 The flexural deflection at the inner edge of the ring is w\ =™-(ln^-l) '' * 8;rZ) o +^ ( b = 16TZ£> 2 + a 2 , 6 1 In —+ a -\ + v )-^ 8sD, a l - v 2 { b 2_ 2 a , ^ 1 In —+ a -l + v r. 2 ) + Pa 4aD, , a l - v v 2 6 l +v m A a 6 l+v 2 2 Therefore, the bending compliance of the circular ring is L2 / =J_( , l A _ i) + — ^ ( Z > + a ) 2 m 2 In—+ a 1+ 2 v ( 6 - a - 2a I n - ) 2 2 2 (E.9) 1 + —r b l+v 2 Eh 3 where D = r - . E is the elastic modulus, and h is the ring thickness. 12(1-v ) 2 5 293 Appendix Figure E.l E: Derivation of Bending Compliance of a Clamped Circular Ring Illustration of a circular ring fully clamped along its outer edge. A transverse uniformly distributed shearing force <2o acts along the inner edge. 294
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An engineering approach to modelling the impact and ballistic response of laminated composite structures Quan, Xiangyang 1998
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Title | An engineering approach to modelling the impact and ballistic response of laminated composite structures |
Creator |
Quan, Xiangyang |
Date Issued | 1998 |
Description | A super finite element program, SENACS, that was previously developed for transient nonlinear analysis of isotropic structures, has been modified to handle the structural response of layered (laminated) composite materials. Both material and geometric non-linearities have been taken into account in the formulation. The structural analysis capabilities of the code have been demonstrated by successfully comparing the predictions with other experimental, analytical, and numerical results in the literature. Impact problems have been subdivided into two groups: nonpenetrating and penetrating. In each case, appropriate contact laws are introduced to evaluate the local impact load on the structure while the structural analysis part of SENACS computes the target global response. In nonpenetrating impact events, where there is only elastic indentation in the targets, the Hertzian contact law has been employed to establish the impact force as a function of the local indentation. Predictions of the nonpenetrating impact response of plain and stiffened laminated composite plate and shell structures have been found to be in good agreement with the experimental measurements previously reported in the literature. Modelling of penetrating impact problems has been guided by experimental investigations. A phenomenological analytical model for static penetration of composite materials has been developed. In this model, three major penetration mechanisms have been accounted for: hole expansion, flexural deformation of a delaminated plate (split-plate), and transverse plugging. The parameters of the model have been determined through material characterisation tests. The penetration model is introduced as a local contact behaviour in SENACS, and used to predict the penetrating impact response of composites to projectiles with different conical nose shapes. Ballistic impact response of two different material systems, IM7/8551-7 Carbon Fibre-Reinforced Polymer (CFRP) and S2-glass/phenolic resin Glass Fibre- Reinforced Polymer (GFRP) laminates provides the physical background and experimental verifications for the present model. Finally, a number of numerical ballistic simulations have been carried out to investigate the influence of structural shapes and sizes, loading conditions, projectile geometry, and impact velocity on the energy absorption capability of composite materials. |
Extent | 12076223 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-07-02 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0228835 |
URI | http://hdl.handle.net/2429/9917 |
Degree |
Doctor of Philosophy - PhD |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1999-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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