A NUMERICAL INVESTIGATION OF BALLISTIC IMPACT ON TEXTILE STRUCTURES by Ali Shahkarami Noori B.Sc. (Civil Engineering), Sharif University of Technology, 1995 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Civil Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1999 © A. Shahkarami Noori, 1999 In presenting degree this at the thesis in partial University of British Columbia, freely available for reference copying of department publication this or of thesis by this his for and study. scholarly or her DE-6 (2/88) O c j - ±C the ejjd O ^ . ^ r ^ , (<=\c^<^ requirements I agree that the purposes may be It thesis for financial gain shall not The University of British Columbia Vancouver, Canada Date of for an advanced Library shall make it I further agree that permission for extensive representatives. permission. Department of fulfilment is granted by the understood be allowed that without head of my copying or my written Abstract The present work focuses on numerical investigation of impact on textile materials. An analytical model is proposed based on the theory of single yarn impact, to evaluate the energy absorbed by a fabric panel. This model is capable of evaluating the total absorbed energy and its components knowing the displacement time-history of the projectile. A numerical model is used to simulate the behaviour of a fabric panel under impact. This model approximates the fabric as an assembly of nodal masses attached to each other by means of string elements in the principal directions. A computer code is developed that is capable of modelling the impact of a blunt cylindrical projectile on rectangular fabric panels with various types of boundary conditions. The code predicts the time-histories of the nodal displacements, as well as forces in the string elements. Based on these, energy stored in the system in the form of strain energy in the strings and kinetic energy of inplane and transverse motion of the masses can also be evaluated. The numerical predictions are successfully compared with instrumented impact test results. Finally, a series of numerical experiments are performed to investigate the sensitivity of the response to different input parameters. The response is found to depend on many geometric and material parameters, emphasizing the importance of having an accurate knowledge of some of the input parameters. In particular, boundary conditions are found to affect the response significantly. It is also observed that increasing the elastic modulus or breaking strain of the yarns affects the energy absorption of the target favourably. Increasing the initial crimp strain is found to decrease the impact energy absorption, similar to increasing the gap between the layers of a multi-layer fabric. 11 Table of Contents Abstract ii Table of Contents iii List of Tables vi List of Figures vii Acknowledgements xi Chapter One: Introduction 1 1.1 Background 1.2 Scope of the work 1 4 Chapter Two: Literature Study 6 2.1 2.2 2.3 2.4 Introduction Transverse impact of a single fibre Analytical models of impact of fabrics Numerical models 6 6 9 17 Chapter Three: Analytical Model 38 3.1 3.2 3.2.1 3.2.2 3.3 3.3.1 38 38 40 41 43 45 Introduction Ballistic impact of a single yarn Theory of longitudinal impact Theory of transverse impact Analytical model for fabrics Panels with fixed boundary conditions 3.3.1.1 Phase-1 3.3.1.1.1 Strain energy 3.3.1.1.2 Transverse kinetic energy of the cone 3.3.1.1.3 In-plane kinetic energy outside the cone 3.3.1.2 Phase-II. 3.3.1.2.1 Strain energy 3.3.1.2.2 Transverse kinetic energy of the cone 3.3.1.2.3 In-plane kinetic energy outside the cone 3.3.2 Panels with free boundary conditions 45 46 48 49 49 51 52 53 55 3.3.2.1 Phase-1 3.3.2.2 Phase-II 3.3.2.2.1 Strain energy 3.3.2.2.2 Transverse kinetic energy of the cone 3.3.2.2.3 In-plane kinetic energy outside the cone 55 55 55 3.4 Results and discussion List of Symbols 60 67 iii 57 58 Chapter Four: Numerical Model 70 4.1 Introduction 4.2 Model basics 4.2.1 Fabric discretization, mass-spring model 4.2.2 Boundary and symmetry conditions 4.2.3 Projectile 4.2.4 Output 4.2.5 Time discretization 4.2.6 Material constitutive model 4.2.6.1 Visco-Elastic Material Models 4.2.6.2 High Strength Aramid Fibers 4.2.6.3 Constitutive Model 70 70 71 75 77 78 79 80 80 82 83 4.2.7 Failure model 4.3 Predictions 4.4 Verification 4.4.1 Verification with the data found in literature 4.4.2 Verification against LS-DYNA List of Symbols 84 85 86 87 87 96 Chapter Five: Results and Discussion 99 5.1 Introduction 5.2 Comparison with experimental data 5.2.1 Experimental set-up 5.2.2 Experimental results 5.2.3 Comparison between numerical and experimental results 5.2.4 Comparison between numerical and analytical results 5.3 Numerical Experiments 5.3.1 Target geometry parameters 99 99 99 101 102 104 105 106 5.3. 1. 1 Boundary conditions 5.3.1.2 Panel size 5.3.1.3 Inter-layer gap 106 107 108 5.3.2 108 Target material parameters 5.3.2.1 Elastic modulus 5.3.2.2 Viscosity 5.3.2.3 Crimp 5.3.2.4 Areal density 5.3.2.5 Failure criteria 108 109 109 110 110 5.3.3 112 Proj ectile parameters 5.3.3.1 Iso-energy impacts 5.3.3.2 Projectile radius 5.4 113 113 Conclusions 114 Chapter Six: Conclusions and Future Work 134 6.1 6.2 134 134 Introduction Summary iv 6.3 Conclusions 135 6.4 Future work 136 References 138 Appendix A: Analytical Model Implementation 141 V List of Tables Table 5-1: Material properties of Kevlar® 129 yarns [Pageau, 1997] 117 Table 5-2: Properties of Kevlar® 129 fabric [Pageau, 1997] 117 Table 5-3 Properties of the projectiles used in the impact tests 118 Table 5-4 Panel properties used in simulating experiments 118 Table 5-5 Experimental data available for different impact specifications 119 Table 5-6 Numerical test matrix used to study the sensitivity of the target response different input parameters 120 Table 5 - 7 Absorbed energy of the target after 100 ms, Vs=148 m/s vi 121 List of Figures Figure 2 - 1: Configuration of the positive half of an impacted yarn, t seconds after impact. [Smithetal., 1958] 28 Figure 2 - 2 : Wave propagation in a transversely impacted yarn [Roylance, 1977] 28 Figure 2 - 3 : Variation in transverse critical velocity due to fracture rate effects [Roylance, 1977] 29 Figure 2 - 4: Broken orthogonal yarns (BOY model) [Wilde et al., 1973] 29 Figure 2 - 5 : Impact geometry [Taylor et al., 1975] 30 Figure 2-6: Propagation front and indentation of orthogonally woven cloth [Leech et al., 1979] 30 Figure 2-7: Comparison of ballistic limit curves obtained for armours made of different basis weight fabric [Parga-Landa et al., 1995] 31 Figure 2-8: Comparison of analytical and experimental results for the 16-grain FSP [Chocron-Benloulo et al., 1997] 31 Figure 2-9: Master curve for impact induced strain at the point of impact (Reproduced from Roylance et al., 1980) 32 Figure 2-10: Amplitude of propagating reflected waves; arrows show the direction of wave [Freeston et al., 1973] 32 Figure 2-11: Predicted wave fronts for in-plane excitation [Leech et al., 1982] 33 Figure 2 - 12: Schematic diagram of the hanging fibre friction experiment [Briscoe et al., 1992] 33 Figure 2-13: Strain development prediction of a direct analysis computer program for impact velocities of 200 m/s and 300 m/s [Cunniff, 1992] 34 Figure 2-14: Plot of out-of-plane positions along the main load bearing fibre for a 9layerpanel [Tingetal., 1993] : 34 Figure 2-15: Plot of Vs vs. Vr using the single layer fabric model for different values of backup spring [Roylance et al., 1995] 35 Figure 2 - 16: Three-element visco-elastic model [Shim et al., 1995a] 35 Figure 2-17: Low velocity deformation and penetration for impact velocity of 150 m/s [Shim etal., 1995b] 36 Figure 2-18: Predicted and experimental results from simulation of normal impact [Lomov, 1996] 36 Figure 2 - 19: Breakdown and geometries of warp and fill fibres [Ting et al., 1998] 37 Figure 3 - 1: Displacement of an arbitrary differential element in a yarn under impact. ..62 Figure 3-2: Waves and their propagation in the fabric after impact vii 62 Figure 3 - 3: (a) Location of waves in a quarter-panel in phase-I, (b) Waves in the central yarn, (c) Waves in a yarn at distance x from the centre 63 Figure 3-4: Longitudinal and transverse wave location in phase-II, before the transition time 64 Figure 3-5: Configuration of waves, phase-II after the transition time 64 Figure 3-6: Deformed shape of an arbitrary yarn with free ends after the reflection of the wave 65 Figure 3-7: Energy absorbed by the target with free boundary conditions; experiment compared with the model prediction 65 Figure 3-8: Energy absorbed by the target with fixed boundary conditions; experiment compared with the model prediction 66 Figure 4 - 1: The impacted panel and symmetry conditions 89 Figure 4 - 2 : Forces applied from the neighbouring elements on an arbitrary node of the mass-string system 89 Figure 4 - 3 : Behaviour of a linear spring and a linear dashpot 90 Figure 4-4: Schematic presentation of visco-elastic material constitutive model 90 Figure 4 - 5 : Constitutive model suggested by Shim et al. [1995] 91 Figure 4-6: Tension-strain relation used for the material 91 Figure 4-7: Comparison of the failure criteria considered in the model 92 Figure 4-8: Displacement, Velocity, and Energy plots for different values of striking velocity 92 Figure 4-9: Comparison of the displacement-time and velocity-time curves for different boundary conditions 93 Figure 4-10: Deformed shape of the fabric with free boundary conditions impacted at 200 m/s, after 50 ms 93 Figure 4-11: Longitudinal and Transverse wave configuration propagating in the xdirection after 30 ms 94 Figure 4-12: Energy absorbed by one-ply Kevlar® 29 panel (Lines: TEXIM; Markers: Roylance and Wnag [1980]). (Abdel-Rahman et al. [1998]) 94 Figure 4-13: Comparison of displacement and velocity time-history of the projectile with LS-DYNA predictions, single-node impact 95 Figure 4 - 14: Comparison of displacement and velocity time-history of the projectile with LS-DYNA predictions, patch impact 95 Figure 5 - 1: The experimental fixture used to support the target (Starratt [1998]) 122 Figure 5-2: Comparison of the impact response of a single layer of Kevlar® 129 with fixed-fixed and free-free boundary conditions; a) normalized velocity vs. projectile viii displacement, b) Normalized Absorbed energy vs. projectile displacement (Cepus et al. [1999]) 122 Figure 5-3: Absorbed energy plotted against the striking energy for various boundary conditions used in experiments on a single layer of Kevlar® 129 (Cepus et al. [1999]).123 Figure 5-4: Post-mortem photographs of a panel with free-free and fixed-fixed boundary conditions (Cepus et al. [1999]) 123 Figure 5-5: Experimental data (fixed-fixed boundary condition, Vs=l 11 m/s) compared with numerical predictions for a single layer of Kevlar® 129; a) projectile velocity vs. time, b) absorbed energy vs. projectile displacement 124 Figure 5-6: Experimental data (free-free boundary condition, Vs=167 m/s) compared with numerical predictions for a single layer of Kevlar® 129 124 Figure 5-7: Comparison of experimental data and numerical predictions for nonperforating impact on a single layer of Kevlar® 129 with fixed-fixed and free-free boundary condition; a) projectile velocity vs. its displacement, b) absorbed energy vs. projectile displacement (Cepus et al. [1999]) 125 Figure 5-8: Comparison of analytical and numerical predictions for impact on a single layer of Kevlar® 129 with Vs=225 m/s; a) total absorbed energy and strain energy vs. time, b) in-plane and transverse kinetic energy vs. time 125 Figure 5-9: Effect of boundary condition on the impact response of a single layer of Kevlar® 129, Vs=148 m/s ". 126 Figure 5-10: Location of elements A and B in the modeled quarter panel of the target, used to study the strain response of the elements 126 Figure 5-11: Predicted deformed shape of a single layer of Kevlar® 129 with fixedfixed, free-free, fixed-free, and infinite boundary conditions 100 ms after impact, Vs=148 m/s 127 Figure 5-12: Impact response of a fixed-fixed single layer of Kevlar® 129 with different sizes for Vs=148 m/s 127 Figure 5-13: Studying the effect of gap between the layer on the response of 8 layers of Kevlar® 129 with fixed-free boundary conditions impacted at Vs=267 m/s 128 Figure 5-14: Investigation of the absorbed energy and longitudinal wave propagation along the central fibre for a fixed-fixed 330 mm by 330 mm single layer of Kevlar® 129, assuming different values of elastic modulus, a) absorbed energy vs. displacement, b) strain as a function of distance from the impact point at 19.5 ms 128 Figure 5-15: Effects of viscosity on the impact response of a fixed-fixed single layer of Kevlar® 129; a) absorbed energy vs. displacement of the projectile, b) strain in element A vs. time 129 Figure 5-16: Effect of crimp on the behaviour of a single layer of Kevlar® 129 impacted atVs=148m/s 129 Figure 5-17: Impact response of a single layer of Kevlar® 129 for different values of areal densities 130 ix Figure 5-18: Prediction of impact behaviour in terms of residual velocity of the projectile, for a fixed-free single layer of Kevlar® 129 with 3% and 3.5% breaking strain.130 Figure 5-19: Prediction of impact behaviour in terms of residual velocity of the projectile, for a fixed-fixed single layer of Kevlar® 129 with 3% and 3.5% breaking strain 131 Figure 5-20: Strain-time plot for element A for various impact velocities, along with the values of absorbed energy for the impact on a single layer of Kevlar® 129 131 Figure 5 - 21: Failure prediction of the model in terms of normalized absorbed energy and time to failure for impact on a fixed-fixed 210 mm by 210 mm single layer of Kevlar® 129 for ebreak=3% 132 Figure 5 - 22: Predictions for iso-energy impact events (Es= 32.5 J) on a fixed-fixed single layer of Kevlar® 129, with different projectile masses and striking velocities. ...132 Figure 5-23: Effect of projectile radius on the impact response of a single layer of Kevlar® 129 with fixed-fixed boundary conditions 133 Acknowledgements I would like to gratefully acknowledge my supervisor, Dr. Reza Vaziri, for all his encouragement and support and more importantly, his unlimited patience throughout this work. I would also like to thank my co-supervisor, Dr. Anoush Poursartip, for his attention and guidance during the life of this project. I would like to acknowledge Ms. Darlene Starratt and Mr. Elvis Cepus, who walked along me throughout this research and showed me the wonderful world of experimental work. Many thanks to Mr. Anthony Floyd for all his valuable help and guidance in the numerical work. Special thanks to Mr. Roger Bennett, who made my experimental effort even more pleasant and joyful. I would also like to thank all the members of UBC Composites Group, who created an enthusiastic environment for research, and for their unlimited support and friendship. Also, many thanks to all my friends, especially Dr. Ismail Lazoglu, for his valuable friendship and his help in writing my thesis. I would like to acknowledge Natural Sciences and Engineering Research Council of Canada (NSERC), the Defence Research Establishment Valcartier (DREV), and Pacific Safety Products (PSP) for their financial, material and technical support in this research. I would like to thank my parents, Fatemeh, and Bijan, who devoted their life for my education. Special thanks to my mother for all her emotional support when I was in need of a helping hand. Also, I would like to acknowledge my sister, Shekoufeh, and my brother, Shahab, for teaching me the meaning of sharing in life. And most importantly, I thank God, who bestowed upon me the power to seek my goals, and for all the success and accomplishments I have made in my life. xi Chapter 1 - Introduction CHAPTER ONE: 1.1 INTRODUCTION BACKGROUND Research on dynamic behaviour of textile materials under impact is relatively a new area of interest. As the need to manufacture a light and effective personal protective system grows, the need to develop an accurate and robust design tool increases. A review of the history and evolution of armour systems in time shows the important role of efficient armours in battles. The following historical review is highly influenced by an article written by H. Nickel, which superbly covers the evolution of armours from the early ages till present time. Armour is referred to any equipment of various materials used to protect a body in combat or against external threats. The earliest body armour was a wide belt to protect the abdomen. Ancient Egyptians (circa 3000 BC) modified this belt into a wraparound garment extending from armpits to knees, reinforced by quilting. Syrians (circa 1400 BC) used a sleeved shirt reinforced with bronze scales to protect charioteers, who with both hands occupied were not able to hold a shield. The scales were sewn into a backing fabric or were laced together in flexible rows of lamellae. Greeks later took over the scale and lamellar armour and developed wraparound armour of reinforced quilts with shoulder flaps. The Roman soldiers wore armours called cuirass, which were a modification of the Greek armour. Roman armour existed in three different types: with bronze scale and leather backing, made of mail (interlinked iron rings), or constructed of horizontally overlapping iron plates. By the 11 century, a knight's armour consisted of a thigh length th 1 Chapter 1 - Introduction shirt of mail with elbow length sleeves and a conical helmet with nose guard. A mail shirt contained up to 250,000 metal rings and might weigh about 11 kg. Mail offered resistance to sword cuts but was not very efficient against the points of spears or arrows and was too yielding against a heavy blow. Therefore shock breakers such as shield and padded undergarment were essential. Armours with deflecting surfaces became necessary when bolts released by the improved crossbows were introduced in the first quarter of the 14 century. The bolts were able to penetrate mail with ease. Thus, body armour made of th small plates riveted inside the surcoat was developed. In construction of armour the weight problem was crucial. Armours were supposed to give maximum protection with minimal weight. A full suit of battle armour was not to exceed about 29 kg, and was expected to provide full mobility. While a knight in full armour was almost invulnerable to pointed or edge weapons, the impact of a soft lead bullet could break an armour plate. Thus, armours began to be modified as guns and gunpowder replaced older weaponry. Thicker plates provided the required protection for the price of greatly increasing the weight of the armour. By the late 16 century, overall th protection was sacrificed in favour of partial armours. In the 18 and 19 centuries, th th cuirasses were still in use as protection against sword cuts. During World War I, a soldier's body was effectively protected inside a trench or foxhole. Steel helmets were still required as protection against shrapnel. Introduction of high-strength polymeric fibres in early 70's evolved the nature of personal armours from rigid to semi-rigid materials. DuPont de Nemours and Co. introduced Kevlar which is made of highly oriented chains of poly-paraphenylene terepthalamide. Kevlar® has found many applications in industry, from sporting goods, to 2 Chapter 1 - Introduction ballistic protections and advanced composites used in aircraft and aerospace structures. Another material highly used for ballistic purposes is Spectra developed by Allied- Signal. When woven into fabric, both Kevlar and Spectra provide superb ballistic resistance and are widely used in personal armour systems. Aside from military use, armour in form of visored helmets, bulletproof vests, and shields is indispensable in modern times for special police work. Armour has its place too in more peaceful, though hazardous, occupations and recreations. Hard hats (a form of helmet) are worn by miners and construction workers, and in contact sports with elaborate additional protection devices. Many efforts have been directed in the past to study the dynamic behaviour of textile materials under impact. Initially, most of the works were focussed on mathematical modelling of single yarns, which are the main constituents of fabrics, under transverse impact. Analytical models were developed from the knowledge obtained in the study of single yarns, to predict the behaviour of fabrics under impact. Due to the complexity of the problem for a fabric, many of the analytical solutions failed to capture the response of fabric within the required accuracy. Introduction of digital computers guided many researchers towards simulating the impact via developing numerical models. Different models were suggested, but due to lack of experimental knowledge, none of them is reasonably verified. Most of the experimental studies are limited to measuring the striking and residual velocity of the projectile, without providing any data during the impact event. Photographic investigations are also performed to visualize the deformation characteristics of fabrics during impact, which provide a limited set of data for the displacements within the impact duration. However, there has always been a desire to 3 Chapter 1 - Introduction capture the on-going response of the target and projectile deceleration during the impact. A measurement technique known as the Enhanced Laser Velocity System, ELVS, developed at the University of British Columbia [Starratt et al., 1999] has made it possible to continuously measure the displacement of the projectile while impacting the target. This method provides the time-history of the projectile displacement, which with mathematical manipulation can be transformed into projectile velocity, deceleration and in-turn contact force data. ELVS has proved to be an excellent source for generating valuable experimental data for studying the dynamic response of targets under impact and for validating numerical models. 1.2 SCOPE OF T H E WORK This research is aimed at contributing knowledge to the impact behaviour of textile materials via numerical simulations of such events. The main focus of the current study is to develop a numerical model to capture the response of impacted fabrics. The results of this research are presented as follows: 1. Chapter Two presents a comprehensive survey of the studies conducted in this area. An overview of the analytical and numerical models proposed to capture dynamic response of single fibres and fabric assemblies is provided. 2. An analytical model is presented in Chapter Three to predict the energy absorption of the target. The input to this model is the projectile displacementtime data, fabric dimensions and density, and the yarn's elastic modulus. The model's output includes the total absorbed energy of the target along with its components including the strain energy of the target, and the kinetic energy due to 4 Chapter 1 — Introduction both in-plane and transverse movement of the particles. The model's prediction of the total energy is compared with the experimental data. This model is capable of determining how each energy component contributes to the total absorbed energy. The results are later compared with the predictions of the numerical model in Chapter Four. 3. In Chapter Four, a numerical model is developed based on the approach originally suggested by Roylance et al. [1973], The code is based on discretizing the fabric into a mass-string system, and solving the equations of motion that result from the application of momentum balance to the masses. The basic assumptions and formulation of this model are presented. The results of the code are also verified by comparing with other numerical results and experimental data from the literature. 4. The numerical code developed is used to perform a series of sensitivity analysis in Chapter Five. Effects of different input parameters, both material and geometric, are investigated and the results are discussed to distinguish the influence of each parameter. These numerical experiments provide a better knowledge of the factors affecting the ballistic response of textile structures. 5. Chapter Six outlines the conclusions that are drawn from this study and the shortcomings of the current numerical model. Future studies and further modifications are suggested to overcome the deficiencies of the model in order to simulate the response of impacted fabric structures more realistically. 5 Chapter 2 - Literature Study CHAPTER TWO: 2.1 LITERATURE STUDY INTRODUCTION This chapter provides a review of the studies found in literature on analytical and numerical modelling of ballistic impact on single fibres and fabrics. In this discussion, an overview of the basics of each model is presented and their advantages and deficiencies are pointed out. 2.2 TRANSVERSE IMPACT O F A SINGLE FIBRE Since World War II, an intensive amount of effort has been devoted to studying the impact behaviour of single fibres as the basic constituents of fabrics used in personal body armours. Taylor and von Karman pioneered working in this field during the war, 1 1 and later were followed by the valuable contributions of Petterson et al. , Shultz et al. , 1 1 Wilde et al. , and Smith et al. . The work done by Smith et al. was the most prolific of all 1 1 the studies performed on this subject. Their study, presented in a series of articles ranging over a period of ten years in the fifties and sixties, contains a wealth of experimental and theoretical contributions. Smith et al. [1958] formulated the transverse impact of a single yarn and presented a closed-form mathematical solution for its response. The differential equation of motion is derived based on considering the positive half of the impacted yarn (Figure 2 - 1 ) . The differential equations are solved through a simple solution that is recapitulated here. Immediately upon impact, longitudinal strain wavelets propagate outwards from the 1 A review of these earlier studies along with the references can be found in Roylance [1977]. 6 Chapter 2 - Literature Study impact point. The outermost wavelet is an elastic strain wave, which travels at speed C, presented as: where M is the mass per unit length of the filament, T is the tension in the filament, and s is the strain. The proceeding wavelets are known as plastic strain waves, which propagate at a slower pace, C : p where e is the strain at the plastic limit of the tension-strain curve. The strain in the p filament increases to e p in the region between the elastic and the plastic wave fronts. As these strain waves pass a given point on the filament, material starts to flow in-plane towards the impact point till it forms itself into a transverse wave, shaped like a tent with the projectile at the vertex. The velocity of this wave, U, is assumed to be smaller than the longitudinal waves, obtained from: Material in the transverse wave tent does not move in the horizontal direction, but only in the direction of the transverse impact. The strain is assumed to be constant in the wake of the plastic strain wave. This simple solution is later expanded to a general case. This solution covers the range of very high impact velocities, where the transverse wave front propagates more rapidly than the plastic wave front and the strain is variable in the transverse wave region. This approach is very valuable in the sense that it provides the 7 Chapter 2 - Literature Study basic knowledge of single yarn impact in a closed-form mathematical solution. However, it ignores the effect of boundary conditions, as the filament is considered to be infinite. Also, short time creep and relaxation are not considered, and the tension-strain cave is not rate-sensitive. A summary of the rate-independent theory of transverse fibre impact is addressed by reference to Figure 2 - 2 . This figure illustrates the deformed shape of an infinitely long fibre, originally straight in the horizontal direction. The configuration of longitudinal strain wave, transverse wave, in-plane motion of the particles between the two waves, and transverse displacement of the deformed tent can be traced according to the figure. Roylance [1977] used the rate-independent theory of single fibre impact to study the effects of different parameters on the behaviour of a single yarn. The paper first reviews the theory, and then proceeds to consider different values of elastic modulus in predicting impact behaviour of single yarns. A failure criterion based on Zhurkov's fracture model [1965] is incorporated to consider the strong temperature and rate dependencies of fibre material. It was observed that the increase in fibre modulus monotonically increases the energy-absorption rate and decreases the impact-generated strain. However, as the increase in stiffness is accompanied by decrease in breaking strain, the reduced ductility overshadows the beneficial reduction of the strain. This study concludes that the energy absorption and thus the impact efficiency of yarns can be improved by finding the balance between the increased elastic modulus and reduced ductility. In addition to this, utilization of the fracture model results in the prediction of the time after impact at which the fibre ruptures. As seen in Figure 2 - 3 , the time-to-break becomes exponentially 8 Chapter 2 - Literature Study longer at lower velocities, and Roylance stated that the failure occurs at the clamp due to wave reflection at times dependent on the wave speed and the fibre length. Morrison et al. [1986] performed impact tests to study the behaviour of single yarns of aramid fibre before and after it is impregnated with two different polymeric resins. The deformation of the yarns was photographed using a camera and a series of six high-speed flashes, which results in a superimposed picture of the deformed yarn. A finite element code was also developed to simulate the behaviour of the impacted yarns. The strain modulus at high strain rate was estimated by determining the qualitative fit between the predicted and observed deformed shape of the yarns. It was observed that the projectile velocity governs the triangular shape of the deformation zone. From this study, it was concluded that the yarns fracture when the strain reaches a threshold, and the failure most likely happens at the impact point or at the boundaries. The dynamic elastic modulus of Kevlar® 49 was estimated to be less than its static modulus, and Kevlar® 29 distinctly performed non-Hookean behaviour in the slow strain rate tests. It was also seen that generally the impregnation of the polymeric matrix increases the density and decreases the elastic modulus of the yarns, resulting in a dramatic drop in the longitudinal wave speed. The strain to fracture of the yarns also reduced, resulting in the reduction of energy absorption of the yarns in the polymer-impregnated yarns. 2.3 A N A L Y T I C A L M O D E L S O F IMPACT O F FABRICS Wilde et al. [1973] presented an analytical model based on three main assumptions. The first assumption was that the total energy absorbed by the fabric is in form of strain and kinetic energy, stored in an area confined within the boundaries of the deformation cone. All the calculations were focused on the study of the cone at the termination time of 9 Chapter 2 — Literature Study energy transfer from projectile to the fabric, which coincides with the projectile perforation. The second assumption was based on the photos of fabric taken during the impact event, examination of the perforated fabrics and some numerical investigations. It was concluded that the strain energy of the fabric is only stored in the broken orthogonal yarns passing through the impact point (Figure 2 - 4 ) . Finally, the kinetic energy of the system was assumed to be only from the transverse motion of either the orthogonal yarns going through the impact point or all of the fibres, within the boundaries of the deformation cone at the instance of perforation. The inputs to the model were material properties such as yarn mass and dynamic breaking energy of the yarns, along with the mass and the striking velocity of the projectile. The output of the code was the final absorbed energy of the target for every striking velocity. The results did not quite agree with the experiments, and it was mentioned by the authors that the model is capable of predicting 50% to 100% of the absorbed energy. The study presented by Wilde et al. [1973] revealed some key features of textile impact. It was observed that the transverse deformation of the fabric is in the shape of a pyramid with variable transverse wave velocity during the early stages of impact and before the projectile perforation. After the penetration of the projectile, it was seen that the speed of the transverse wave front stays constant and the shape of the transversely deformed fabric changes to an expanding cone. A formula was also presented to calculate the transverse wave velocity for different impact velocities upon the projectile perforation. However, the analytical model presented neglected the yarn-to-yarn crossover points and consequently the strain energy transferred to the other yarns of the fabric. The strain 10 Chapter 2 - Literature Study energy calculation was also conducted without considering the strain in the yarns outside the deformation cone, which results in an unrealistic evaluation of strain energy of the system. It has been claimed that considering the whole length of the yarns within the cone being under the breaking strain compensates for the underestimation of strain energy mentioned above. In calculating the kinetic energy of the system, it appears that considering only the broken orthogonal yarns results in underestimation of the kinetic energy. On the other hand, considering the kinetic energy of all the material in the deformation cone leads to an overestimation of kinetic energy. It was concluded that the true kinetic energy is somewhere between these two values. The model ignores the in-plane motion of the particles outside the deformation cone. This might be the reason for the discrepancy in the absorbed energy prediction especially at higher velocities. The model is also based on the assumption that the strain and velocity is uniform and constant in the deformation cone, and the suggestion to consider the strain and velocity gradient to obtain a better analytical model. In a paper published in 1975, Vinson and Zukas presented an analytical model to predict the behaviour of fabrics under ballistic impact. The model is based on the basic elements offibremechanics and the theory of linear conical shells under axially symmetric loads, as it was observed that the deformed shape of an impacted single-ply of nylon cloth is conical. The concept is to obtain the transverse wave propagation characteristics from the strain wave equations in a single yarn and calculate the forces and strains from the conical shell theory. A quasi-static analysis was used for a statically loaded conical shell, as the impact time-frame is such that the natural frequencies and modes of the shell are not excited noticeably. The inputs to this model are the projectile mass, radius and nose 11 Chapter 2 - Literature Study shape, its striking velocity, elastic modulus of the material as a function of strain and strain-rate, fabric thickness and density, and the breaking strain of the fibre (ultimate strain of the fabric) for the relevant strain-rate. In the case where there is no data available for the modulus of the material, a plot of transverse wave front location as a function of time from an experiment can be used. Using the formulae derived for the model, the velocity of the projectile during the impact is calculated as a function of time. The comparison of the results with experimental data shows a reasonable agreement. Plotting the maximum strain vs. projectile striking velocity shows a linear relationship between these two parameters. This observation agrees with other studies by Wilde et al. on Kevlar® 49. It should be noted that the maximum strain (failure strain) is obtained by enforcing the predicted residual velocity of the projectile to be equal to the one from the experiments. Some points about the above model are worth mentioning: • The model is based on the conical shell theory, which cannot realistically represent the behaviour of the fabric under impact, especially when the fibre undergoes large deformation. • The photographic information on the deformed shape of the target during the impact, as well as the numerical simulations, predict a pyramidal deformation zone before perforation of the projectile, not conical. • The strains predicted from this method are much higher than the capacity of the fibres. It was claimed that this discrepancy is because of the nature of the fabric, which exhibits more deformation to failure than a single yarn. 12 Chapter 2 - Literature Study The same model was used again to compare with the experimental data generated from the impact tests on 1 to 5 layers of Kevlar® 29 [Taylor et al., 1990]. The model (Figure 2 - 5) simulates the multi-layer targets by adding compressibility (from 0% to 100%) to the material in each layer. The predictions of the model for the deformed shape of the cone, the residual velocity of the projectile, and the number of perforated layers are then compared to the actual cases. Parametric studies on the effects of the striking velocity, target thickness, and projectile type were performed and the results were discussed in comparison with the test data. It was concluded that the model has good agreement with the experiments and is a useful tool to predict the impact behaviour of textile structures. The only major discrepancy was observed in predicting the arrest of the projectile. It was stated that the deformation of the arrested projectiles ("mushrooming") is the reason for the error in the prediction of number of perforated layers. Leech et al. [1979] proposed an analytical model using a variational approach to capture the response of woven cloth and nets to impact. From a previous paper by the same authors, it was concluded that the deformation shape of the transversely moving particles of an impacted fabric is rhomboidal. This rhombus (Figure 2 - 6), limited by the transverse wave fronts of the localized impact is referred to as the "excited zone". Upon the application of the Hamilton's principle, the conditions for conservation of the total energy in the system, in the form of strain energy and kinetic energy of the excited zone and the projectile kinetic energy is applied. Consequently, the differential equation of the motion for the nodes of the excited zone is obtained. For this analysis, linear elastic material along with both linear (small deflections) and non-linear (large deflections) geometry effects are considered. Since the method is based on a variational approach, it 13 Chapter 2 - Literature Study can ultimately lead to the exact solution, depending on the accuracy aimed for the analysis. Results of the model are ultimately presented in the form of non-dimensional parameters that include system properties as fabric density and pre-strain, projectile mass and its velocity. Comparison of the results with experimental data shows good agreement between the experiment and the non-linear model. Leech's analytical model is valuable in the sense that it uses the fundamental laws of physics to capture the impact behaviour of fabrics. The outcome of the model is a set of differential equations, which can be solved either in closed form or by means of some numerical manipulations. However, the model lacks some basic requirements, as it neglects the strain and kinetic energy of the particles and masses outside the excited zone. Also, the results for a system without any pre-strain are not investigated, as the model is based on the presence of initial strain in the yarns. Parga-Landa et al. [1995] presented an analytical model to predict the impact behaviour of fibre assemblies (i.e. fabrics). The model is developed from expanding the knowledge of the impact on a single yarn into layers of crossed, unwoven yarns. The assumptions are then extended into woven fabrics by correcting the longitudinal strain velocity, as suggested by Roylance et al. [1980]. From a series of studies on fabric impact, Roylance concluded that the longitudinal strain wave velocity in a woven fabric is a fraction of the velocity in the constituent yarns. A parameter, a, was introduced to specify the relation between the two wave speeds, which usually has a value greater than one: 14 Chapter 2 — Literature Study In the same study, it was stated that in a square woven fabric the linear density of a fibre, along which the wave is propagating is effectively doubled. As a result, the strain wave velocity, c= —, will be reduced by a factor of V2 . Based on the above assumption, VP the strain wave velocity used in the model was set to be: c' = — The tension forces in the fibres were calculated considering the friction between the yarns. The frictions between the projectile and the fabric and between the individual fabric layers are assumed to be negligible. By solving a system of equations in which the projectile velocity, acceleration, and the tensions are the unknowns, the response of the fabric is obtained. The results were compared with the experimental data and showed reasonable agreement (Figure 2 - 7). Chocron et al. [1997] presented a simple analytical model based on the knowledge of the single fibre impact and fundamental laws of physics. In their model, the fabric is assumed to behave as two independent orthogonal unidirectional fibres, i.e. no interaction between the fibres is considered. Effect of crimp and weave is ignored and the strain is assumed to be the same in all layers. The longitudinal wave speed in fabric is considered to be the same as the one in a single yarn. It is assumed that the projectile is rigid and the deceleration force on the projectile results from the tension in the yarns that are in contact with it. The model is based on solving the Newton's second law along with the basic differential equation of wave propagation in a single yarn extended to the fabric. The unknowns, 15 Chapter 2 - Literature Study being transverse wave speed, projectile velocity, strain in yarns, angle between the fabric plane and deformed part of the yarns, and tension in the yarns are obtained by solving these equations simultaneously. A failure model is incorporated to account for the perforation of the projectile. This failure model is based on the accumulation of damage in the yarns, rather than considering the instantaneous strain value. The authors argued that it is not enough to strain a yarn beyond its failure strain; but it is necessary that it remains at that level for a duration of time. A damage parameter based on the strain energy stored in the system during a period of time is introduced and checked against a critical value. This critical value is obtained from the state of energy stored at the ballistic limit, multiplied by a factor a. Comparing the results of the model with experiment, good agreement was seen using a = 1 for Dyneema fabric and a = 0.7 for Kevlar® 29. The model is compared with the experimental data available and seems to be able to predict the behaviour reasonably well. One of the problems with this model is that due to the nature of its formulation, the predicted strain in the yarns diminishes in time, which contradicts with the fact that the failure occurs later in the process. Thus the failure might happen at a point when the strain in the yarns is much less than the failure strain of material. A sample curve of the striking velocity vs. residual velocity of projectile can be found in Figure 2 - 8 . Unfortunately, the model is strictly local and focuses on the impact zone only. However, as it is meant to be a simple analytical model, it sacrifices some accuracy to avoid complexity. 16 Chapter 2 — Literature Study 2.4 NUMERICAL MODELS Capturing the impact behaviour of fabrics via closed-form mathematical procedures is not easily feasible due to the complexity of the problem. For the impact of a single fibre, although closed-form mathematical analysis can be applied to obtain the initial ballistic response, late-time effects resulting from the interaction of strain waves and reflections makes these closed-form analyses intractable. In fabrics, the presence of yarn crossovers and crimp change the pattern of strain wave propagation in such a way that the problem can no longer be treated by a closed-form solution. With the advent of computers, numerical methods were also established to study the dynamics of impact on textiles. Roylance et al. [1973] introduced a numerical model for simulation of impact on single fibres and fabrics. In this paper, a numerical model was introduced, which was an outgrowth of a technique established by Mehta and Davids [1966], referred to as "the direct analysis." Davids et al. developed a dynamic form of finite-element analysis that was applied successfully to a variety of wave propagation problems. The mechanics of wave propagation is usually formulated by the application of impulse-momentum balance and satisfying continuity condition for an incremental volume of material. This results in a system of hyperbolic partial differential equations by reducing the size of the volume element to the limit, which in conjunction with the boundary values and the material constitutive law describes the space-time response of the physical system. Using the above analogy, Roylance et al. developed a numerical model for impact on single fibres and fabrics. Direct analysis was first used by Lynch (1970) to analyze the transverse impact of a single fibre. The results of analysis for single fibre was checked against the analytical 17 Chapter 2 — Literature Study solution as well as the experimental data, and showed good accuracy in predicting the behaviour. Same algorithm was applied to the impact of the fabrics. In this model, the panel is idealized by pin-jointed flexible visco-elastic fibre elements, along with masses at the joints, which makes the areal density of the idealized mesh equal to that of the panel being simulated. The projectile mass and initial velocity is then assigned to the nodes in the contact area, which result in the development of strain in the adjacent elements. From material visco-elastic constitutive law, the tensile forces in the elements are calculated and are later used to evaluate the acceleration and velocity of the masses. The calculation is repeated for all the nodes and the nodal coordinates are updated. At the end, the projectile velocity is re-evaluated from the tensions in the elements in the contact area. The same procedure is followed afterwards for a new increment of time. The results from this model could not be easily verified, as there is no mathematical solution available to compare with. However, the model prediction was close to the experimental data, which provides a rather convincing demonstration of the codes capability. The same model was further developed and compared with the experiments in a study published in 1980 by Roylance and Wang. The failure model incorporated in this model was based on the criterion due to Zhurkov (1965), who considers the lifetime of a solid subjected to a constant tensile stress. The lifetime is calculated from the parameters based on the information of temperature and some material constants related to the dissociation kinetics of the atomic bonds and the internal defect structure of the material. A constitutive model was also introduced based on the visco-elastic behaviour of the material and due to the available evidence on the occurrence of relaxation in the fibres within the ballistic time frame. The constitutive model included an array of Newtonian 18 Chapter 2 - Literature Study dashpots and Hookean springs to capture the strain-rate-sensitive nature of the polymeric material. The numerical results were checked against the experimental data by plotting the residual velocity of the projectile against the initial impact velocity, and by studying the cone size both predicted from the model and measured in the photographic study of the impacted fabric. In most cases, the agreement between the numerical model and the experimental seemed to be satisfactory. Effect of various input parameters on the impact response of the fabric material was also studied using the predictions of the model for four different materials. It was stated that as the penetration generally occurs before the arrival of stress waves at the clamped edges, the nature of the boundary conditions is relatively unimportant. It was also observed that the level of strain is usually higher at the impact point, which grows continuously with time. The fibre modulus was known to be one of the factors governing this behaviour, as higher modulus decreased the strain level and increased the wave-speed. Among the materials studied (Graphite, Kevlar® 29, Kevlar® 49, Nylon), Kevlar® 29 was concluded to have the best combination of energy absorption rate and longer time to penetration, and thus predicted to be the best ballistic material in the above set. Finally, a master curve (Figure 2 - 9 ) was produced for the evaluation of impact-induced strain at the point of impact. The curve was the plot of normalized strain at the impact point against time factored by the fourth root of the fibre modulus, as a linear relation was observed between the first peak in the strain and the fourth root of the fibre modulus. Freeston et al. [1973] presented a numerical scheme for the calculation of longitudinal strain wave attenuation along a yarn of an impacted fabric. The cases of wave reflection 19 Chapter 2 — Literature Study at fixed and free boundaries were adopted as the two extremes for the idea used in the model. It was assumed that in a fabric, yarns crossovers behave as partially fixed ends that reflect a part of the strain wave and transmit the rest of it, ignoring any energy dissipation in the system. It was further assumed that the strain wave velocity in a yarn is given by the infinite medium plane wave speed I—, in which E is the elastic modulus VP and p is the material density of the yarns. This implies that all creep and relaxation effect along with the crimp in the system is neglected. The computational scheme can be seen in Figure 2 - 1 0 , for wave amplitude of A, a reflection coefficient of K, and transmission coefficient of P. The wave reflection and transmission is shown for five time-steps, where each time-step is equal to the time needed for wave to travel the length between two successive nodes. Following the same algorithm for a fabric results in the amplitude of strain wave in each point along the yarns. The initial strain amplitude, £ , generated 0 upon transverse impact by a projectile traveling at velocity V, taken from single yarn study by Ringleb [1957], was assumed to be: 4 where c is the strain wave velocity in the yarn. Using the above initial strain and the aforementioned algorithm, the strain in the yarns at every time step is calculated. Using this model to predict the failure of ballistic yarns, it is concluded that the reflection coefficient in these types of fabrics is very low (about 1%), meaning that the crossovers offer little resistance to the wave transmission. It was also mentioned that the model describes the increasing strain portion of the impact event in non-perforating tests, and 20 Chapter 2 - Literature Study predicts the strain increase to yarn failure in perforating ones. Although the model gives a good insight to the nature of wave reflection in the fabric system, it is not capable of predicting any of the projectile or fabric responses during the event. Moreover, the relation used to calculate the initial strain might not be valid for all the cases and for different materials. Furthermore, it seems to neglect the presence of transverse wave and the effect of perpendicular yarns in the wave reflections. Leech and Adeyefa [1982] employed the method of characteristics to develop a code to numerically simulate the dynamic behaviour of impacted fabrics. In this method, the dynamics of single yarn is applied and equation of motion for fibres and joints are obtained using the Hamilton principle. Motion of the nodes is determined, which leads to the estimation of the fabric behaviour. The zones of influence for in-plane excitation are shown in Figure 2 - 11. This method provides reasonable agreement with the experimental data. However, it is not applicable to visco-elastic materials since it does not support the material characteristic discontinuities. Briscoe and Motamedi [1992] studied the impact behaviour of fabrics considering the effect of interface friction between the yarns. The friction coefficient of the yarns is estimated by performing hanging-fibre friction tests (Figure 2 - 12). The calculated coefficient of friction was found to be a function of the applied normal load, for a given yarn-contact combination. A simple first-order model of the ballistic impact-penetration process was also introduced on the basis of a quasi-static indentation process. The model assumes that the geometry of the deformation is controlled by the configuration of the supports, and not by the velocity of wave propagation. Based on the assumption that the weave is a thin coherent Chapter 2 - Literature Study homogeneous solid plate, the deformation of the fabric under quasi-static loading is considered. Using the above approach, Briscoe et al. modelled the ballistic deformation of a fabric by this quasi-static model of indentation, since they are both related to the effective stiffness of the material. Effect of friction within the fabric system was found to be important for both cases of normal indentation and ballistic deformations, as the extent of yarn-yarn sliding is a major factor contributing to the fabric effective stiffness. It was found that the energy dissipated in fabrics with high friction (and hence low effective moduli) is higher than fabrics with low friction, for both quasi-static and ballistic deformation process. This study serves to emphasize the critical role of the interfacial friction in the deformation of the fabric. In 1992, Cunniff developed a conceptual framework to relate the single yarn impact mechanics to fabric impact mechanics [Cunniff, 1992]. Experimental data from the impact on single-layer fabric systems of Spectra®, Kevlar® 29 and Nylon with different yarns denier and weave types were used along with numerical simulation of single-layer impact to support this study. A numerical code, named "the Natick direct analysis method model", was developed by reformulation of the Roylance model [1973] applied to the ballistic impact of woven fabrics. The result was a code that allows for a better definition of the projectile geometry, dispenses with the requirement for a point impact approximation, and provides a variety of new edge conditions. The model also facilitates parametric studies because of the way it is coded. Figure 2 - 13 is a plot of strain in principal yarns for two different striking velocities at about 10 and 20 ms after impact. From the figures, it can be seen that the strain value developed in the yarns scale with the impact velocities. It was also stated that the transverse wave velocity does not scale with 22 Chapter 2 - Literature Study the projectile velocity, leading to a much steeper strain gradient in the region of transverse deflection at the higher impact velocities. Cunniff used the numerical results to explain the observations made in the experimental efforts. The numerical model presented by Roylance et al. ([1973], [1980]) was modified and further studied by Ting et al. [1993]. The numerical code developed previously was extended to account for fibre slippage at the crossovers and clamped ends, and to simulate the behaviour of multiple layers of fabric. In addition, a new computer interface was developed that provides the user with a graphical view of the full-field strain at every time-step in the fabric. This new display was claimed to offer the user the ability to monitor the development of strain in the central fibre and determine the behaviour of an entire fabric panel during the impact event, sacrificing some code execution speed. Simulation of multi-layer targets was performed through introducing an adjustable parameter for the initial separation between the layers. The nodes in each layer are free to slide in-plane, and the contact between the layers is checked when the nodes move outof-plane. In case of contact, the effective slowdown due to the resulting inelastic collision was computed. Nodal displacements were evaluated by summing the forces due to all of the fibre segments in contact. Results of this approach can be seen in Figure 2 - 14. In order to simulate the effect of yarn slippage, the maximum amount of in-plane tension due to two components of friction is calculated. The first component represents the friction due to the construction of the weave and the second component is equivalent to the dynamic friction due to the normal forces during impact. However, the extent of friction in an actual case depends on many factors, including the tightness of the fabric weave, the yarn friction coefficients, and moisture. A series of numerical experiments Chapter 2 - Literature Study was performed to see the effect of yarn friction and number of layers on the response of fabric targets. It was seen that the ballistic resistance (or the value of V50) decreases with the decrease of friction parameter. Also, it was concluded from the results of studies on multi-layer targets that V50 is nearly a linear function of the number of layers. In a paper published in 1995, Roylance et al. [1995] reviewed the work presented earlier by Ting et al. [1993] and discussed the results of simulation of impact on resin impregnated fabrics. Initially, backup springs were added to the mesh-nodes in order to simulate the effect of human torso behind the fabric, which provided a force in the direction opposite to the projectile velocity. Later, these springs were used to simulate the effect of the impregnating polymer matrix in a composite material on the ballistic resistance of the composite. Figure 2-15 shows the result of analysis for different values of backup spring. It can be seen that the ballistic resistance decreases with increasing values of spring stiffness. According to this study, the reduction of V50 with the increase in backup spring stiffness appears to be related to the tendency of these springs to impede 2-direction movement of the fabric, and impart a transverse impact to fibres other than those passing through the impact point. Shim et al. [1995] studied the deformation and damage characteristics of fabric panels under small projectile impact. A series of tests was conducted on panels of Twaron® and the impact and residual velocities were recorded to study the energy absorption of the target. Plotting absorbed energy by the target against the projectile striking energy, two behaviours were noticed; a "low velocity" range in which the absorbed energy increases linearly with the striking energy, and a "high velocity" response in which the absorbed Chapter 2 - Literature Study energy remains almost constant after a sudden drop. A set of equations was derived from a linearfitto the experimental data for these two regions. A numerical model was developed to predict the ballistic behaviour of the material. The fabric is idealized as a network of pin-jointed, visco-elastic fibre elements and discrete masses. A three-element visco-elastic constitutive model of Hookean springs and Newtonian dashpots (Figure 2 - 16) is incorporated in the model to represent the polymer material. Crimping effect was considered enforcing the following equation: f ^actual & total Utal E l-e ^ crimp v \ Ecrimp J The failure criterion in the model was based on instantaneous rupture of each of the two springs in the three-element constitutive model. Results of this numerical model were compared with the experimental data and seemed to agree quite well including the failure patterns. It was observed that the penetration occurs via a combination of yarn breakage at the impact point and yarn slippage around the projectile (Figure 2 - 17). It was also concluded that the deformation characteristics also depends on boundary conditions. Lomov [1995] presented a numerical model to simulate the impact behaviour of multilayer targets of fabric. He later [Lomov, 1996] extended his model to account for obliquity of projectile impact. The model is based on the assumptions of the projectile rigidity, linearity of yarn deformation, not slippage between the yarns, and deformation dependent yarn failure. The obliquity of impact is considered via the angle between the projectile velocity vector and the plane of the fabric. The equation of motion is solved for 25 Chapter 2 - Literature Study the yarns and deceleration of the projectile is calculated from the sum of the forces in the contacting yarns. Input parameters to this model consists of projectile mass and geometry, impact velocity and its angle, target geometry including yarn density, weave, crimp and count, and elongation to break. In modelling the behaviour of multi-layer stacks of fabric, it was assumed that deformation properties of the pile of undamaged layers is the sum of the properties of individual yarns, and that layers rupture one after another. The predictions of this model are compared with experimental data for both oblique and normal impact (Figure 2 - 18). Good accuracy was obtained in the predictions for design purposes. It was observed that the shape of projectile and transmission conditions on the boundaries affect the behaviour of the target. In the oblique impact, slippage of the projectile on the fabric was observed. In a recent study published by Ting et al. [1998], a numerical model was presented to simulate the transverse yarn interaction using an empirical model of crossed yarn stiffness. This model simulates the yarns interaction at the crossovers of warp and fill, considers the coupling of multiple layers, takes into account the effect of yarn bending and clamp imperfection, and models the impact of circular and rectangular projectiles. A fabric is represented by a collection of mass nodes connected by mass-less fibre spring elements. The crossover points are each represented by two nodes, each attached to either fill or warp yarns (Figure 2 - 19). Springs acting in the transverse direction couples the motion of each pair of warp and fill mass nodes. This modelled interaction between fill and warp deviates the speed of longitudinal wave from the theoretical velocity calculated for a single yarn. A torsion spring is placed at each crossover to introduce bending stiffness in the fibres. The code also allows the user to consider any, or limited, 26 Chapter 2 - Literature Study compressive loads developed in the fibres. The limit for the compressive force in elements can be set to be the buckling load of the yarns. For the case of multiple layers, each layer of fabric is coupled with the adjacent one via a series of compression elements. Numerical studies were performed using the described model to see the effect of crimp and warp-fill coupling of the yarns. It was seen that the introduction of crimp significantly affects the behaviour of the fabric, especially strain wave propagation and transverse wave speed. The coupling effect was studied considering rigid, linear elastic and non-linear elastic model for coupling elements. The panel with empirically based non-linear model was found to be in good agreement with the experimental data. Chapter 2 - Literature Study Figure 2-1: Configuration of the positive half of an impacted yarn, t seconds after impact. [Smith et al., 1958] Figure 2-2: Wave propagation in a transversely impacted yarn [Roylance, 1977]. 28 Chapter 2 - Literature Study = 0.140 750 775 800 825 Impact Velocity, tn/sec Figure 2-3: Variation in transverse critical velocity due to fracture rate effects [Roylance, 1977]. Boundary of I CR I J H I Missile Penetration Figure 2-4: Broken orthogonal yarns (BOY model) [Wilde et al., 1973]. 29 Chapter 2 — Literature Study Figure 2-5: Impact geometry [Taylor et al., 1975]. Figure 2-6: Propagation front and indentation of orthogonally woven cloth [Leech et al., 1979]. 30 Chapter 2 — Literature Study Figure 2-7: Comparison of ballistic limit curves obtained for armours made of different basis weight fabric [Parga-Landa et al., 1995]. 22 ply K29,16 gt. 400 600 800 1000 1200 1400 Vs (m/s) 16 Ply K29,16 gr. 400 600 800 1000 1200 1400 Vs (m/s) Figure 2-8: Comparison of analytical and experimental results for the 16-grain FSP [Chocron-Benloulo et al., 1997]. 31 Chapter 2 — Literature Study t (^s)[E(gpd)] o; Figure 2-9: Master curve for impact induced strain at the point of impact (Reproduced from Roylance et al., 1980). Time Mode 1 Step 1 A Step 2 KA PA Step 3 KA KPA KA K PA 2 2 111111111111 PA 2 2 Step 4 KP A 2 KP A 2 5 KPA K PA KPA K PA KPA KP A KP A 3 2 PA 3 2 KA 2 2 Step 5 KP A 2 2 KP A 3 PA 4 2 3 2 Figure 2 -10: Amplitude of propagating reflected waves; arrows show the direction of wave [Freeston et al., 1973]. Chapter 2 — Literature Study y /—Front of ' w t a k ' t x t t n s i o n a l signal* Front of 'y' txcitatiort Front of ' x ' • xcitation Figure 2 - 11: Predicted wave fronts for in-plane excitation [Leech et al., 1982]. Direction of hangingfibremotion Hanging fibre Height h / ^A Hanging fibre -tr^fc- Tension T Horizontal fibre Horizontal fibre i Distance d ! Mass. ra Figure 2 - 12: Schematic diagram of the hanging fibre friction experiment [Briscoe et al., 1992]. 33 Chapter 2 - Literature Study DISTANCE FROM IMPACT POINT (cm) DISTANCE F R O M IMPACT POINT (cm) Figure 2 - 13: Strain development prediction of a direct analysis computer program for impact velocities of 200 m/s and 300 m/s [Cunniff, 1992], n o o CL, OL, O O nBBBaOB0B8oB8BB00080e0( iqoaeoooo OOOOOOOaOOBBCBBBOBBBBBaeflQeOBCOOOBOBOO oooooooeoooe>oeeoBaaoo60«6«oe6eaoao«»<»« BCOOOBOBBOBOBflBBBOBOBBaBBBBBeBaBBBB OOOBOOOBOOOOBBBBetBOBeaBeOBBOflBeBeflB >oo»o« maa»BBgB«at 3 4 5 6 7 8 Distance From Impact Point (cm) 10 11 Figure 2 - 14: Plot of out-of-plane positions along the main load bearing fibre for a 9-layer panel [Ting et al., 1993]. 34 Chapter 2 - Literature Study i . . . . i . . . . i . . . . i . . . . i . . . . i . . . . i .. . I .100 0 100 200300400500600 700 800900 Vs, Single layer (lyr*3) Figure 2 - 15: Plot of V vs. V using the single layer fabric model for different values of backup spring [Roylance et al., 1995], s r •OEH2 K1 K 2 Figure 2 -16: Three-element visco-elastic model [Shim et al., 1995 ]. 3 35 Chapter 2 - Literature Study Figure 2 - 17: Low velocity deformation and penetration for impact velocity of 150 m/s [Shim et al., 1995 ]. b Figure 2 -18: Predicted and experimental results from simulation of normal impact [Lomov, 1996]. 36 Chapter 2 — Literature Study Warp Fibers —>• Figure 2 - 19: Breakdown and geometries of warp and fill fibres [Ting et al., 1998]. 37 Chapter 3 - Analytical Model CHAPTER THREE: ANALYTICAL M O D E L 3.1 INTRODUCTION An analytical model is presented to compute the total absorbed energy and its components from the knowledge of the projectile's position in time. The model uses the basics of single yarn impact and extends it to a fabric via simple assumptions. In this chapter, an overview of transverse impact on a single yarn is presented based on the paper by Smith et al. [1958]. The model assumptions and formulation are presented and its predictions are compared with the experimental data. 3.2 BALLISTIC IMPACT OF A SINGLE YARN A fabric is composed of single yarns of woven in two directions. The overall impact behaviour of a fabric is influenced by the response of single yarns. The complexity of the fabric weave and the way the individual yarns interact affect the nature of the impact response of a fabric from being simply a superposition of yarns in two directions. However, studying impact behaviour of a single yarn still gives insight into the response of textiles. Many studies have been performed to capture the impact behaviour of single yarns [Smith, I960], Results of a comprehensive study by Smith et al. was published in 60's, which discussed a closed-form mathematical solution for impact on an infinitely-long single yarn. The basics of single fibre impact are reviewed here. 38 Chapter 3 — Analytical Model Assume an infinitely long filament with uniformly distributed mass being impacted in the middle (Figure 3 - 1). Two co-ordinate systems are introduced to study the response of the yam. A global co-ordinate system (or laboratory co-ordinate system), x' and y', is set to monitor the horizontal and vertical location of all the points along the yarn at any time, / . Also, a Lagrangian co-ordinate system, x, is considered to denote the distance of each point from the impact point on the unstrained yarn, which coincides with x' of the global co-ordinate system at time t = 0. The horizontal and vertical displacements of any point x on the undeformed yarn are expressed by the functions £(x,t) and rj(x,i), respectively. Thus, the location of an arbitrary point at time t is: x' = x + £(x,t) (3.1) y' = rix,t) (3.2) For a differential length, dx, of the unstrained yarn, the strained length, ds, at any time t after the impact is given by: If ds = dxA\ { c^x,,)V fdrjxj) | +| '/ | dx J ^ dx | ( 3 3 ) The strain in the yarn can then be calculated knowing the new length of the elements: 1+^ 1 + dx a ( 3 4 ) Also, the angle between every segment of the yarn and the horizontal axis, 6, can be determined knowing the displacement functions (equations (3.1) and (3.2)) as follows: 39 Chapter 3 - Analytical Model cosO- dn 1 sin 0 (3.5) 1 + e dx 1+s 1+ df dx . (3.6) Satisfaction of the dynamic equilibrium for the element dx in two directions (Figure 3-1) results in the differential equation of motion: c\Tcos9), ,,d £, 'dx = {Mdx)-^2 2^F = ma -> X U/f x M 2 y y M y c\Tsin9) 2_ F =ma ^>— l (3.7) ^ = ^(T.cosd) dt dx dx } ,s&-n -dx = (Mdx)- ' dC Ujr ^ L = —(T.sine) dt 2 dx y ' (3.8) where T is the tension in the strained yarn and M is the mass per unit length of the yarn. Substituting for angle 6 from equations (3.5) and (3.6), results in the following set of differential equations: dn d dt ~ dx M(\ + e) dx a 7/_ 2 2 d%_ d 2 T dt ~ dx M(\ + e) V 2 3.2.1 dx (3.9) (3.10) THEORY OF LONGITUDINAL IMPACT For the case of a longitudinal impact, the vertical component of displacement, ?j, would be equal to zero. Substituting TJ= 0 in equation (3.4) gives: 40 Chapter 3 -Analytical Model e J i <'> 3 n ox which reduces the system of equations (3.9) and (3.10) to: 1 dT d% 2 M dt 2 dx (3.12) The answer to the above differential equation, assuming a completely elastic behaviour for the material, is a pair of waves propagating outwards from the impact point with a velocity C, as expressed below: The material behind the outgoing elastic wave front moves uniformly in the impact direction with a velocity, w, given by: w \lC{s)ds Jo In case of a linear elastic behaviour for the material, the longitudinal strain wave speed, C, is a constant, resulting in: w = Cs (3.14) 3.2.2 THEORY OF TRANSVERSE IMPACT Solution of the system of equations (3.9) and (3.10), in general, is the answer to the transverse impact response of a single yarn. Applying some special conditions further simplifies the solution. Studies of general cases can be found in literature [Smith et al.]. This section focuses of the response of a single yarn with linear elastic stress-strain 41 Chapter 3 - Analytical Model relationship, in which no large stress relaxation or creep effects occur in small timeintervals. Upon the transverse impact, a longitudinal strain wave is generated and propagated along the yarn. This wave travels with a speed C, defined by equation (3.13). As the material is (dT\ assumed to posses a linear-elastic stress-strain behaviour, the value of — being the \ds) =o E slope of the tension-strain curve would be a constant, equal to EA, multiplication of the elastic modulus and the yarn area. This would simplify equation (3.13) even further into the well-known relation of strain wave speed in material (note that M is the mass per unit length of the yarn): C= - (3.15) The longitudinal strain wave introduces an in-plane motion in the material, as it travels along thefilament.The in-plane velocity of the particles in the wake of this strain wave, w, can be determined from equation (3.14). A transverse wave follows the longitudinal strain wave, which converts the in-plane motion of the particles to a transverse displacement with the same velocity of the projectile. The transverse wave front, assuming a Lagrangian co-ordinate system, travels with a speed U, defined below: U= ]M{l J . + e) (3.16) The propagation velocity of the transverse wave U, in the laboratory co-ordinate system would be equal to (7 + s)U - w. As the transverse wave propagates, the out-of-plane 42 Chapter 3 - Analytical Model motion of material forms a transversely deformed tent, in which the particles move with a velocity V, equal to: (3.17) Other material models would result in different responses of the transverse impact. Other analogies might be applicable regarding the problem and the intended application. 3.3 A N A L Y T I C A L M O D E L F O R FABRICS Based on the theory of the single yarn impact, an analytical model is proposed to approximate the energy absorption of the fabric, knowing the displacement-time record of the projectile. This model calculates the fabric's total absorbed energy in terms of strain and kinetic energy components. It is assumed that upon impact, a longitudinal strain wave is generated in the strained yarns. The propagation velocity of this wave, C, is assumed to be smaller than the one in a single yarn, due to the nature of weave and yarn crossovers. A study performed by Roylance et al. [1980] optimized this factor and showed that the value of 42 results in stability and convergence of the solution. This value is adopted in this study and the strain wave velocity is formulated as follows: (3.18) where E and p are the elastic modulus of the yarns and mass density, respectively. The elastic modulus is divided by a constant factor of 2, in order to account for reduction of wave speed due to the fabric weave, yarn crossovers, and the crimp, which exist in a 43 Chapter 3 - Analytical Model fabric system [ref#]. This longitudinal wave strains the yarns and introduces an in-plane motion in the material as it propagates. The material within the longitudinal wave fronts travel in-plane towards the impact point at a speed w given by equation (3.14) The longitudinal wave is then followed by a transverse wave, which causes the material to move in a direction perpendicular to the fabric's plane. The transverse displacement of the material within this wave forms a pyramid (deformation pyramid or cone) within which all the material move out-of-plane with the speed of the projectile (Figure 3-2). The propagation velocity of the transverse wave in the Lagrangian co-ordinate system, U, can be derived from equation (3.16) replacing T and M with EAe and pA, respectively: (3.19) In the laboratory co-ordinates, the transverse wave travels at a speed U, which can be derived from U . U = (l + e)U- Simplification of the above relation results in: (3.20) The strain, s, is assumed to be constant along each yarn and is calculated using the yarn length within the longitudinal wave fronts. This assumption implies that only the yarns, which go through the deformation cone, are strained due to their transverse displacement. 44 Chapter 3 - Analytical Model Thus, it can easily be concluded that the strain wave travels only in those individual yams that go through the deformation cone in two directions (Figure 3-2). The model predicts the absorbed energy and its components in time, for the duration before the first reflected longitudinal strain wave returns to the impact point. Analysis is divided into two phases; phase-I corresponding to the time when the longitudinal strain wave travels outwards to the boundaries, and phase-II corresponding to the period when the wave has reflected from the boundaries (either partially or fully) and is coming back to the impact point. This analytical model is established for panels with two different boundary conditions, fixed all-around, and free all-around. The approach is explained separately for each of these two cases. 3.3.1 PANELS WITH FIXED BOUNDARY CONDITIONS The panels discussed here are considered to be square targets with clamped edges, meaning that all the degrees of freedom of the nodes on the boundary are restrained. Two time periods are considered: phase-I starting from the impact time and ending just as the longitudinal strain wave front reflects from the boundaries, and phase-II, which immediately follows phase-I and ends when the reflected strain wave returns to the impact point. Formulation of these two phases is presented below. 3.3.1.1 Phase-I As mentioned above, phase-I starts from t = 0 to t = T , where 7) is the termination time 1 for phase-I and is calculated as follows: 45 Chapter 3 - Analytical Model where L is the panel size. The total absorbed energy of the target is broken down into three components, strain energy of the yarns, transverse kinetic energy of the cone, and in-plane kinetic energy outside the cone. 3.3.1.1.1 Strain energy Strain is determined using the deformed length of a yarn. Figure 3 - 3(b) shows the deformed shape of a yarn going through the impact point (i.e. central yarn), for which, the deformed length, /, is equal to: I = 2(Ct-b) + 2b^l + ^jj (3.21) where d is the projectile displacement and b is the cone size in the global co-ordinate system, calculated from the transverse wave velocity, U, as follow: b = \'lJdt (3.22) Considering the undeformed length of the yarn covered by the longitudinal strain wave, 2Ct, the strain in the central yarn, e , can be calculated as follows: 0 o=- £ l-2Ct 2Ct after substituting equation (3.21) and simplifying: Ct (d^ 1+ (3.23) In general, the same approach can be used to determine the strain of a yarn in the ydirection at a distance x from the impact point (Figure 3 - 3(c)): 46 Chapter 3 - Analytical Model s - b-x Ct -x (3.24) As it can be seen from the above relation (eq. (3.24)), the strain is a function of the cone size, b. Meanwhile, the cone size b is directly dependent on the transverse wave velocity, U (eq. (3.22)), which in turn is a function of strain (eq. (3.19)). Solving these three equations (eqs. (3.19), (3.24), and (3.22)) simultaneously, results in the values of U, b, and s in terms of d and t. Finally, from the above equations, the strain energy of the system can be calculated as follows: (3.25) where v is the volume of the yarns. Due to the existing symmetry in the fabric, the strain energy is calculated for half-width of all the v-running yarns and then is multiplied by four to get the total strain energy: (3.26) where h is the thickness of the fabric, and hence — is the yarn thickness. 47 Chapter 3 - Analytical Model 3.3.1.1.2 Transverse kinetic energy of the cone Same as the single yarn impact, it is assumed that the material in the area covered by the transverse strain wave moves transversely with the speed of the projectile. This creates a transversely deformed pyramid or cone with the projectile being at its centre. In order to find the total mass of the material in this region, areal density of the fabric before impact is used. The areal density which is equal to the mass of the fabric per unit area, is valid only for the undeformed fabric. Thus, the transverse wave velocity in the Lagrangian system of co-ordinates, U, (from eq. (3.20)) is used to consider the area of the cone, and then calculate the mass of the particles within the cone. The area within the transverse wave front, A c o n e , is then calculated using the following equation: A = 4(0.5P) = 2b 2 cone where b is the cone size and is determined by the following relation: (3.27) b=j' Udt o Kinetic energy of the cone would then be equal to: v p cone 1 =—m V ^ 7 cone proj 2 [ft areal ^"corteWproj substituting for A : cone K.E. cone where p areal = (bV ) Pareal proJ 2 (t<T ) } (3.28) is the areal density of the fabric, and V • is the projectile velocity. 48 Chapter 3 — Analytical Model 3.3.I.1.3 In-plane kinetic energy outside the cone Same as in the single yarn impact, it is assumed that part of the absorbed energy goes into the in-plane motion of the material between the two strain waves (longitudinal and transverse waves). The in-plane velocity of the masses, w, is a function of the strain in the yarns (equation (3.14)), which changes with the distance from the impact point. The total energy is then derived by integration of the yarns' kinetic energies over the entire area within the two waves. The in-plane kinetic energy of a group of yarns for a width dx is equal to: d(K.E. ) = ±w dm 2 out ^{^[jPareal x2(Ct-b)dx^ y P (Ct-b)e dx = 2 2 area The total kinetic energy of the fabric would then be: K.E. out = 4{f o d{K.E. )y2p C {Ct-b)\ 8 dx mt areal b 2 2 o (tzT,) (3.29) The total energy absorbed by the target in phase-I is calculated by summation of all energy components from equations (3.26), (3.28), and (3.29): E total 3.3.1.2 = SlM +K.E. +K.E. cone out (> < 7)) (3.30) Phase-H Phase-II starts from 2), when the longitudinal strain wave first reaches the boundary to the time T , when the reflected wave returns to the impact point. Assuming the same u velocity for the out-going and returning longitudinal wave, it can be concluded that 49 Chapter 3 - Analytical Model T - 2Tj. Upon reflection of the wave, the state of strain and stress in the yarns change u according to the boundary conditions. From the fundamentals of wave reflection, it is known that the strain wave reflects at a fixed boundary with the same sign, while on a free boundary, it returns with the opposite sign. In other words, for a fixed boundary the reflected wave doubles the existing strain, while the reflected wave at a free boundary cancels the existing strain on its way back to the impact point. Similarly, the in-plane motion behind the longitudinal strain wave is cancelled in reflection at a fixed boundary and is doubled at a free boundary by the reflected strain wave. The analysis in phase-II is carried out in two steps. Figure 3 - 4 shows the configuration of waves in early stages of reflection. The width of the reflected portion of the wave, r, can be expressed by the following relation: r = Ct-— 2 (3.31) As can be seen in Figure 3-4, initially the width of the reflected portion of the wave, r, is smaller than the cone size, b. Due to higher speed of the longitudinal strain wave, this reflected width increases in size faster than the cone size. There would be a transition time after which r becomes greater than b, indicating that the outgoing wave has completely reflected from the boundaries and is returning to the impact point (Figure 3 5). The energy calculation in phase-II is performed for the duration before and after this transition time, T , separately. The transition time is determined as follows, knowing that tr at time T the width of the reflected wave equals the cone size: tr 50 Chapter 3 - Analytical Model Therefore, T =— 3.3.1.2.1 (3.32) Strain energy Before the transition time, t <T , the strain energy is computed for two distinct areas, tr marked Al (including both Al-a and Al-b) and A2 as shown in Figure 3 - 4. Al is the area of the yarns in which the strain wave has reflected from the boundaries. As the yarns in this area are completely under strain, total length of the yarn is used to determine the strain: (3.33) Due to the method used for evaluating the strain in the yarns, the calculated strain is an average value over the entire length of the yarn. For this reason, the strain energy is integrated over the entire length of the yarns. The strain energy stored in the area Al is then equal to: For the yarns located in A2 (Figure 3 - 4), the strain is calculated using the length covered by the longitudinal strain wave: 51 Chapter 3 — Analytical Model b-X Si = 1+ Ct-x a- (3.35) J The strain energy is again calculated using the length within the longitudinal wave fronts: = 2Eh^e (Ct-x)dx (T < t < TJ 3 2 } (3.36) The total strain energy before transition time (i.e. t < T ) is equal to the summation of the tr energies of the two areas from equations (3.34) and (3.36): n*- =^.+n^ W, *tzT„) (3.3?) After the transition time, as can be seen in Figure 3 - 5 , the yarns are all swept by the longitudinal strain wave and the wave front returns to the impact point. For this case, the strain is determined using the total length of the yarns: 2(7) .V) 1+ d (3.38) The strain energy is also calculated over the entire length of the yarns: a. EhL\ b 3.3.1.2.2 JO e dx 2 (T <t<T ) tr u (3.39) Transverse kinetic energy of the cone The kinetic energy of the particles in the cone moving out-of-plane with the projectile is calculated in the same way as in phase-I. Any effect on the wave velocities resulting from 52 Chapter 3 - Analytical Model the interference of the returning longitudinal strain wave and transverse wave is neglected. The formulation would be identical to what was obtained in phase-I, equation (3.28): 3.3.1.2.3 In-plane kinetic energy outside the cone Same as the strain energy calculation, the kinetic energy is evaluated over two distinct areas, Al and A2 (Figure 3 - 4), for the duration before the transition time, / < T . It is tr assumed that the strain wave after reflection from the fixed boundary cancels the in-plane motion of the material on its way to the impact point (Al-a in Figure 3-4). The kinetic energy is derived by integration over the remaining areas that are not yet swept by the reflected strain wave. For area A 1-b, the yarn length / ' , not swept by the returning strain wave is: (3.41) The kinetic energy of the moving material of length / ' i s : p C [e (L areal 2 2 + 4x-2b-2r)dx {T^tzTJ (3.42) where s is calculated from equation (3.33). x For area A2, the kinetic energy is integrated over the current length of the yarns: 53 Chapter 3 - Analytical Model K.E. A2 x2(Ct-b)ckj = *l\{Cs ) {±p 2 2 areal (7)<^<rj (3.43) = 2p C le (Ct-b)dx 2 2 areal 2 where e is determined from equation (3.35). 2 The total in-plane kinetic energy is obtained by summation of the kinetic energies of the two areas, equations (3.42) and (3.43): K.E. = K.E. ,+K.E. ou! A A2 (7)</<rj (3.44) As mentioned before, after the transition time, t>T , the strain wave fully reflects from tr the boundaries, and approaches the impact point. The kinetic energy is integrated over the area between the two waves, only over the yarn lengths not yet covered by the returning strain wave: K-E. out --p C \ e (L areal 2 b o 2 + 4x-2b-2r)dx (3.45) (T <t<T ) tr n in which e is given by equation (3.38). Similar to phase-I, the total energy in phase-II is calculated by adding the three energy components from equations (3.37) and (3.39) for strain energy, equation (3.40) for transverse kinetic energy of the cone, and equations (3.44) and (3.45) for kinetic energy out-of-the-cone: K tal =n +K.E. K.E. strain cone+ out (T, <t<T )(3A6) u 54 Chapter 3 - Analytical Model 3.3.2 PANELS WITH FREE BOUNDARY CONDITIONS For the targets with free boundary conditions, the same approach can be followed. In this case, however, the edges of the fabric are completely free to move and have unrestrained degrees of freedom in all three directions. The effect of this motion on the boundaries is discussed in phase-II. 3.3.2.1 Phase-I As there is no boundary condition influence on the energy absorption of the fabric in phase-I, the same calculations are valid for a target with free boundaries. 3.3.2.2 Phase-H Phase-II starts with the reflection of the longitudinal strain wave from the free boundaries. As mentioned before, the strain wave reflects with the opposite sign of the original wave, thus cancelling the strain of existing wave on its way back to the impact point. On the other hand, the reflected wave doubles the in-plane velocity of the particles in the areas that it covers. These phenomena are considered in calculation of the in-plane kinetic energy. 3.3.2.2.1 Strain energy Figure 3 - 6 shows the deformed shape of a yarn with free ends. As there is no restraining force on the two ends of the yarns, they easily move in-plane towards the impact point upon the reflection of the longitudinal strain wave. Same as the panel with fixed boundaries, a transition time prevails. The calculation is performed for two distinct areas, Al and A2 (Figure 3 - 4). The deformed length of a yarn within the reflected zone (Al) at a distance xfromthe centre yarn is equal to: Chapter 3 - Analytical Model l=L-2(b-x) + 2(b-x)jj+ £) -4w(t-T )^J--^ 1 where T is the termination time for phase-I. The strain can be calculated as follows: t 2{b-x) L 4w{t-T ) ( x 1-L 1 1+ Since w, is itself a function of the strain in the yarn (eq. (3.21)), we can rewrite the above relation as: 2(6-x) (d 1+ (3.47) -1 I + 4C(*-V,)(l-7) The strain energy associated with the yarns in this area can be expressed as follows: x L \dx J = EhL[ s, dx {T,<t<T ) 2 o tr (3.48) For the yarns located in A2, in which the longitudinal wave has not reached the boundaries, the strain is calculated using the length travelled by the strain wave: 2(b-x) Ct fd" 1+ -1 (3.49) The strain energy of this zone is then determined by the following equation: 56 Chapter 3 — Analytical Model = 2Ehj £ {Ct-x)dx b (3.50) (Tj<t<T ) 2 2 tr The total strain energy for this time period (i.e. t>T ) is calculated by addition of the tr energies for the two areas: (3.51) (Tj<t<T ) n * - = ^ , + ^ 2 lr For the period after the transition time, T , as the longitudinal strain wave in all the yarns tr reflect, the strain in a yarn at distance x from the middle can be calculated as: 2(b-x) 4w{t-T ) 1 1+ L L 2{b-x) L + \ 7-* r (d (3.52) 1+ 4C{t-T ){^l-^ I And similarly, the strain energy is: (T <t<T ) EhL [ s dx Jo 2 3.3.2.2.2 tr (3.53) n Transverse kinetic energy of the cone Similar to phase-I, the kinetic energy of the transversely moving material in the deformation cone can be calculated from the following relation: K-E. cone = p (b area! V ) pro 2 (T <t< t T) n (3.54) Chapter 3 - Analytical Model 3.3.2.2.3 In-plane kinetic energy outside the cone As discussed before, the energy calculation is divided into two time-periods, before the transition time and after the transition time. For the time when t<T , the in-plane kinetic tr energy of the material is calculated for the two areas shown in Figure 3 - 5 . For Al, the area is again divided into two sub-areas, Al-a and Al-b, to consider the part covered by the reflected wave separately: Z-E-Ai-a x 2{r-x)dx^j = [^{2Ce ) ^p 4 2 1 areal = C p \ {8r-8x)dx 2 areal r (3.55) 2 o£l And, K-E-A,- = b ^ ( C e ^ p a r J ' d x ^ Substituting /' from equation (3.41) and simplifying the above relation results in: K. E. _ = C p A1 2 b area! I e (L 2 + 4x-2b- (3.56) 2r)dx So, the total kinetic energy of area Al would be the summation of equations (3.55) And (3.56): K.E. Al = 2C p j (L-4x-2b+6r)dx 2 r areal 2 o£} tr (3.57) (T <t<T ) (3.58) (Tj <t <T ) For A2, the kinetic energy is calculated as follows: x2(Ct-tyfr) K-E-A2 = ^\{Cs2)2[^pareal = 2C p {Ct-b)[e dx 2 areal 2 2 } tr Chapter 3 - Analytical Model The total in-plane kinetic energy of the target is equal to: 1 K.E. = K.E. ,+K.E. out After the transition time (t>T ), A (T^t^TJ A2 (3.59) the in-plane kinetic energy is calculated again by tr adding the energies in the two sub-areas, the kinetic energy of the part covered by the reflected wave (K.E. _ ) 0Ut and the energy of the area not covered by the reflecting wave a (K.E. _ ): out b K-E. _ out = 4\ L(2Cs) [^p b a x2[r-x)dx^ 2 o areal = C p \ e {8r-8x)dx 2 b areal (3.60) 2 o and. =Cp 2 £ s {L + 4x-2b2 ared 2r)dx (3.61) Therefore, the total kinetic energy is: K-E. out = K.E. _ K.E. „ =C out a+ out b \ S {L-4x~2b+6r)dx 2 b Pared 2 o (T <t<T ) tr u (3.62) Finally, the total absorbed energy of the target is calculated by the summation of all the above energy components, strain energy from equations (3.51) and (3.53), transverse kinetic energy of the cone from equation (3.54), and in-plane kinetic energy from equations (3.59) and (3.62): 59 Chapter 3 - Analytical Model E =Cl +K.E. +K.E. total 3.4 strain cone out (7, <t<T ) n (3.63) R E S U L T S A N D DISCUSSION The model explained above is implemented as a program in the MATHCAD environment, presented in Appendix A. Results and predictions of the model against the experimental data for the two different boundary conditions are presented in Figure 3 - 7 and Figure 3 - 8. It can be seen that the prediction of the model is in good agreement with the experimental results. It should be noted that the experimental data are obtained using the Enhanced Laser Velocity System (ELVS), which is discussed in Chapter Five. To compare the trend of the ELVS results with the analytical model, the experimental data are over-smoothed to eliminate noise. From the results of the model it can be seen that different boundary conditions change the energy absorption behaviour of the target. Tliis change starts from the instant when the longitudinal strain wave first reflects from the boundary. It is observed that in the panels with free boundaries, most of the energy stored in the fabric is in the form of kinetic energy, and the strain energy in the fabric decreases with time. On the other hand, in the panels with fixed boundary conditions, most of the energy is stored in the yarns in the form of strain energy. The values of transverse and inplane kinetic energy are smaller compared to strain energy. Overall, at a given instant of time, the panels with fixed boundary conditions absorb more energy than the panels with free boundary conditions. Although the model seems to be capable of capturing the nature of impact in a fabric, it is only useful over a short span of time; from the impact incident till the moment when the first reflection of the longitudinal strain waves returns to the impact point. After that time, the interaction of the waves makes the study more complicated and the formulations 60 Chapter 3 -Analytical Model more extensive. However, one can extend this model to calculate the response for a longer time in the impact event. Even though the input to the analytical model and experimental calculations is common (i.e. displacement-time data), the experimental total energy absorbed is obtained by simply calculating the instantaneous kinetic energy of the projectile and subtracting it from the incident energy. Given the simplifying assumptions made in the analytical model, it is expected that the total energy computed as the sum of kinetic energy and strain energy of the target panel, will be different from the experimentally obtained total energy. One possible source of discrepancy is the fact that there are dissipative energies that are not accounted for in the analytical model. A shortcoming of the present analytical model is its dependency on the displacementtime data from experiments. It would be of great interest to establish an independent relation between the current state of strain in the yarns and the deceleration of the projectile at every time-step. This will enable the code to predict not only the energy absorption of the target, but the displacement and velocity time-histories of the projectile for a specific striking velocity. 61 Chapter 3 - Analytical Model Element "dx" at time "t" dT C~ dx il (x.t) Unstrained Yarn x' Figure 3 - 1 : Displacement of an arbitrary differential element in a yarn under impact. Longitudinal Strain Wave Front "1 x r Figure 3 - 2 : Waves and their propagation in the fabric after impact. 62 Chapter 3 - Analytical Model Ct-x (C) Figure 3 - 3 : (a) Location of waves in a quarter-panel in phase-I, (b) Waves in the central yarn, (c) Waves in a yarn at distance x from the centre. 63 Chapter 3 -AnalyticalModel L J Figure 3 - 4 : Longitudinal and transverse wave location in phase-II, before the transition time. Reflected Wave Figure 3 - 5 : Configuration of waves, phase-II after the transition time. 64 Chapter 3 — Analytical Model 2w(t-t )(l-x/r) t x/'b)d b-x\ u -x Figure 3 - 6 : Deformed shape of an arbitrary yarn with free ends after the reflection of the wave. Figure 3 - 7 : Energy absorbed by the target with free boundary conditions; experiment compared with the model prediction. 65 Chapter 3 -Analytical Model 8 Impact velocity: 153 m/s 7 - Panel size: 330mm x 330mm Mc del Predit:tion Material: Kevlar 129 6 Areal density: 204 gr/m Elastic modulus: 96 GPa 5 Boundary condntions: Fixed 2 S> 4 c * mmental I Expe Data 3 10 20 30 40 50 60 70 80 Time (u,s) Figure 3 - 8 : Energy absorbed by the target withfixedboundary conditions; experiment compared with the model prediction. 66 Chapter 3 -Analytical Model LIST O F SYMBOLS a Acceleration of the yarn particles in horizontal direction. a Acceleration of the yarn particles in vertical direction. x A Base-area of the deformation pyramid at time t in Lagrangian co-ordinate system. b The diagonal of the diamond-base deformation pyramid (or deformation cone) in the global co-ordinate system. b The diagonal of the diamond-base deformation pyramid (or cone deformation cone) in the Lagrangian co-ordinate system. C Longitudinal strain wave-front velocity. d .: Projectile displacement at time t. ds Deformed length of the element dx at time t. dx A differential length on the undeformed yarn. E F Elastic modulus of the yarn material. Forces acting on the yarn in horizontal direction. F Forces acting on the yarn in vertical direction. h Fabric thickness, assumed to be twice the thickness of a single yarn. x y K.E. Kinetic energy stored in area,4/ at time /. A! K.E. Kinetic energy stored in area A2 at time t. A2 K. E. Transverse kinetic energy of the deformation cone at time t. K.E. Kinetic energy of the particles outside the deformation cone. / The deformed length of yarns that are transversely displaced. com out 67 Chapter 3 -Analytical Model V Length of that part of the yarns in area Al which is not covered by the returning strain wave. L Fabric panel size. L new Deformed length of the yarn after the reflection of the longitudinal strain wave from the boundaries. m Mass of the differential element dx. m Mass of the material in the deformation cone at time t. cone M Mass per unit length of the yarn. r Width of the reflected portion of the longitudinal strain wave. t Time. T Tension in the yarn. T The time when the longitudinal strain wave fully reflects from the boundary. Tj Termination time of phase-I. U Transverse wave speed in the global co-ordinate system. U Transverse wave speed in the Lagrangian co-ordinate system. Projectile velocity at time In-plane velocity of the particles outside the cone, swept by the longitudinal wave. The distance of points on the undeformed yarnfromthe centre. Horizontal distance of the points from the centre in a global co-ordinate system. tr V w x x' y' Vertical distance of the points from the centre in a global co-ordinate system. — First derivative with respect to x. dx 68 Chapter 3 — Analytical Model s Strain in a yarn located at distance x from the centre at time t. s Strain in the central yarn, which goes through the impact point. 0 Strain in the yarns located in area A J. e Strain in the yarns located in area A2. n(x,t) Vertical displacement of a point initially at distance x from 2 the centre at time 0 Angle between the deformed yarn and horizontal axis. p Density of the yarn material. p Areal density or mass per unit area of the fabric. a Stress in the yarns at time t. areal Cl A1 Strain energy stored in area A J as shown in Figure 3 - 4 C1 Strain energy stored in area A2 as shown in Figure 3 - 4 Q Strain energy stored in the fabric at time t. £(x,t) Horizontal displacement of a point initially at distance x from the centre at time t. A2 strain 69 Chapter 4 - Numerical Model CHAPTER FOUR: NUMERICAL M O D E L 4.1 INTRODUCTION A numerical code is developed to predict the impact behaviour of textile panels. The model, based on the model by Roylance et al. [1973] simulates impact on single and multiple layers of fabric and predicts the response of the system. The basic assumptions and formulation of the model is presented in this chapter. 4.2 M O D E L BASICS A fabric is a two-dimensional assembly of yarns, which because of its specific layout behaves like an orthotropic shell/membrane structure. However, the orthotropic conditions cease to apply during impact, since the yarns undergo significantly large inplane rotations and no longer remain at right angles. A fabric panel can be reasonably modeled as a net-like structure that is tightly knit and has certain known properties in the weft and warp directions. A numerical model is proposed to discretize a multi-layer fabric panel into a net-like mass-string system, with quasi-2D mesh configuration for each layer. The code, TEXIM, which has been developed for this purpose, is an extension of the Roylance model first introduced in 1973. In this approach, a quarter of the fabric is modeled as a finite number of nodes with three translational degrees of freedom in space (Figure 4 - 1). In each layer, the nodes are attached to the four neighbouring nodes by means of strings or cable elements in two global directions. These mass-less cable elements are basically similar to 70 Chapter 4 - Numerical Model the one-dimensional bar elements in a way that they can only carry tensile forces and offer no resistance to compression. Every node is also assigned a lumped mass to account for the mass of the fabric. The mechanical properties of every element reflect the viscoelastic behaviour of the fabric, while the masses simulate the fabric's inertia. The masses are set in such a way that the model and the real fabric have the same areal density. Multi-layer targets are modeled as a stack of individual fabric layers separated by a nominal gap. The gap controls the interaction between the layers, contacting each other at different times and locations. At the beginning of the impact, the projectile only contacts the first layer, while the other layers are still at rest. As the impact event proceeds and the deformation of the first layer exceeds the gap value, it activates the second layer, and this continues till the point when all the layers are in contact with each other and contribute to the overall resistance of the target. The principle of conservation of momentum is applied to the nodal masses of different layers in contact with each other or with the projectile, assuming an inelastic collision. Detailed discussion of the model basics is presented in the following sections. 4.2.1 F A B R I C DISCRETIZATION, MASS-SPRING M O D E L Simulation of the impact event is carried out for a certain number of discrete time intervals, At. Figure 4 - 2 shows a general node of layer k with mass m subjected to a set of internal forces from the surrounding strings at time t. The coordinates of each node are updated from the calculated nodal displacements at every time-step, and are stored in a vector as follows: 71 Chapter 4 — Numerical Model (4.1) \~ h.jM y M.j.k where (r') * 'i.j.k is the vector of coordinates of the node (ij,k), and x', y', and z' are the location of the node with respect to x, y, and z-axis, respectively. Similarly, the velocity vector of the same node would be: M= ^ (4.2) 'i.j.k i.i.k where (Y!). I S K the velocity vector of the node (JJ,k) with components v^, v' , and v[ in y each the three global directions. The same approach is applied to the forces in the strings running in x or_y-direction. Each force vector is considered as the summation of three components, which are basically its projection on the three main axes. This results in the following force vector: Tx[ (TV) Tx' Tx' Where \ Tx j ^ ' i.j.k (4.3) y i.j.k is the force vector in the x-running string attached to the node (ij,k) (see Figure 4 - 2), and Tx' , Tx' , and Tx\ are the projection of that force in the x, y and zx y directions. The magnitude of the tensile force in the cable element would then be: (4.4) i.j.k 72 Chapter 4 - Numerical Model Similar equations are derived for the forces in ^-running elements. At each time-step, the impulse-momentum balance is applied to all the nodal masses to calculate the new velocity-vector of the nodes, knowing the applied forces in the previous time-step, t-At: m ~At Knowing the velocities, the incremental displacements of the nodes for the interval At can be easily calculated. The nodal coordinates are then updated using these incremental displacements: After determining the location of the nodes in each layer, the deformed length o f the strings can be calculated. For jc-running elements we can write: Lx' x (be') v = Lx' y \ 'i.j.k Lx' -x' •^i+l.j.k yi+I,j,k z' x *i,j,k (4.7) yi.j.k -z' j.k Lx' , >i,j,i i.j.k x' where [Lx') \ ^i.j.k is the length of the string between nodes (ij,k) and (i+JJ,k) at time t, and Lx' , and Lx[ are the projection of this length on x, y, and z-axis. Similar equations y apply for the strings in the_y-direction: 73 Chapter 4 — Numerical Model Ly' x x* -x' i,j+l,k A i,j,k A yi,j+i,k (4.8) ytj.k -z' z' ^i.j+l.k i.j,k L From the above relations and knowing the length of the elements in the previous timestep, the incremental change in strain can be evaluated for each string. The updated Lagrangian elongation strain of the x-running strings at time t is defined as: 'i.j.k V x V At V where { ). £ x l s m k (i+lj,k), and (/^) e (4.9) (l'- \ 'i.j.k x 'i.j.k x updated strain of the x-running string between nodes (ij,k) and is the total length of the same string, calculated as follows: (4.10) i.j.k Similarly, the strain of the ^-running strings can be calculated as: ((/') \ y)i.j.k b with (/') \ y / \ b y )i .k 4 -(l'- ) At ) (4.11) tf-toj being given by: i,j,k (4.12) i.j.k 74 Chapter 4 — Numerical Model From the visco-elastic constitutive relations implemented in the model, the forces in the strings are related to their non-negative strain and strain rate as follows: ( T x % , k = ( M < ^ ) + h , J ^XA^'^X, k ( 4 1 3 ) (414) where E is the elastic modulus of the fibers, A is the cross-sectional area of the fibers represented by the string elements, and n is a viscosity term. The strain rate is calculated as follows: (K) = (K) = '' " M k (4.15) (4.16) It should be noted that a negative strain corresponds to a zero force. Detailed explanation of the constitutive model can be found in section 4.2.6. 4.2.2 BOUNDARY A N D S Y M M E T R Y CONDITIONS The next step is to apply the boundary and symmetry conditions. As mentioned before, only a quarter of the panel is modeled due to the existing symmetry (Figure 4 - 1). Symmetry conditions are applied to the nodes along the x and _y-axes. Symmetry implies that the nodes on x-axis do not have any motion in the _y-direction, and similarly, the nodes on y-axis do not move in the x-direction. In the code, the x-displacement of the nodes on the v-axis (z'=0) and .y-displacement of the nodes on the x-axis (j=0) are set to zero at each time-step. 75 Chapter 4 - Numerical Model Different types of boundary conditions can be modeled in the code, among which the fixed and the free boundaries are the most practical ones. To model a fixed boundary condition, the masses in the fixed zone (a line or an area on the clamped edge) of the panel are restrained in all directions, x, y and z, by setting their displacement to zero at each time-step. For the free boundary condition, no constraints are applied, and the masses preserve all their translational degrees of freedom. Other types of boundary conditions can also be considered in the code, which include non-reflecting boundary to simulate the infinite panels, and non-active direction to simulate complete slippage of the yarns in one direction. To simulate the non-reflecting boundary condition, in each time-step the in-plane velocity of the masses on the boundary were set to be equal to the in-plane velocity of the neighbouring nodes at the previous time-step. Hence, to simulate a non-reflective boundary in x-direction, the following conditions are applied to the masses located on the edges parallel to they-axis: V >Nx,j,k V V INx.j.k V y x y x 'Nx-l.j.k lNx-l,j,k where Nx is the number of nodes in x-direction. This is done to reduce the effect of nonuniformity on the boundary and prevent the reflection of the wave on the boundaries. Assuming that the wave travels about one element-length in each time-step, the above assumption is not farfromreality. To simulate a non-active direction in the fabric, the stress in the elements of that direction is set to zero in each time step. This is to model the complete slippage of the yarns in one 76 Chapter 4 - Numerical Model direction, so that only the elements in the opposite direction maintain the system's energy absorption. 4.2.3 PROJECTILE The projectile is assumed to be a blunt right circular cylinder (RCC) with mass m , p impacting the centre of the fabric at normal incidence and with a velocity V . The contact area of the projectile and the fabric is determined from the projectile radius. To simulate the blunt face of the projectile, all the nodes within its radius are assigned the same velocity and, consequently, the same displacement. Assuming zero obliquity in the projectile's motion before and after impact, it is concluded that the projectile's decelerating force results from the z-component of the forces in the strings that are in contact with it. As the nodes within this area move with the projectile and they have no transverse displacement relative to each other, the z-component of the string forces in this contact zone is zero. However, the strings crossing the edge of the projectile, which connect the nodes within the contact area to the ones outside, are the ones that contribute to the projectile's deceleration. At each time-step, the code keeps track of the masses and the strings that fall within the projectile's presented area and adds up the z-component of the tension in the edge-strings to compute the total force on the projectile. Then the impulse-momentum balance is applied to the mass of the projectile and the nodes in the contact area: —7f~(K a l ~Vr) = 1(7*; +7X) (4.18) edge 11 Chapter 4 — Numerical Model where V is the velocity of the projectile and the masses located in the contact area, and p A is the area of contact. The term ^ / w denotes the summation of the masses within the c projectile radius, and ^{Tx[ + Ty[y edge is the summation of the projected tensions in the ^ z-direction for the strings that cross the periphery of the projectile. Calculation of the projectile's velocity and displacement concludes the calculations for the current time-step. The next time step starts by adding another increment of time and going through the same procedure outlined above. 4.2.4 OUTPUT The output consists of the updated location of the nodes, in-plane and transverse velocity of the nodes, strain in the yarns, velocity of the projectile, and components of the absorbed energy by the target. The projectile energy loss at each time-step, AE , is p calculated from: (4.19) M,=±m,(V,y-±m (v;f p where m is the projectile mass, V is its striking velocity (initial velocity), and V is the p s projectile's velocity at time t. For non-deforming projectiles and barring any other sources of energy dissipation, this energy should be equal to the energy absorbed by the fabric, E'^. The latter consists of the strain energy of the strings, E' , and the kinetic s energy of the masses, E' . These two energy components can be calculated as: k 78 Chapter 4 - Numerical Model z i.j.k ''' z i.j.k , J ' (4.21) i.j.k i.j.k 4.2.5 T I M E DISCRETIZATION The stability of the numerical solution in this model is very similar to that of finite difference solution of wave propagation problems using an explicit time-integration. This stability condition, whose origin goes back to a paper by Courant, Friedrichs and Lewy [1928], restricts the time-step size, At, to be less than the time required for the stress wave to travel along the length of the smallest element. In other words, the time step should satisfy the following condition: Al <- c where / is the element length and c is the wave speed in that element. The above relation shows that the choices for time-step size and element length are not independent of each other. To satisfy the stability condition, a stability parameter X, usually referred to as CFL parameter, is introduced as follows: X= -f C (4.22) Application of this stability condition is the same as matching the computer solution rate to the stress wave propagation rate. The condition X<\ must be satisfied to obtain a stable solution, depending on the constitutive model used. A value of X = 1 results in a stable solution for linear elastic systems. Adding viscosity to the system lowers the maximum allowable value for X. 79 Chapter 4 — Numerical Model 4.2.6 M A T E R I A L CONSTITUTIVE M O D E L In order to relate the strains in the yarns to the forces induced, a constitutive model needs to be established. Before choosing a model, a basic knowledge of the material behaviour and constitutive models are essential. 4.2.6.1 Visco-Elastic Material Models Visco-elasticity is usually concerned with materials that exhibit strain rate sensitivity in response to the applied stresses. The relation between the stresses and strains and time in these solids is usually expressed through a constitutive model. The degree of complexity of such a model depends on the application of the model, availability of the experimental data, and the required precision in predicting the material behaviour. Basically all visco-elastic models are made of linear (Hookean) springs and linear (Newtonian) viscous dashpots. In a linear spring, as shown in Figure 4 - 3, we have: o = Re (4.23) where R is the Young's modulus. Equation (4.23) implies that the strain is linearly related to the applied stress. For a linear viscous dashpot: ds ° = V-r = Ve dt , (4.24) where TJ is the coefficient of viscosity. This equation states that the stress is proportional to the strain rate, e.g. the dashpot deforms continuously at a constant rate when it is subjected to a constant stress (Figure 4-3). 80 Chapter 4 — Numerical Model Different combinations of springs and dashpots can be considered. Maxwell [1890] suggested a model in which the relation between the stresses and strains in a solid is more closely predicted by the following equation: ds dt CT 1 da •+ - 7] R dt or * =rj- R+ £ (425) The above differential equation can be solved to obtain the stress-time relations under various strain conditions and strain-time relations for a given strain input. The above equation implies that if the stress is applied for a short time, the material behaves elastically and the strain is linearly related to the stress through the elasticity modulus. In contrast, if the stress is applied for a long time, the material behaves like a viscous liquid. This model can simply be idealized by a mechanical system consisting of a spring and a dashpot in series (Figure 4 - 4(a)). The spring follows the Hooke's law for elastic material and the dashpot is based on the Newton's law of viscosity implying that the velocity is proportional to the applied force. Another type of coupling between elastic and viscous properties of solids was first developed by Meyer [1874] and then modified by Voigt [1892]. This model, known as 1 1 Kelvin or Voigt model, assumes that the stress component is the summation of two sets of terms, one being proportional to the strain and the other being proportional to the rate of strain changes. As can be seen in Figure 4 - 4 (b), a Hookean spring and a Newtonian A review of these models along with the references can be found in Kolsky [1963]. 81 Chapter 4 - Numerical Model dashpot in parallel represent this model schematically. A simplified relation to express this kind of behaviour would be: 1 dt or a = Rs + 7]e (4.26) The response of this model to an abruptly applied stress is that the stress is carried by the dashpot in the beginning. The constant stress in time is then transferred to the spring as the strain increases in the dashpot, and this continues until all the stress is entirely carried by the spring. This behaviour is often called "delayed elasticity". Although these models may be useful in describing the nature of material's viscous behaviour qualitatively, they are often too simplistic for quantitative applications. A general case would be to consider a slightly more complicated arrangement, combining the features of Maxwell and Kelvin/Voigt models (Figure 4 - 4 (c)). In the literature, many different models can be found which may consist a number of Maxwell models and/or a number of Voigt models in series [Kolsky, 1963]. Application of these models depends on the complexity of the problem, material behaviour and characteristics, and degree of accuracy needed. More discussion on different types of constitutive models and their application can be found in references [Mase, 1970] and [Findley, 1976]. 4.2.6.2 High Strength Aramid Fibers Incorporation of visco-elasticity in the constitutive model of polymeric materials is essential due to their highly visco-elastic behaviour. The morphology of polymers shows 82 Chapter 4 - Numerical Model that highly oriented polymeric fibres (like aramids) consist of long chains of macromolecules. The intermolecular bonds (primary bonds) are mainly hydrogen bonds, while the forces between the chains (secondary bonds) are covalent in nature. Under stress, these chains slide on each other and as a result, the material exhibits a viscous behaviour. A suitable constitutive model for these materials is the one that considers the strain from both inter-molecular and intra-molecular bonds and the viscous property of the sliding chains. A study by Shim et al. [1995], based on the findings of Termonia and Smith [1988] indicates the efficiency of a three-element model, which consists of a system of parallel dashpot and spring in series with another linear spring as shown in Figure 4 - 5. In this system, the first spring in series with the other elements, KI, represents the primary bonds in a polymer chain, and the second spring, K2, parallel to the dashpot simulates the effect of secondary bonds. 4.2.6.3 Constitutive Model In the present model, a system of two elements is considered; a spring, representing the displacement in both primary and secondary bonds, parallel to a dashpot in order to model the viscous behaviour of the fibres. Thus, the mathematical formulation for such a constitutive model would be: T = Ke + ne (4.27) where T is the tension in the string, K is the tensile stiffness (elasticity modulus multiplied by the element cross-sectional area), and r/ is a viscosity parameter. 83 Chapter 4 — Numerical Model 4.2.7 FAILURE MODEL Failure of the target is probably one of the most crucial tasks to be accomplished. The failure criteria currently implemented in the code is based on the comparison of a representative parameter against an allowable threshold. The force-strain curve adapted is shown in Figure 4 - 6. As can be seen, the material behaviour is assumed to be completely linear elastic before the strain reaches the breaking limit and the ensuing catastrophic failure. Also, the effect of weave and initial crimp in the fabric can be modeled by introducing e , which specifies the threshold of strain in the elements, crimp which triggers their resistance (Figure 4 - 6(b)). The crimp strain can be obtained experimentally by comparing the actual density of the fabric with the density of the raw material. The first failure model of the code is based on the measure of the instantaneous strain in the individual yarns. When the strain in an element exceeds the breaking strain, it is considered failed. The code assigns a zero force to the failed elements, eliminating them from the system for the rest of the calculations. The overall failure of the fabric is detected by checking the total resisting force on the projectile. The erosion of the failed elements eventually leads to a zero force acting on the projectile from the target, which implies the perforation of the projectile. At this point, the code stops and reports the residual velocity. Due to the numerical oscillation in the elements' strain-time data, the instantaneous failure model might not be able to predict the failure of the target. To overcome this deficiency, the strain in an element is averaged over the time that element is excited. For an element (ij,k) at time t = nAt the average strain is calculated as follows: 84 Chapter 4 -'NumericalModel C^ZC (4-28) i=0 This average strain is then compared against the threshold considered for failure. Averaging the strain overcomes the numerical oscillations and results in a better failure prediction in some cases. Finally, a third failure criterion is introduced which is based on the two previous criteria. In each time-step, the strain is averaged over the values of the last five time-steps. The following relation shows the formulation of this average strain for an element (ij,k): <r*=4ix£ (-> 42 9 •> i=0 The average strain obtained is then compared against the failure strain. This method of averaging is best to eliminate the effect of local high-frequency oscillations. Figure 4 - 1 shows a comparison of results from the above failure criteria. Further discussion on the results of the failure models is presented in Chapter Five. 4.3 PREDICTIONS The model discussed above was assembled together in a Fortran77 code. The output generated by this code for the case of impact of a projectile weighing 2.78 grams, with a diameter of 5.56 mm flying at 100, 150, and 200 m/s on a 330 mm by 330 mm square panel of Kevlar 129 with fixed boundaries is shown in Figure 4 - 8 . This figure shows the plots of the displacement-time, velocity-time, and energy-displacement of the projectile for the first 300 (is of the impact. These graphs are useful to track the displacement of the projectile in time and study the energy absorption of the target in an 85 Chapter 4 - Numerical Model on-going impact event. Comparison between these results and the experimental data are presented in Chapter Five. Also effect of different boundary conditions on the energy absorption of the target is shown in Figure 4 - 9 . The deformed shape of the target can be plotted at each time-step using the updated coordinates of the nodes. Figure 4 - 1 0 shows the deformation of the target at 50 ps after an impact of an RCC projectile with the striking velocity of 200 m/s on a panel with free boundary conditions. Also, the longitudinal and transverse waves in the target can be traced by creating a contour plot of the in-plane and out-of-plane nodal velocities, as shown in Figure 4 - 1 1 . The deformed shape of the target and the waves' contour plots are generated using TECPLOT™. TECPLOT™ is a software useful for visualizing a wide range of data. It provides the environment to create a variety of plots, among which the 2D and 3-D mesh and contour plots were used in this study to investigate the deformation and wave propagation characteristics of the target. More discussions about the validity of the results, and their comparison with experimental data are presented in Chapter Five. 4.4 VERIFICATION As there is no closed-form solution available for the response of a fabric under impact, it is not easy to check the accuracy of the model predictions. Thus, the model is compared with other numerical models and codes, and experimental information. In this section, the code is verified against the results available in the literature and previous works, and also results from other finite element codes. 86 Chapter 4 — Numerical Model 4.4.1 VERIFICATION W I T H T H E D A T A FOUND IN L I T E R A T U R E The predictions of the present model were verified by comparison with the results of other numerical solutions for impact of textile materials. Figure 4-12 shows the resulting absorbed energy for the case of an elastic square panel of single layer Kevlar® 29, along with the results obtained by Roylance and Wang [1980]. The figure shows the total absorbed energy up to 12 \xs of the impact, and the partition of this total energy into strain energy, transverse kinetic energy and in-plane kinetic energy. Good agreement was found between the results from Roylance and Wang model and the current code. 4.4.2 V E R I F I C A T I O N AGAINST L S - D Y N A LS-DYNA is a multi-purpose explicit finite element code capable of performing static and dynamic analysis of different structural systems. Normal impact of a blunt cylindrical projectile on a single layer of fabric is simulated in LS-DYNA using the cable material model [MAT71], which breaks down the fabric into a combination of masses and strings (or cables). Predictions of this code were used to further verify the results of the model presented here. Figure 4-13 shows the results from the two codes for the impact of a projectile weighing 2.87 gr flying at 158 m/s and impacting a 330mm by 330mm panel of one-ply Kevlar® 129 with two different boundary conditions: completely free all-around, or fixed. The radius of projectile was assumed to be so small that it only affected the central node (single-node impact). Figure 4-14 shows the same impact for a larger value of projectile radius. In this case, the projectile contacts a number of nodes in the contact area (patch impact). As can be seen, the results from the two codes are almost identical. This, 87 Chapter 4 - Numerical Model considering the fact that both of the codes share the same basis for analysis, is further proof for the validity of the present code results. 88 Chapter 4 - Numerical Model Ay i Figure 4-1: The impacted panel and symmetry conditions. Level i-1 • B Level i • i Level i+1 > t Level j+ • Level j — •• • Level j-1 ii i Figure 4-2: Forces applied from the neighbouring elements on an arbitrary node of the mass-string system. 89 Chapter 4 - Numerical Model Figure 4 - 3 : Behaviour of a linear spring and a linear dashpot. (a) (b) (c) Figure 4 - 4 : Schematic presentation of visco-elastic material constitutive model. Chapter 4 — Numerical Model 1 4 — Primary Bond Secondary Bond O KI -WVVH K2 Monomer (a) (b) Figure 4-5: Constitutive model suggested by Shim et al. [1995]. (a) (b) Figure 4-6: Tension-strain relation used for the material. 91 Chapter 4 - Numerical Model \lnstantar eous Strain E •£ 0.015 c I Hi U \ Average ov rer 5 tlmeste A 0.010 OT 50 100 [Average o /er W/ne [ 150 200 250 300 Time (us) Figure 4 - 7 : Comparison of the stress calculation used in failure criterion of the model. 45 \Vf200mh | 40 35 E 30 E c | Q 25 20 15 - 100 mh | 10 5 0 50 100 150 200 250 300 350 '— „ 120 ^ 100 ^ jUs = 150Tl/J | |V5 = » 0 0 m & | 10 15 20 35 40 « Displacement (mm) Figure 4 - 8 : Displacement, Velocity, and Energy plots for different values of striking velocity predicted by the model. 92 Chapter 4 - Numerical Model Figure 4-9: Comparison of the displacement-time and velocity-time curves for different boundary conditions. Figure 4-10: Deformed shape of the fabric with free boundary conditions impacted at 200 m/s, after 50 u.s. 93 Chapter 4 - Numerical Model •Strain Wave Propagation in x-running Elements. After30 micro-s" I I Transverse Wave Propagation. Time=30 micro-s" Figure 4-11: Longitudinal and Transverse wave configuration propagating in the jc-direction after 30 ps. Figure 4 - 12: Energy absorbed by one-ply Kevlar 29 panel (Lines: T E X I M ; Markers: Roylance and Wnag [1980]). (Abdel-Rahman et al. [1998]) 94 Chapter 4 - Numerical Model Figure 4 - 13: Comparison of displacement and velocity time-history of the projectile with LS-DYNA predictions, single-node impact. Figure 4 - 14: Comparison of displacement and velocity time-history of the projectile with LS-DYNA predictions, patch impact. 95 Chapter 4 — Numerical Model LIST OF SYMBOLS A Cross-sectional area of an element. c Strain wave velocity in an element. — Differentiation with respect to time. dt E Elastic modulus of the material. E[ Kinetic energy of the system at time t. E' Strain energy of the system at time t. i An integer representing the position of each node in xdirection. j An integer representing the position of each node in_y- s direction. k An integer representing the layer number of each node. K Tensile stiffness of each element. [Lx') Vector containing the projection of an element length at time t on the three global axes. Lx' Projection of the element length at time t on x-axis. Lx' Projection of the element length at time t on_y-axis. Lx\ Projection of the element length at time t on z-axis. x y (l' ) x m k Length of element (ij,k) at time Nodal mass at each crossover. m Projectile mass. Nx Number of elements in x-direction. p 96 Number of elements in ^-direction. Vector of nodal coordinates. Slope of stress-strain curve, or Young's modulus. Tension in the string elements. Vector of tension in x-running elements at time t. x-component of tension in an x-running element at time t. v-component of tension in an x-running element at time t. z-component of tension in an x-running element at time t. Velocity vector of node (ij,k) at time t. x-component of nodal velocity at time t. .y-component of nodal velocity at time t. z-component of nodal velocity at time t. Projectile velocity at time t. Impact striking velocity. Component of nodal coordinate on x-axis. Component of nodal coordinate on^-axis. Component of nodal coordinate on z-axis. Projectile's energy loss, equal to the absorbed energy by the target at time t. Time-step size. Initial crimp strain. Average strain in element (ij,k) at time t. Chapter 4 — Numerical Model (e' y k Vector of strain in x-running element (ij,k) at time t. [e ) k Strain-rate vector of the x-running element (ij,k) at time t. x x X Courant-Friedrichs-Lewy (CFL) number for stability condition. 77 Coefficient of viscosity. 98 Chapter 5 - Results and Discussion C H A P T E R F I V E : R E S U L T S A N D DISCUSSION 5.1 INTRODUCTION In this chapter, the numerical predictions are compared with available experimental data, to investigate the validity of the model. The experimental set-up and procedure for the ballistic impact tests are introduced and the data obtained are presented. Finally a series of numerical experiments are performed to investigate the effect of various input parameters on the ballistic response of fabric panels. 5.2 COMPARISON WITH E X P E R I M E N T A L DATA A novel experimental measurement technique is used to obtain the displacement-time history of the projectile during the impact. In this section, the experimental set-up and the technique used for simulating the ballistic impact of small projectiles on textile panels are briefly reviewed. The data obtained from the impact tests are then presented and finally the numerical predictions for the impact on single and multi-layer Kevlar® 129 panels are compared with the experimental data. 5.2.1 E X P E R I M E N T A L SET-UP Ballistic (high velocity) impact tests are usually performed using a low-mass projectile flying at a velocity in the range of 100 to 1000 m/s. These tests are designed to study the impact of bullets, fragments or other flying objects on composite protective armours. Textile-based personal protective systems are a good example of this study. 99 Chapter 5 - Results and Discussion In order to launch projectiles, a powder gun made of a universal receiver attached to a 5.58 mm (0.22") diameter Remington rifle barrel is used. To decrease the amount of debris and blow-by smoke, mainly resulting from the unburned powder, a blast deflector is placed at the end of the barrel. The projectile hits the fabric panel at the centre, which is held in a fixture as shown in Figure 5 - 1 . The fixture is made of aluminium and designed to impose fixed boundary condition on the fabric by means of a series of rods, with a mechanism shown in the same figure. The clamping force at the fixed boundary is applied at a distance 105 mm (4.1") from the centre of the target, resulting in a panel size of 210 mm (8.2") in the fixed direction. To achieve a free boundary condition these rods are removed and the fabric is held in place by lightly taping its corners to the fixture. The panel size for free boundary conditions would be the fabric size itself, which can vary in the range between 210 mm (8.2") and 330 mm (13"). A catchment chamber was placed behind the target to stop the projectiles that perforate the target. The powder gun was calibrated to determine the amount of charging powder and placement of projectile in the barrel for a specific striking energy. For more information on the experimental equipment and set-up, see Starratt [1998]. The experimental set-up was instrumented with a displacement-tracking device called the Enhanced Laser Velocity System (ELVS). This system, developed in-house, is capable of tracking the projectile before and during impact, resulting in a continuous measurement of projectile displacement. The concept of ELVS is based on the measurement of the laser intensity in time, during the motion of the projectile towards the target. Details on this measurement device are provided in Starratt et al. [1999] and Starratt [1998]. 100 Chapter 5 - Results and Discussion Figure 5 - 2 shows the normalized velocity of the projectile and energy absorbed by the fabric versus projectile displacement for two typical impact conditions on single layer of Kevlar® 129 with fixed and free boundary conditions all-around. The mechanical properties of Kevlar 129 yarns and fabrics can be found in Table 5-1 and Table 5 - 2 . Table 5 - 3 presents the properties of the two types of projectiles used in all the experiments reported here. Information on the panels used and their sizes corresponding to each boundary condition can be found in Table 5 - 4 . The currently available experimental database consists of impact data on 1, 3, 8, and 16 layers of Kevlar® 129, for a wide range of striking velocities. 5.2.2 E X P E R I M E N T A L RESULTS The data acquired from ELVS is processed to determine the time histories of the projectile displacement, velocity, and acceleration during impact. Discussion about the treatment of raw data is out of the scope of this thesis, and can be found in Starratt et al [1999]. The outcome of the process is only used to investigate the physical behaviour of textile materials and verify the numerical model. Figure 5 - 3 shows the energy absorbed by the target against the initial striking energy of the projectile for three different boundary conditions. These types of curves, usually found in the ballistics literature, provide information on the energy absorbed by the target (textile) after being perforated by the projectile. Before perforation, the target absorbs all the striking energy of the projectile, E (shown by the line at 45 degrees in the figure). s Upon perforation, occurring at a velocity termed the ballistic limit, the energy absorbed by the target drops suddenly. The energy absorbed increases slightly at higher velocities, 101 Chapter 5 - Results and Discussion as can be seen in the figure. The sudden drop in the total absorbed energy after the ballistic limit is the result of yarn failure and target perforation. This sudden drop indicates that just above the ballistic limit the target does not reach its full potential for absorbing the energy presumably because some of the yarns in contact with the projectile before all the yarns in the fabric have had the opportunity to deform. In post-mortem examination of the fabrics showed fibre pullout in some areas. This effect was more pronounced in the panels with free-free boundary conditions (Figure 5 - 4), where extensive pullout was observed in the orthogonal yarns intersecting the impact point. Examining the ELVS results for these tests (Figure 5 - 2), it can be seen that the velocity of the projectile is reduced by slightly more than 10%. Neglecting the friction between the yarns, a simple kinetic energy balance correlates the projectile velocity reduction with the additional mass of the orthogonal yarns that are pulled out of the fabric. 5.2.3 COMPARISON B E T W E E N N U M E R I C A L A N D E X P E R I M E N T A L R E S U L T S The experimental data available was used to evaluate the numerical predictions of TEXEvl. Table 5-5 provides a list of all the experiments that are used to compare against the model's predictions. The discussion here is focused on both single and multi-layer targets. The prediction of the model has been studied for a series of tests on single-layer targets with different boundary conditions. Single-layer tests are valuable in the sense that they eliminate the interaction between adjacent layers, which exists in multi-layer stacks. Simulating single layers of fabric helps verify the main assumptions of the model without 102 Chapter 5 - Results and Discussion introducing the complicated effect of the contact between the various layers in multilayer targets. Examining the fabric panels after impact and observation of the fibre slippage at the fixed edges reveal that applying a perfect clamping condition at the boundaries is a difficult task. Also, the test fixture itself absorbs a fraction of the impact energy when reflecting the strain waves, while in the model the energy is assumed to be conserved within the projectile-fabric system. Considering the difficulty of providing a perfect clamping condition at the boundaries and the interaction between the fabric and the testfixture,the code predictions based on the assumption of fixed boundary conditions are questionable. Therefore, the behaviour of the fabric is compared with the numerical simulations of the panels with completely fixed and completely free edges, in anticipation that the experimental response lies between these two extreme cases. Figure 5 - 5 presents the experimental results for a panel fixed on all four edges, superposed on the predicted results forfixed-all-around(referred to as "fixed-fixed") and free-all-around (referred to as "free-free") boundary conditions. The measured response of the fabric is seen to be somewhere between the model's prediction for fixed-fixed and free-free cases. The numerical simulation of an experiment with free-free boundary conditions is also shown in Figure 5 - 6. In the panels with free boundaries, extensive amount of fibre pullout was observed in the post-mortem examination. As the numerical model is not capable of considering the fibre slippage, the results are expected to be different from the experimental data. However, the prediction of the model is in good agreement with the 103 Chapter 5 — Results and Discussion experiments, especially for the initial part of the curves where the effect of slippage is not significant. The predictions for both the fixed-fixed and free-free boundary conditions are superposed on the corresponding experimental data in Figure 5 - 7. It can be seen that the model predictions agree fairly well with the results obtained from the experiments. 5.2.4 COMPARISON B E T W E E N N U M E R I C A L A N D A N A L Y T I C A L RESULTS Figure 5 - 8 shows a comparison of the predictions using the numerical model and the analytical model presented in Chapter Three. Energy absorption characteristics for impaict of a single layer of Kevlar® 129 with fixed-fixed boundary conditions impacted by a type-I projectile travelling at 225 m/s are compared. It can be seen that the predictions using these two approaches are close, not only in estimating the total absorbed energy, but also in evaluation of the energy components. The difference in the predicted values from the two methods can be attributed to the different assumptions made in the two models. In particular, the analytical model makes many simplifying assumptions such as neglecting any interaction between the warp and weft yarns. The velocities of the longitudinal and transverse waves in formulation of the analytical model are assumed to be constant, and are calculated based on the relations derived for singlefibreimpact. The numerical model, however, considers the interaction of the warp and weft yarns and the wave velocities are obtained intrinsically within the mechanics of the system. Moreover, the analytical model is capable of estimating the energy absorption for only one reflection of the wave from the boundaries, while the numerical model has no such limitation. Generally, results of the two models seem to be in good agreement. The analytical model 104 Chapter 5 — Results and Discussion is also considered to be a good tool to understand the characteristics of wave propagation in an impacted fabric. 5.3 NUMERICAL EXPERIMENTS A series of numerical experiments was performed to study the sensitivity of the model predictions to different input parameters. The results of the parametric studies are presented in three main categories. The first set of results considers the parameters that specify the geometry of the panel. The second set covers the parameters related to the material properties of the target, and the third category relates the response of the fabric to the projectile properties and impact conditions. The failure criteria implemented in the code are examined and the advantages and disadvantages of each failure model are discussed. All the parametric studies performed are summarized in Table 5 - 6 , along with the other input information used in the computer runs. It should be noted that in studying each parameter, all the other input parameters were held fixed at their baseline values. These baseline parameters are nominal values that are known from the literature and are those that were used in numerical prediction of the experiments presented in the previous section. The test used for the studies on a single-layer of fabric is the impact of a type-I projectile on a panel withfixed-fixedboundary condition and a striking velocity of 148 m/s. For multi-layer targets, a type-I projectile flying at 267 m/s and impacting an eightlayer fabric panel with fixed-free boundary condition (fixed on two edges and free on the other two) was simulated. 105 Chapter 5 — Results and Discussion 5.3.1 TARGET GEOMETRY PARAMETERS Sensitivity of the panel response to the parameters related to the geometry of the panel was studied by specifically considering different boundary conditions and panel sizes in single-layer targets and also the effect of gap between layers in multi-layer targets. 5.3.1.1 Boundary conditions Figures 5 - 9 shows the impact response of panels with different boundary conditions. The strain in element A (as shown in figure 5 - 10) is also presented in the same figure. Upon impact, the strain in the element suddenly increases due to the generation of the strain waves in the system. After that, the strain increases almost at a constant rate, until the time that the reflected strain wave passes through the element, at which time another discontinuity occurs. These sudden changes in the strain can be additive or subtractive depending on the type of the boundary conditions. In the case of reflection from a fixed boundary, the strain wave doubles the value of strain in the element on its way back to the impact point. On the other hand, the reflected strain wave from a free edge cancels the existing strain in the element as it travels back to the centre. This behaviour is clearly observed infixed-fixedandfree-freepanels (Figure 5-9). However, for the panels with fixed-free boundaries (fixed on two parallel sides and free on the other two) the behaviour is somewhat different. The strain decreases to zero after the passage of the first reflected strain wave from the free edge; however, due to the restraining force from the crossing elements fixed in the perpendicular direction, the strain value in the elements increases later in the event. Considering the energy absorption of the panels with different boundary conditions, it can be seen from Figure 5 - 9 that the panel with fixed-fixed boundary absorbs more o 106 Chapter 5 - Results and Discussion energy for a given displacement of the projectile than other boundary types. Table 5 - 7 compares the components and the total energy absorbed by similar panels with different boundary conditions (fixed-fixed, fixed-free, free-free and infinite) 100 \is after impact. It can be seen that in the fixed-fixed case, most of the energy absorbed is in the form of strain energy. Also the kinetic energy of the transversely moving masses is significant due to the large size of the deformation cone (Figure 5 - 11). The kinetic energy from the in-plane motion of the masses is very small due to the restraining forces at the boundaries. On the other hand, in the free-free case, the strain energy stored in the fabric is almost negligible and all the absorbed energy is in the form of kinetic energy, both inplane and out-of-plane. The panel with fixed-free boundary type exhibits a behaviour that falls between the free-free and fixed-fixed cases, where the strain energy, in-plane and transverse kinetic energies are all of the same order. For an infinite panel, the energy components calculated by the code is only valid for the part within the modeled fabric (i.e. meshed part), as there is no information available for material points outside the mesh. 5.3.1.2 Panel size Results of the study on the effect of panel size, shown in Figure 5-12, imply that initially the response of the fabric for all boundary types follows the same trend, diverging at a time referred to here as "bifurcation point." Further investigation reveals that this time is a function of the panel size and corresponds to the time when the first reflection of the strain wave reaches the impact point. At this time, the panel's boundary condition affects the projectile deceleration and the response bifurcates from the original path. It can be concluded that the original path is followed in the case where there is no reflection of the 107 Chapter 5 — Results and Discussion wave from the boundaries. This is the governing condition for an infinite panel or a panel without any boundaries. 5.3.1.3 Inter-layer gap Modeling the response of multi-layer fabrics using TEXIM introduces a new geometrical variable, which is the gap between the layers. Effect of this gap between the layers has been studied by assigning different numerical values for the spacing between the layers. As shown in Figure 5 - 1 3 , increasing the gap between the layers decreases the energy absorption of the target. The reason is that with increasing gap, the impacted layer continues to deform through the gap distance without experiencing the additional resistance that it would experience from its contact with the next layer when the gap is small or nonexistent. For the same reason and because each layer has to undergo a larger deflection till it reaches the next one, the strain in each layer increases for increasing values of the gap. 5.3.2 T A R G E T M A T E R I A L P A R A M E T E R S The parameters related to the material used in the target are limited to the elastic modulus, strain-rate sensitivity (viscosity) factor, initial crimp strain, fabric's areal density, and elements' breaking strain. Effects of each of the above parameters are studied and discussed separately. 5.3.2.1 Elastic modulus Increasing the elastic modulus of the fabric directly increases its energy absorption, as can be seen in Figure 5 - 1 4 . Also, monitoring the strain development in a yarn going through the impact point (or x-axis) reveals that there is a relationship between the elastic 108 Chapter 5 - Results and Discussion modulus and the velocity of strain wave front. It was observed that increasing the elastic modulus leads to a higher velocity of the strain wave. However, the sensitivity of the panel response to elastic modulus is not very high, as a 20% increase in elastic modulus increases the absorbed energy by about 10% after 15 mm displacement of the projectile. 5.3.2.2 Viscosity It was observed that using a completely linear elastic constitutive model results in numerical instability later in the analysis, when the reflected and on-going waves interact. This effect is more noticeable in the panels with fixed boundary conditions. Adding viscosity to the system dampens these numerical oscillations. Results of the analysis performed for viscosity parameters of 1%, 5%, 10% and 25% show that regardless of the viscosity parameter value used, the displacement and velocity of the projectile and energy absorption of the fabric remains the same (Figure 5-15). Adding strain rate sensitivity damps out the oscillation in the element strain significantly compared to the linear elastic case. The change in the strain history becomes less significant for those values of the viscosity parameter larger than 5%. Hence, a viscosity value of 5% was chosen to perform all the analyses reported here. 5.3.2.3 Crimp Sensitivity of the results to the value of crimp, which is not a material property but is related to the fabric weave, was investigated as shown in Figure 5-16. Adding crimp to the system appears to make the system softer initially, thus decreasing the energy absorption of the fabric for a given projectile displacement. After the strain develops in the elements and the slack is taken out, the fabric behaves like a taut system. It is worth noting that the crimp affects the time for the first strain wave to reach the boundary. This 109 Chapter 5 - Results and Discussion can be observed on the strain plot of element A (Figure 5 - 10) by comparing the reflection times of the strain wave. It is interesting to note that the time delay is proportional to the crimp value. Also, addition of the crimp to the model increases the numerical oscillations in the strain-time response, which may be attributed to the reflection of the wave at each member before the slack is taken out. 5.3.2.4 Areal density Similar impacts on panels with different areal densities (pareai) were simulated, and the results are plotted in Figure 5 - 1 7 . From the results obtained, it can be concluded that the fabrics with higher areal density absorb more of the projectile energy for a certain displacement of the projectile. The strain values in the elements also increase considerably with decreasing areal density. However, it should be noted that using fabrics with higher areal density results in heavier body armours, which reduces the efficiency and manoeuvrability of the personnel using the armour. 5.3.2.5 Failure criteria The failure criteria considered in the code are based on the instantaneous and the average values of the strain in time. The most direct way of simulating failure is monitoring the strain value at every time-step and comparing it to a threshold. This method assumes that the yarns fail at the moment when their strain exceeds a pre-assigned breaking value. However, in some cases, this approach results in unrealistic determination of failure, as the instantaneous value of strain in elements might be artificially increased due to numerical oscillations and do not represent the true state of strain. Adding strain-rate sensitivity and averaging the strain in elements over a period of time results in a more realistic evaluation of strain and hence, a better failure model. 110 Chapter 5 — Results and Discussion One of the averaging methods implemented in the code is averaging the strain over the total impact time. This method results in a strain value that is very different from the current state of strain in the elements. Some studies suggest failure criteria based on the accumulation of strain energy in the yarns, rather than just looking at the strain value in time [Chocron-Benloulo et al., 1997, Roylance et al., 1980]. The overall average strain presented above can be a good measure of the energy in the yarns to be used for failure considerations. The second averaging scheme focuses on averaging the value of strain in a window of five time-steps. Using this approach, the numerical oscillations in the strain response are locally suppressed and a mean value for the strain in each element can be obtained. This method of strain evaluation is useful when considered with a breaking strain to determine the failure of elements. The threshold considered for failure is taken from the value of elongation at break for Kevlar® 129 yarns. The nominal breaking strain (sbreak) is reported to be between 3.3 and 3.5% for Kevlar® 129 yarns [Pageau ,1997, Du Pont, Hexcel]. Different values of breaking strain were tested in the code to find the sensitivity of the results to this parameter. Figure 5 - 1 8 and Figure 5 - 1 9 show plots of residual velocity vs. striking velocity and the plot of the absorbed energy by the target against the striking energy of the projectile. These graphs are produced for the impact of type-I projectile on 210 mm x 210 mm panel of Kevlar® 129 with fixed-fixed and fixed-free boundary conditions, assuming two breaking strains of 3% and 3.5% for the constituent yarns. It can be seen that the absorbed energy by the target dramatically decreases immediately after the ballistic limit, increasing slightly thereafter at higher impact velocities. Figure 5 - 20 111 Chapter 5 - Results and Discussion shows the strain-time plot of element A. It can be seen that at higher impact velocities, both the magnitude and the rate of strain increases and therefore the failure threshold is reached faster than at lower impact velocities. However, the strain energy value (as a ratio of the total absorbed energy) does not change significantly at higher impact velocities (in this case, for impact velocities between 180 and 250 the change in the ratio is less than 10%, as shown in Figure 5 - 20). Another representation of these data can be found in Figure 5-21, where the result for the panel withfixed-fixedboundary condition and 3% breaking strain is plotted. To get a better qualitative sense of energy absorption, this value is normalized against its relevant striking energy. Also, the values of time-to-failure for the same impact events are overlaid on the same plot. It can be seen that the time-to-failure decays almost exponentially with the striking velocity. The reduction in the time-to-failure becomes less pronounced at higher impact velocities (i.e. above 200 m/s). For the same reason the portion of the striking energy absorbed by the target reaches a plateau at about 10% (Figure 5 - 21). In other words, at higher impact velocities the energy absorbed increases linearly with striking energy. Similar observations were made by Roylance [1977] for impact on single yarns, where the value of time-to-failure was seen to vary exponentially with the impact velocity. 5.3.3 PROJECTILE PARAMETERS The projectile is introduced to the system via its mass, radius and striking velocity. The focus of this study is on the change in projectile mass while maintaining a constant 112 Chapter 5 - Results and Discussion striking energy. Also, the impact of projectiles having the same mass and velocity, but different radii is studied here. 5.3.3.1 Iso-energy impacts Simulation of the iso-energy impacts for different projectile masses was performed to investigate the effect of this parameter on the fabric response. In order to maintain the same impact energy for all the masses, striking velocities were calculated as shown in Table 5-6. Figure 5-22 shows the energy-displacement and strain-time curves for such an impact. It can be seen that for the lighter projectile, the absorbed energy by the target (or energy loss by the projectile) is higher, while the heavier projectiles result in a smaller rate of energy loss. Considering the strain values in different elements reveals that in the case of light projectiles, as they have higher striking velocities for the same energy, the initial strain that develops in the yarns is significantly higher. This can easily lead to failure of the yarns in the early stages of impact, before the projectile energy is transferred to the entire fabric. It should also be noted that the lighter, and thus smaller, projectiles might penetrate the fabric by going through the spacing between the woven yarns and pushing them aside, rather than straining them to failure. This might introduce a greater threat to the system of fabrics, and the current model is not capable of considering such phenomenon. 5.3.3.2 Projectile radius To investigate the effect of projectile radius (or contact area) on the ballistic response of fabric panels, a series of numerical simulations were performed by changing the radius of the projectile to cover 1, 2, 3, and 4 nodes of the target in each of the principal directions, x and y. Choosing the correct projectile radius is critical, as the interaction between the 113 Chapter 5 - Results and Discussion projectile and the panel is through the elements that cross the projectile perimeter. Choosing a smaller value for the projectile radius results in higher strain value, mainly in the elements crossing the projectile perimeter. This will then result in a premature failure of the fabric. Figure 5 - 2 3 shows the difference in the strain value and also energy absorption of the target, considering various values for the projectile radius. It should be noted that the strain value plotted is in the fifth element away from the centre of the fabric on the assigned x-axis. This element was chosen to make sure that in all the cases the monitored element is outside the projectile contact area. It was observed that the strain in the elements, especially the ones around the contact area of the projectile and the fabric, increases dramatically with the reduction in the projectile radius. This is the result of the fact that the elements crossing the perimeter of the projectile are the only links between the projectile and the rest of the fabric. Smaller projectiles are linked to the fabric via fewer number of elements, which results in higher force and strain in those elements. In this study, the element size (1.05 mm) is chosen to be as close as possible to the length of the yarns within two consecutive crossovers (0.91 mm). The only limitation in using smaller values of element size is the computational restrictions due to the increase in the number of elements, and thus increases in calculation time. 5.4 CONCLUSIONS The numerical studies show the importance of most of the input parameters. This implies that accurate information on the geometric and material properties is needed to simulate an impact event. Boundary conditions are found to play a major role in the energy 114 Chapter 5 - Results and Discussion absorption of the fabric layers. Thus it is beneficial to realistically model the boundary condition of the fabrics in personal body armours, and apply boundary types that are seen to be more efficient in decelerating the projectile. The decrease in the gap between layers is found to increase the ability of multi-layer targets to arrest the projectile. Therefore it is recommended that the spacing between the layers be reduced as much as possible by means of stitching the layers together. This is also true for the case of yarn crimp in the fabric, where less crimp (i.e. straining the yarns) improves the energy absorption of the fabric. Elastic modulus directly affects the ballistic response of the fabric. It should be j noted that although increasing the elastic modulus has a favourable effect on energy absorption, the ductility of the system might reduce due to the reduction in the breaking strain. However, a medium balance should be sought to find the best combination of elastic modulus and breaking strain of the yarns [Roylance, 1977]. Areal density of the fabric is of high importance in the behaviour of target, as well as the serviceability of the armour and comfort of the person wearing it. Higher areal density results in a better ballistic performance, while it results in higher weight and consequently, less manoeuvrability. Instantaneous strain-based failure criterion seems to be able to predict the perforation phenomenon fairly well. The effect of strain rate sensitivity of the material is not very obvious, however, application of numerical damping helps reduce the oscillations in the response. There are also some shortcomings in this model, of which the user should be aware. The crossovers are modeled as pinned joints, which prevents the yarns from slipping. This might result in unrealistic predictions in the cases where the amount of yarn slippage is significant. The algorithm employed to simulate the interaction between the individual 115 Chapter 5 - Results and Discussion layers neglects any coupling between the masses in the in-plane direction. Also, the projectile is assumed to be non-deformable and is represented by its distributed mass over the contact area. A more realistic model would have to consider the contact of the projectile and its deformation during impact. 116 Chapter 5 — Results and Discussion Table 5-1: Material properties of Kevlar® 129 yarns [Pageau, 1997]. Tensile Elastic Modulus, N/tex* (GPa) 66.8 (96) Tensile Strength, N/tex* (MPa) 2.35 (3378) Strain to Failure, % 3.3 Yarn Density, Denier** (dtex***) 840 (930) Specific Gravity, gr/cm 1.44 3 * tex: The basic property to define the fineness (and conversely, the cross-sectional area) of a yarn. It is defined as the weight in grams of 1,000 meters of the material [Dupont]. ** denier: Property unique to the fibres industry to describe the fineness of a filament, yarn, rope, etc. It is defined as the weight in grams of 9,000 meters of the material. *** dtex: Standard abbreviation for "decitex." Table 5 - 2: Properties of Kevlar* 129 fabric [Pageau, 1997]. Weave Plain l x l Count, yarns/cm 11 Areal Density, gram/m 204 Yarn Crimp < 3% difference between warp and weft Fibre Tenacity, gram/denier* 20 (warp and weft) 2 117 Chapter 5 — Results and Discussion Table 5 - 3 Properties of the projectiles used in the impact tests. Properties Projectile Type-I Projectile Type-II Section Circular Circular Radius, mm 2.87 2.87 Length, mm 45.7 46.5 Material Aluminium Steel Mass, gram 2.99 8.8 Table 5 - 4 Panel properties used in simulating experiments. Boundary Condition Fixed-Fixed Fixed-Free Free-Free Shape Square Rectangle Square Horizontal Dimension, mm 210 330 330 Vertical Dimension, mm 210 210 330 Material Kevlar® 129 Kevlar® 129 Kevlar® 129 118 Chapter 5 - Results and Discussion Table 5 - 5 Experimental data available for different impact specifications. Test Number of Boundary Layers Conditions Velocity (m/s) Perforation Layers Projectile Penetrated Type 8428-lp-fixed-fixed-01 l Fixed-Fixed 153 Yes l 8428-lp-fixed-fixed-03 l Fixed-Fixed 217 Yes 1 I 8428-1 p-fixed-fixed-04 l Fixed-Fixed 217 Yes l I 8428-1 p-fixed-fixed-06 l Fixed-Fixed 111 No 0 I 8428-lp-fixed-fixed-10 l Fixed-Fixed 136 No 0 I 8428-lp-fixed-fixed-13 1 Fixed-Fixed 208 Yes 1 I 8428-lp-free-free-02 l Free-Free 167 No 0 I 8428-lp-free-free-03 l Free-Free 206 Yes 1 I 8428-lp-free-free-04 l Free-Free 105 No 0 I 8428-lp-free-free-05 1 Free-Free 141 No 0 I 8428-lp-free-free-06 l Free-Free 162 No 0 I 8428-lp-free-free-07 l Free-Free 143 No 0 I 8428-lp-free-free-15 l Free-Free 242 Yes 1 I 8428-lp-fixed-free-02 l Fixed-Free 200 Yes 1 I 8428-lp-fixed-free-03 l Fixed-Free 182 Yes 1 I 8428-lp-fixed-free-04 l Fixed-Free 187 Yes 1 I 8428-lp-fixed-free-05 1 Fixed-Free 98 No 0 I 8428-lp-fixed-free-06 l Fixed-Free 167 Yes 1 I 8428-lp-fixed-free-07 l Fixed-Free 93 No 0 I 8428-lp-fixed-free-09 l Fixed-Free 121 No 0 I 8428-lp-fixed-free-10 1 Fixed-Free 152 No 0 I 8428-lp-fixed-free-ll l Fixed-Free 134 No 0 I 8428-lp-fixed-free-12 l Fixed-Free 173 Yes 1 I 8428-lp-fixed-free-13 1 Fixed-Free 137 No 0 I 8428-lp-fixed-free-14 l Fixed-Free 166 Yes 1 I 8428-lp-fixed-free-15 l Fixed-Free 126 No 0 I 8428-lp-fixed-free-16 l Fixed-Free 146 Yes 1 I 8428-lp-fixed-free-17 l Fixed-Free 123 No 0 I 8428-lp-fixed-free-18 1 Fixed-Free 177 Yes 1 I 8428-3p-fixed-free-01 3 Fixed-Free 198 Yes 3 II 8428-3p-fixed-free-02 3 Fixed-Free 210 Yes 3 II 8428-3p-fixed-free-03 3 Fixed-Free 244 Yes 3 II 8428-3p-fixed-free-04 3 Fixed-Free 264 Yes 3 II 8428-8p-fixed-free-01 8 Fixed-Free 315 No 1 I 8428-8p-fixed-free-03 8 Fixed-Free 378 No 1 I 8428-8p-fixed-free-04 8 Fixed-Free 350 No 1 I 8428-8p-fixed-free-08 8 Fixed-Free 272 No 0 I 8428-16p-fixed-free-01 16 Fixed-Free 379 No 0 I 8428-16p-fixed-free-02 16 Fixed-Free 366 No 0 I 8428-16p-fixed-free-03 16 Fixed-Free 365 No 0 I 119 Chapter 5 - Results and Discussion Table 5 - 6 Numerical test matrix used to study the sensitivity of the target response different input parameters. Sensitivity Analysis Pond Size Boundary Conditions Gap Elastic Modulus Viscosity Factor (K) Areal Density (Portal) Projectile Mass Projectile Radius Geometry-Related Parameter Panel Boundary Number Halflength Condition of Layers 63 fixed, free 1 105 fixed, free 163.8 Material-Related Parameters El. E Size N/tex 0 2.1 66.8 0.05 1 0 2.1 66.8 0.05 fixed, free 1 0 2.1 66.8 210 fixed, free 1 0 2.1 105 fix 1 0 105 free 1 105 fix 105 Infinite 105 fixed 105 105 Projectile Parameter Areal Mp Radius Vp Density (l?r) (mm) (m/s) 0 204 2.97 2.78 148 0 204 2.97 2.78 148 0.05 0 204 2.97 2.78 148 66 8 0.05 0 204 2.97 2.78 148 1.05 66.8 0.05 0 204 2.97 2.78 148 0 1.05 66.8 0.05 0 204 2.97 2.78 148 1 0 1.05 66.8 0.05 0 204 2.97 2.78 148 1 0 1.05 66.8 0.05 0 204 2.97 2.78 148 . s 0 1.05 66.8 0.05 0 204 2.97 2.78 267 fixed 8 0.2 1.05 66.8 0.05 0 204 2.97 278 267 fixed S 0.5 1.05 66 8 0.05 0 204 2.97 278 267 105 fixed S 0.8 1.05 66.8 0.05 0 204 2.97 278 267 105 fixed 8 1 1.05 66.8 0.05 0 204 2.97 2.78 267 105 fixed 1 0 1.05 53.44 0.05 0 204 2.97 278 148 105 fixed 1 0 1.05 60.12 0.05 0 204 2.97 2.78 148 105 fixed 1 0 1.05 66.8 0.05 0 204 2.97 2.78 148 105 fixed 1 0 1.05 73.48 0.05 0 204 2.97 2.78 148 105 fixed 1 0 1.05 80.16 0.05 0 204 2.97 2.78 148 105 fixed 1 0 1.05 66.8 0 0 204 2.97 2.78 148 105 fixed 1 0 1.05 66.8 0.01 0 204 2.97 2.78 148 105 fixed 1 0 1.05 66.8 0.05 0 204 2.97 2.78 148 105 fixed 1 0 1.05 668 0.1 0 204 2.97 2.78 148 105 fixed 1 0 1.05 66.8 0.2 0 204 2.97 2.78 148 105 fixed 1 0 1.05 66.8 0.05 0 204 2.97 2.78 148 105 fixed 1 0 1.05 66.8 0.05 0.005 204 2.97 2.78 148 105 fixed 1 0 1.05 66.8 0.05 0.01 204 2.97 2.78 148 105 fixed 1 0 1.05 66 8 0.05 0.02 204 2.97 278 148 105 fixed 1 0 1.05 66 8 0.05 0 100 2.97 278 148 105 fixed 1 0 1.05 66.8 0.05 0 204 2.97 278 148 105 fixed 1 0 1.05 66.8 0.05 0 300 2.97 278 148 105 fixed 1 0 1.05 66 S 0.05 0 400 2.97 2.78 148 105 fixed 1 0 1.05 66.8 0.05 0 700 2.97 278 148 105 fixed 1 0 1.05 66 8 0.05 0 204 2.97 2.78 148 105 fixed 1 0 1.05 668 0.05 0 204 5 2.78 114.1 105 fixed 1 0 1.05 66.8 0.05 0 204 10 2.78 80.7 105 fixed 1 0 1.05 66.8 0.05 0 204 1 2.78 255.1 105 fixed 1 0 1.05 66.8 0.05 0 204 2.97 1 148 105 fixed 1 0 1.05 66.8 0.05 0 204 2.97 2 148 105 fixed 1 0 1.05 66 8 0.05 0 204 2.97 2.78 148 105 fixed 1 0 1.05 66.8 0.05 0 204 2.97 5 148 Gap K Scrimp 120 Chapter 5 - Results and Discussion Table 5 - 7 Absorbed energy of the target after 100 jas, V=148 m/s. s Boundary Condition Current Velocity [m/s] Projectile Displacement [mm] Strain Energy [J] Kinetic Energy Transverse Fixed-fixed 101.8 13.25 9.38 7.61 0.16 17.15 Fixed-free 119.6 13.71 3.63 4.52 3.09 11.23 Free-free 137 14.21 0.03 1.41 3.13 4.56 Infinite 125.5 13.88 N/A* N/A* N/A* 8.83 m Kinetic Energy In-plane m Total Absorbed Energy m * As there is no information available on the portion of strain wave, which is outside the meshed panel, the energy calculation is not accurate. The total absorbed energy is calculated from the current velocity of the projectile. 121 Chapter 5 — Results and Discussion BACK PLATE Figure 5-1: The experimentalfixtureused to support the target (Starratt [1998]). (a) (b) Figure 5-2: Comparison of the impact response of a single layer of Kevlar® 129 with fixed-fixed and free-free boundary conditions; a) normalized velocity vs. projectile displacement, b) Normalized Absorbed energy vs. projectile displacement (Cepus et al. [1999]). 122 Chapter 5 - Results and Discussion Dashed vertical lines represent x Fixed-Fixed • Fixed-Free A Free-Free E for each boundary condition c | Free-Free \ ~ 30 Fixed-Free \ / x LU 1 ^r A 1 / Fixed-Fixed x x 0 0 10 20 30 4 0 50 6 0 70 8 0 E (J) 9 0 S Figure 5-3: Absorbed energy plotted against the striking energy for various boundary conditions used in experiments on a single layer of Kevlar® 129 (Cepus et al. [1999]). Figure 5-4: Post-mortem photographs of a panel with free-free and fixed-fixed boundary conditions (Cepus et al. [1999]). 123 Chapter 5 - Results and Discussion (a) (b) Figure 5 - 5 : Experimental data (fixed-fixed boundary condition, V = l l l m/s) compared with numerical predictions for a single layer of Kevlar® 129; a) projectile velocity vs. time, b) absorbed energy vs. projectile displacement. s (a) (b) Figure 5 - 6 : Experimental data (free-free boundary condition, V =167 m/s) compared with numerical predictions for a single layer of Kevlar® 129. s 124 Chapter 5 — Results and Discussion 5 10 15 Projectile Displacement (mm) (a) 5 10 15 Projectile Displacement (mm) (b) Figure 5-7: Comparison of experimental data and numerical predictions for nonperforating impact on a single layer of Kevlar® 129 with fixed-fixed and free-free boundary condition; a) projectile velocity vs. its displacement, b) absorbed energy vs. projectile displacement (Cepus et al. [1999]). Time (p,s) (a) Time (u.s) (b) Figure 5-8: Comparison of analytical and numerical predictions for impact on a single layer of Kevlar® 129 with Vs=225 m/s; a) total absorbed energy and strain energy vs. time, b) in-plane and transverse kinetic energy vs. time. 125 Chapter 5 - Results and Discussion (a) (b) Figure 5-9: Effect of boundary condition on the impact response of a single layer of Kevlar® 129, V=148 m/s. s Nodal Mass Projectile Contact Element A Element B Figure 5 - 10: Location of elements A and B in the modeled quarter panel of the target, used to study the strain response of the elements. 126 Chapter 5 - Results and Discussion Fixed-Fixed Boundary Condition Free-Free Boundary Condition Fixed-Free Boundary Condition Infinite Boundary Condition Figure 5-11: Predicted deformed shape of a single layer of Kevlar 129 with fixedfixed, free-free, fixed-free, and infinite boundary conditions 100 u.s after impact, V=148 m/s. s 127 Chapter 5 — Results and Discussion (a) (b) Figure 5 - 14: Investigation of the absorbed energy and longitudinal wave propagation along the central fibre for a fixed-fixed 330 mm by 330 mm single layer of Kevlar® 129, assuming different values of elastic modulus, a) absorbed energy vs. displacement, b) strain as a function of distance from the impact point at 19.5 u.s. 128 Chapter 5 - Results and Discussion (a) (b) Figure 5 -15: Effects of viscosity on the impact response of afixed-fixedsingle layer of Kevlar® 129; a) absorbed energy vs. displacement of the projectile, b) strain in element A vs. time. (a) (b) Figure 5 -16: Effect of crimp on the behaviour of a single layer of Kevlar 129 impacted at V=148 m/s. s 129 Chapter 5 - Results and Discussion 100 Projectile Displacement (mm) (a) 150 Time (|is) (b) Figure 5 - 17: Impact response of a single layer of Kevlar 129 for different values of areal densities. 100 150 200 Projectile Striking Velocity (m/s) (a) Striking Energy (J) (b) Figure 5 - 18: Prediction of impact behaviour in terms of residual velocity of the projectile, for a fixed-free single layer of Kevlar® 129 with 3% and 3.5% breaking strain. 130 Chapter 5 - Results and Discussion (a) (b) Figure 5 - 19: Prediction of impact behaviour in terms of residual velocity of the projectile, for a fixed-fixed single layer of Kevlar® 129 with 3% and 3.5% breaking strain. Figure 5 - 20: Strain-time plot for element A for various impact velocities, along with the values of absorbed energy for the impact on a single layer of Kevlar® 129. 131 Chapter 5 — Results and Discussion Figure 5-21: Failure prediction of the model in terms of normalized absorbed energy and time to failure for impact on afixed-fixed210 mm by 210 mm single layer of Kevlar® 129 for Sbreak=3%. 5 10 15 100 20 150 Time (as) Projectile Displacement (mm) (b) (a) Figure 5 - 22: Predictions for iso-energy impact events (E = 32.5 J) on a fixed-fixed single layer of Kevlar® 129, with different projectile masses and striking velocities. s Chapter 5 — Results and Discussion 133 Chapter 6 —Conclusions and Future Work C H A P T E R SIX: C O N C L U S I O N S A N D F U T U R E W O R K 6.1 INTRODUCTION Summary of the research presented in the previous chapters and their conclusions are presented here and a framework is drawn for the future studies and further investigations on the topic. 6.2 SUMMARY 1. An overview of the analytical and numerical models developed previously to capture the impact response of textile structures is presented. 2. An analytical model is developed to break down the absorbed energy of impacted fabric structures into its components. This model is based on the theory of single fibre impact and is capable of predicting the total absorbed energy of the target and its components from the displacement-time information of the projectile. 3. A numerical code is developed to capture the behaviour of textile structures under the transverse impact of blunt cylindrical projectiles. This code is able to predict the preand post-failure behaviour of fabric panels and can be used as a preliminary tool to optimize the design of personal armour systems. 4. Predictions of the numerical model are verified against the experimental data. A series of numerical studies is also performed to investigate the sensitivity of the target response to various input parameters. 134 Chapter 6 —Conclusions and Future Work 6.3 CONCLUSIONS 1. Studying the predictions of the analytical model, it is seen that the panels with fixed boundary conditions absorb the impact energy mostly in the form of strain energy, while the panels with free boundary conditions absorb the energy mainly in the form of kinetic energy. 2. Availability of the ELVS measurements during an impact event allows us to further verify the validity of our numerical model. It also extends our knowledge of the fabric response during ballistic events. 3. Results of the numerical studies and the comparison of the model prediction with the experimental data demonstrate the complexity of the ballistic impact response of textile structures. Effect of the parameters involved in the numerical simulations is so diverse that presenting a simple curve or relation that covers all the aspects of impact seems to be impossible. Yet, depending on the application and the required accuracy, one can make reasonable assumptions to come up with simple solutions. 4. Numerical studies performed show that many parameters influence the overall behaviour of textile structures under impact. Boundary conditions are found to affect the response considerably, emphasizing the importance of a realistic simulation for the case of a real body armour system. Decreasing the gap between the layers of fabric or decreasing the initial crimp strain, increases the energy absorption of the target. The elastic modulus is found to have a favourable effect on absorption of energy by the target. Increasing the yarn's breaking strain increases, the time-tofailure of the fabric, and consequently, the energy absorption characteristics and V :50 135 Chapter 6 -Conclusions and Future Work value of the target material. Areal density is also found to have a positive effect on the energy absorption of the target. An instantaneous strain-based failure model is found to capture the material characteristics of Kevlar® 129. 5. Knowing the sensitivity of response to various input parameters, one can adjust them in a way that enhances the performance of the target. This knowledge can be used as the preliminary design tool for an assembly of rigid and semi-rigid materials in an armour system to reduce the experimental cost and time. 6.4 FUTURE WORK Although the numerical code presented in this study is very helpful in giving insight to the impact behaviour of fabric materials, there is still much to be accomplished to come up with a robust analysis/design tool. In order to obtain such a model, the following future works are suggested: 1. To predict the dynamic response of a textile structure, realistic input parameters should be chosen that accurately represent the material and geometry of the target. Available information on the dynamic and visco-elastic material properties is not extensively studied and well established for the domain of the material used. These geometric and material properties such as dynamic elastic modulus, viscosity, crimp, etc. requires more investigations. 2. The pin-jointed mesh does not seem to be able to exhibit some of the behaviours observed in the experiments. A more realistic model that considers the friction between the orthogonal yarns should be developed to capture phenomena such as yarn pullout. 136 Chapter 6 -Conclusions and Future Work 3. The algorithm currently employed to simulate the contact of individual layers in multi-layer targets is oversimplified. In order to simulate the interaction of adjacent layers in the overall energy absorption of the target, a realistic cable-to-cable contact algorithm should be implemented. 4. Implementation of a robust contact model for the contact between the mesh nodes and other finite elements provides the ability to study the behaviour of a complete armour system containing semi-rigid and rigid materials, along with a backing material (e.g. human body). This algorithm would also be capable of simulating the impacting projectiles explicitly, and facilitates the consideration of deformable projectiles. 5. The available experimental database should be extended to include different materials with various properties. Also, the existing data for a certain type of material and target specifications should be completed. 137 References REFERENCES Abdel-Rahman, N., Starratt, D., Pageau, G., Zhang, Y., Vaziri, R., and Poursartip , A., "Experimental and Theoretical Investigation of the Ballistic Impact of Textile Materials", Proceedings of Personal Armour Systems Symposium (PASS 98), Colchester, U. K., September 1998, p.p.213-222. Brisco, B. J., and Motamedi, F., "The Ballistic Impact Characteristics of Aramid Fabrics: the Influence of Interface Friction", Wear, Volume 158, No. 1-2, October 1992, p.p. 229-247. Cepus, E., Shahkarami, A., Vaziri, R., and Poursartip, A., "Effect of Boundary Conditions on the Ballistic Response of Textile Structures", Proceedings of International Conference on Composite Materials (ICCM12), July 5-9, 1999, Paris, France. Chocron-Benloulo, I. S., Rodriguez, J., and Sanchez-Galvez, V., "A Simple Analytical Model to Simulate Textile Fabric Ballistic Impact Behaviour", Textile Research Journal, Volume 67, No. 7, 1997, p.p. 520-528. Cunniff, P. M., "An Analysis of the System Effects in Woven Fabrics Under Ballistic Impact", Textile Research Journal, Volume 62, No. 9, September 1992, p.p. 495-509. Du Pont, "Kevlar Aramid Fiber. Technical Guide". December 1992. Findley, W. N., Lai, J. S., and Onaran, K., "Creep and Relaxation of Non-linear Viscoelastic Materials", North-Holland Publishing Company, 1976. Freeston, W. D., Claus, W. D., "Strain-Wave Reflections During Ballistic Impact of Fabric Panels", Textile Research Journal, Volume 43, No. 6, June 1973, p.p. 348-351. Hexcel Fabrics, "Fabric Handbook". 1997. Kolsky, H., "Stress Waves in Solids". New York: Dover Publications Inc., 1963. Leech, C. M., Adeyefa, B. A., "Dynamics of Cloth Subject to Ballistic Impact Velocities", Computers and Structures, Volume 15, No. 4, 1982, p.p. 423-432. Leech, C , Hearle, J. W. S., and Mansell, J., "A Variational Model for the Arrest of Projectiles by Woven Cloth and Nets", Journal of Textile Engineering, Volume 43, No. 11, 1979, p.p. 469-478. Lomov, S. V., "Modelling of High-Velocity Impact on the Textile Woven Targets' Clothing and Personal Equipment for the Combat Soldier Post 2000 Conference, Science 138 References and Technology Division, Defence Clothing and Textiles Agency, Ministry of Defence, Colchester, October 1995, p.p. 212-221. Lomov, S., "Oblique High-Velocity Impact on a Textile Woven Target: Mathematical Simulation", Proceedings of Personal Armour Systems Symposium '96, Colchester, U. K., September 3 - 6, 1996, p.p. 145-156. Mase, G. E., "Theory and Problems of Continuum Mechanics", Schaum's Outline Series, McGraw-Hill, 1970. Mehta, P. K., and Davids, N., "A Direct Numerical Analysis Method for Cylindrical and Spherical Elastic Waves", AIAA, No. 4, 1966, p.p. 112-117. Morrison, C , Bader, M . G., "Behaviour of Aramid Fibre Yarns and Composites Under Transverse Impact", Fulmer Research Institute, 1986, p.p. 706-712. Nickel, H., "Armor", Microsoft® Encarta® Encyclopedia. Pageau, G., The Defence Research Establishment Valcartier (DREV), personal communication, 1997. Parga-Landa, B., and Hernandez-Olivares, F., "An Analytical Model to Predict Impact Behaviour of Soft Armours", International Journal of Impact Engineering, Volume 16, No. 3, 1995, p.p. 455-466. Prosser, R. A., "Penetration of Nylon Ballistic Panels by Fragment-Simulating Projectiles, Part I: A Linear Approximation to the Relationship Between the Square of the F50 or V Striking Velocity and the Number of Layers of Cloth in the Ballistic Panel", Textile Research Journal, February 1988, p.p. 61-85. c Ringleb, F. O., "Motion and Stress of an Elastic Cable Due to Impact", Journal of Applied Mechanics, No. 24, 1957, p.p. 417. Roylance, D., and Wang, S. S., "Penetration Mechanics of Textile Structures. Ballistic Materials and Penetration Mechanics", Elsevier Scientific Publishing Co., 1980, pp. 273-292. Roylance, D., Wilde, A., Tocci, G., "Ballistic Impact of Textile Structures", Textile Volume 43, No. 1, January 1973, p.p. 34-41. Research Journal, Roylance, D., "Ballistics of Transversely Impacted Fibers", Textile Research Journal, October 1977. Shim, V. P. W., Tan, V. B. C , and Tay, T. E., "Modeling Deformation and Damage Characteristics of Woven Fabric under Small Projectile Impact", International Journal Impact Engineering, Volume 16, No. 4, 1995, p.p. 585-605. 139 References Shim, V. P. W., Tan, V. B. C , and Tay, T. E., "Perforation of Woven Fabric by Spherical Projectile", AMD, Volmue 205, 1995. Smith, J. C , Blandford, J. M., Schiefer, H. F., "Stress-Strain Relationships in Yarns Subjected to Rapid Impact Loading, Part VT: Velocities of Strain Waves Resulting from Impact", Textile Research Journal, Volume 30, October 1960, p.p. 752-760. Smith, J. C , McCrackin, F. L., Schiefer, H. F., "Stress-Strain Relationships in Yarns Subjected to Rapid Impact Loading, Part V: Wave Propagation in Long Textile Yarns Impacted Transversely", Textile Research Journal, Volume 28, No. 4, April 1958, p.p. 288-302. Starratt, D., "An Instrumented Experimental Study of the Ballistic Response of Textile Materials", Masters Thesis, Department of Metals and Materials Engineering, The University of British Columbia, April 1998. Starratt, D., Sanders, T., Cepus, E., Poursartip, A., and Vaziri, R., "An Efficient Method for Continuous Measurement of Projectile Motion in Ballistic Impact Experiments", International Journal of Impact Engineering, In Press, 1999. Taylor, W. J., and Vinson, J. R., "Modeling Ballistic Impact into Flexible Materials", AIAA Journal, Volume 28, No. 12, December 1990, p.p. 2098-2103. Termonia, Y., Smith, P., "A Theoretical Approach to the Calculation of the Maximum Tensile Strength of Polymer Fibers", High Modulus Polymers, 1988, p.p. 321-362. Ting, C , Ting, J., Cunniff, P., Roylance, D., "Numerical Characterization of the Effects of Transverse Yarn Interaction on Textile Ballistic Response", Proceedings of SAMPE Symposium, October 1998. Ting, J., Roylance, D., Chi C. H , Chittangad, B., "Numerical Modeling of Fabric Panel Response to Ballistic Impact", 25 International SAMPE Technical Conference, October 1993, p.p. 384-392. th Vinson, J. R., and Zukas, J. A., "On the Ballistic Impact of Textile Body Armor", June 1975, p.p. 263-268. Journal of Applied Mechanics, Wilde, A. F., Roylance, D. K., and Rogers, J. M., "Photographic Investigation of HighSpeed Missile Impact upon Nylon Fabric, Part I: Energy Absorption and Cone Radial Velocity in Fabric", Textile Research Journal, Volume 43, No. 12, December 1973, p.p.753-761. Zhurkov, S. N., "Kinetic Concept of the Strength of Solids", International Journal of Volume 1, No. 311, 1965. Fracture Mechanics, 140 Appendix A APPENDIX A : A N A L Y T I C A L M O D E L IMPLEMENTATION A s stated before, the analytical model described in Chapter Three was implemented into a M A T H C A D file. A sample calculation along with the computer program developed for a panel with free boundary conditions is presented here. The following program determines the components of the absorbed energy by the target for the duration before the first longitudinal strain wave reaches back to the impact point. In this model, the basic analytical theories of the impact on a single yar is used. The boundary conditions are free all-around. INPUT: Material Properties and the impact data E := 96 10 Pa ,3 kg P 1.44 10 3 m (Elasticity modulus of the yarns) (Density) pa (Areal density of the fabric) PL = 330 mm (Square panel dimension) Reading the Displacement, Velocity, and Energy time-histories from the input file (experimental data): y U:V.\fr-fr-01.xls 141 Appendix A Add units to the arrays of Time, Displacement, Velocity and Energy to be used in calculations: T := time_disp < X> 10" s 6 Disp := time_disp 10 m Vel := time_disp — s Energy := time_disp <4> J Find the size of array, which stores the input data imported from the excel file: n_data := rows(T) n_data = 392 Calculation of the basic parameters: c= 5.774*10 »m»s (Longitudinal strain wave speed) h = 0.142 °mm (Approximate thickness of the fabric) The wave propagation time is divided into two phases; Phase-I, corresponding to the time before the longitudinal wave reaches the boundaries, and Phase-II, for the time after the reflection at the boundaries till it arrives at the impact point. The duration of Phase-I can be calculated knowing the panel-size and the longitudinal wave speed: t l := PL t l = 2.858.10 -5 °s Assuming that the longitudinal wave speed is constant, the duration of Phase-II would be equal to Phase-I: t2 := 2 t l t2 = 5.716.lO^.s 142 Appendix A PHASE-I Knowing the termination time f o r this phase, calculate the number of elements of the input data f o r this phase: tl = 2.858-lO" ^ 5 n _ p h _ l := for i e 1.. n_data n<- i t _ p h _ l := T n _ p h _ l = 72 if tl> T. t _ p h _ l = 2.84-10 n_ph_l -s This section calculates the transverse wave velocity and the strain at the impact point using the a trial and error approach. For more information on th relationships used go to the theoretical chapter. U := £ _ 1 < - 0.05 for ne 2.. n _ p h _ l for i e 1.. 100 u«— c ^ E _ l • ( £_1 -t-l) 1 + if —< — - / Disp \ \ j U T £_1 - 1 0.05 i<-100 v <— u n m v — s 143 Appendix A The strain vector is then calculated from the transverse wave speed: for i e 2.. n_ph_l U Oisp \ i i i 1 -t- 2 - 1 U T i i 0.004 6 0.002 Q I 0 I 5'10 I 6 1*10 I 5 1.5*10 I 5 T 2«10 5 L_ I 2.5'10~ 5 3»lo" Strain Energy Calculation: Strain energy stored in the target is calculated by integrating the strain (which itself is a function of the distance from the center) over the area covered by the longitudinal strain wave: 144 Appendix A for i e 2.. n_ph_l E strain 1 := kl<— 2-E-h 12 k2« Disp. \ 1+ ru k3< ' T ' lUT.-x' 2 i i cT. - x c T. - x) d x i k.«-kl k2 k3 k J Plot the strain energy for Phase-I: 0.4 0.3 E_strain_l 0.2 0.1 Q I 0 1 5»10 I 6 1*10 I 5 1.5'lOr I 5 2'10 I 5 2.5*10 I 5 3*10 145 Appendix A Transverse kinetic energy of the cone: To determine the kinetic energy of the cone, the masses in the cone are calculated using the transverse wave velocity with respect to the masses in the unstrained fabric: U cone 1 := for i e 1.. n_ph_l Ubar.<— c 1 + Ubar™ KB cone 1 := for i e 1.. n_ph_l K.<-pa (u_cone_t -T.-Vcl. KJ Kinetic Energy outside the cone: The nodes within the longitudinal strain wave front and outside the cone travel in-plane towards the impact point with a velocity**'assumed to be equal to c.s. KE out 1 := for i € 2.. n_ph_l kl«— 2 p a ( c - U_cone_l.) c T fU cone 1. T. Ui Ti - x k2< c-T. dx JO Disp \ k3< 1 - 1 -tU .- ,/ T kl k2 k3 kJ 146 Appendix A The total energy of the system would easily be the summation of the energies calculated above: E_total_l := ((E_strain_l + KE_cone_l -t- KE_out_l)) • kg.m »s E total 1 2.256.10 The total absorbed energy predicted by the model is compared to the energy loss of the projectile in the following graph: Energy E_total_l E_strain_l KE_cone_l KE_out_l 0 5*10 6 1*10 5 1.5-10 5 2'10 5 T 2.5*10 5 3»10 5 3.5'10 147 Appendix A PHASE-II: In This part, the strain energy, in-plane and transverse kinetic energy of th fabric is calculated considering the reflection of the longitudinal strain wav on the free edges of the target. Due to the nature of the wave reflection on free boundaries, the strain wave reflects with the opposite sign and cancels out to the existing wave. On the other hand, that returning wave doubles th inplane motion of the waves on its way to the impact point. n_ph_2 := for i e 1.. n_data n_ph_2= 143 n<-i if t2> T. t_ph_2 := Tn_ph_2 t_ph_2 = 5.68-10 -s Transverse strain wave speed for this phase, assuming the whole fiber length being under strain: U_ph2 e_l<- 0.05 for ne n_ph_l+ 1.. n_ph_2 for i e 1.. 100 j <— n - n_ph_l u<— C[^E_1(E_1 -I- 1) - £_1 1+ 2-u-T uT 4c- T - T n W PL n_ph_iy if - - - < 0.01 £ 2 £ 1 i<- 100 v <— u j m 148 Appendix A 380 360 h U_ph2 340 h 320 0 1'10 5 3»10 5 3*10 5 Now, add up the new U-vector (U_ph2) to the old transverse velocity vector, U U := stack \ U-i.,U_ph2-i)-m m / s 400 U 200 0 2»10 5 4^10 5 6«10 5 As the transverse wave speed is not constant in phase-II, the cone size is calculated incrementally, as follows: bj^Om for i e 2.. n_ph_2 At«- V b.<-b. 1 1 - 1 T i - i + u.— -At 1 m bm 149 Appendix A When the wave reaches the boundary, the wave traveling in the fiber going through the impact point reflect first, and the reflected area increases till all the outgoing wave reflects and returns to the impact point. The time corresponding to this period is refered to as the transition point. for n_ph_2_tr := i e n_ph_l.. n_ph_2 PL if c T . - —< b. 2 ' n<— i n_ph_2_tr = 77 1 So, the transition time would be: T tr := Tn_ph_2_rr T tr 3.04'lcf ' S 5 Strain Energy: The strain calculated above is being used to evaluate the strain energy of the system. In doing so, the returned portion of the strain is also taken into accoun Before r>Ut, the strain is calculated using the full length for the yarns in the reflected area, and the current length for the rest of them: Strain energy of the yarns in the zone where the longitudinal stain energy has reflected at the boundary: E strl for i e n_ph_l +• 1.. n_ph_2_tr j « - i - n_ph_l cT. 1 PL 2 Disp. kl< 1* - 1 2 K2< b. - x •kl dx JO K3«-E-h-PL E.«-K3-K2 E-J 150 Appendix A Strain energy in the zones where the longitudinal strain energy is still travelling outward to the boundaries is equal to: E_str2 := for ie n_ph_l + 1.. n_ph_2_tr j *— i - n_ph_l r<—c T - — ' 2 rb b. - Kl< c T. - x dx 12 K2<-2Eh 1+ Disp. \ - 1 E.<-K1 K2 EJ i The total strain energy vector for the transition period is: E_ph2a := E_strl + E_str2 0.375 0.376 E_ph2a 0.373 0.368 0.364 i g«m «s'k 2 2 Appendix A After the transition time, when r > b, the longitudinal strain wave is all travelling back to the impact point. The strain is then calculated the full length of the yarns as it is fully covered by the strain wave. The energy is summed over the distance traveled by the wave: E str3 for i e n_ph_2_tr +• 1.. n_ph_2 j<— i - n_ph_2_tr r«—c-T. - — Disp\ 1• kl< 2 - 1 rb. - x 2-fb. K2. P L , 4 . c . f T 1 - T j h -kl dx _ , ) . K3«-E-h-PL E.<-K3 K2 E-J E_strain_ph2 := stack ESTRAIN := stack | E E_ph2a E_str3 S t r a m - 1 E_strain_ph2 ) J 152 Appendix A ESTRAIN 0.2 Transverse kinetic energy of the c o n e : Same as phase I, the area within the deformation cone is determined and assuming that the masses within the cone travel with the speed of the projectile, kinetic energy is calculated: First, the propagation velocity of the transverse wave with respect to the Lagrangian coordinates is evaluated: U cone 2 := for ie n _ p h _ l - t - 1 . n_ph_2 - n_ph_l U bar.<—c j 1 + U bar ™ / s U cone := stack U cone 1 — , U m s \ m cone 2— •— m/ s 153 Appendix A Knowing the propagation velocity, the cone size can be easily calculated in the same coordinate system: b bar := b_bar«—0 for i € 2.. n_ph_2 At«- 'T i - T i - l b bar.<—b bar. . + U cone •— At 1 - 1 1 mi b bar m KE cone 2 for ie n_ph_l +• 1.. n_ph_2 j<— i - n_ph_l KE.«-pa- b_bar.-Vel. ,2 J KE-J Now, add the vector of kinetic energy to the one calculated for Phase-I: . / KE_cone_l KE_cone_2 \ KE cone := stack , •J Kinetic Energy out of the cone: Same as Phase-I, the masses outside the cone and within the longitudinal strain wave is considered. As the reflected part of the wave cancels the in-plane veocity of the masses covered by it, the kinetic energy is predicted to decrease as the wave progresses: Appendix A for ie n_ph_l +- 1.. n_ph_2_tr | - n_ph_l r<— c T - — ' 2 kl«— pa-c 2 k2< 1 +- / Disp. LN k3< PL +• 4 c - KE - 12 2-(b. - x V k 2 T. - T •[ PL + Ax - 2 b bar. - 2 r ) dx 1 - la«-kl-k3 J 2-b. - x k2 k4< 12 (8-r - 8 x)dx PL + Ac- J. i - T n_ph_l KE l b . « - k l - k 4 J - k5<— 2-pac ^c-T. - b_bar. 2 b / b. k6< c-T. - x i dx KE_2 « - ( k 2 ) -k5 k6 KE.<— KE la. +• KE lb. + KE 2. j j J J — KE-J 155 AppendixA for i e n_ph_2_tr +- 1.. n_ph_2 j< — i - n_ph_2_tr r«—c-T. ' 2 kl<— pac Disp. k2< 1 + b. 2 b . - x k2 12 k3< PL + 4-x - 2-b bar. - 2-r dx PL + 4 c - T. - T i . , • 1 - n_ph_iy KE a.<-kl-k3 j b. 2-b. - x k2 k4<- 12 (8-r - 8 x ) d x KE b.<-klk4 KE.<— KE a. + KE b. j J J — KE.«-0 j — if KE.< 0 j KEJ KE out 2 := stack K2a K2b J KE_out_2 = 0.178 0.182 * 9* k m 2 * s -2 Appendix A Add the kinetic energy computed in Phase-II to the one calculated in Phase-I „r- • , , / KE_c-ut_l KE_out_2 \ KE_inplane := stack , J t i •k g«m «s " KE_inplane = 2 2 3.562.10 The total absorbed energy is determined by adding all the energies (strain energy, in-plane and transverse kinetic energy) together: E_TOTAL := ESTRAIN +• KE_cone + KE_inplane 1-10 2»10 5 3'10 5 T 4«10 5 5«10 -5 157 Appendix A Save the data in an EXCEL file: for i e 1.. n_ph_2 output out.1,1,«-T. i out. <— ESTRAIN out o u t i out «- KE_cone 4 « - KE_inplane. E_TOTAL out U:\.mode-01.xls output 0 I4.10" output 0 0 0 0 7.526.10" 1.147.10" 3.562.10" 2.256.10" 5 7 4 4 5 3.009.10" 4.587.10" 1.424.10" 9.02.10" 8.10" 7 4 4 4 4 1.2.10" 6.768.10" 1.032.10" 3.203.10" 2.029.10" 6 4 3 4 3 1.6.10" 1.203.10" 1.834.10" 5.692.10" 3.606.10" 6 2.10" 6 3 3 4 3 1.879.10" 2.865.10" 8.892.10" 5.633.10" 3 3 4 3 158
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A numerical investigation of ballistic impact on textile structures Noori, Ali Shahkarami 1999
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Title | A numerical investigation of ballistic impact on textile structures |
Creator |
Noori, Ali Shahkarami |
Date Issued | 1999 |
Description | The present work focuses on numerical investigation of impact on textile materials. An analytical model is proposed based on the theory of single yarn impact, to evaluate the energy absorbed by a fabric panel. This model is capable of evaluating the total absorbed energy and its components knowing the displacement time-history of the projectile. A numerical model is used to simulate the behaviour of a fabric panel under impact. This model approximates the fabric as an assembly of nodal masses attached to each other by means of string elements in the principal directions. A computer code is developed that is capable of modelling the impact of a blunt cylindrical projectile on rectangular fabric panels with various types of boundary conditions. The code predicts the time-histories of the nodal displacements, as well as forces in the string elements. Based on these, energy stored in the system in the form of strain energy in the strings and kinetic energy of inplane and transverse motion of the masses can also be evaluated. The numerical predictions are successfully compared with instrumented impact test results. Finally, a series of numerical experiments are performed to investigate the sensitivity of the response to different input parameters. The response is found to depend on many geometric and material parameters, emphasizing the importance of having an accurate knowledge of some of the input parameters. In particular, boundary conditions are found to affect the response significantly. It is also observed that increasing the elastic modulus or breaking strain of the yarns affects the energy absorption of the target favourably. Increasing the initial crimp strain is found to decrease the impact energy absorption, similar to increasing the gap between the layers of a multi-layer fabric. |
Extent | 8668067 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-06-27 |
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 | 10.14288/1.0228831 |
URI | http://hdl.handle.net/2429/9768 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1999-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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