Nonlinear mechanical behavior of automotive airbag fabrics: an experimental and numerical investigation by Steven Edward Zacharski B.Sc., Villanova University, 2008 A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF APPLIED SCIENCE in The Faculty of Graduate Studies (Materials Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) December 2010 © Steven Edward Zacharski, 2010 Abstract ii Abstract Over the past two decades, the airbag has become an essential safety device in automobiles. The airbag cushion is composed of a woven fabric which is rapidly inflated during a car crash. The airbag dissipates the passenger’s kinetic energy thereby reducing injury through biaxial stretching of the fabric bag and escaping gas through vents. Therefore, the performance of the airbag is greatly influenced by the mechanical properties of the fabric. Unlike traditional engineering materials, airbag fabrics are composed of discrete constituents and have highly nonlinear mechanical behavior that arises from both geometric deformations and material nonlinearity. Henceforth, airbag designers are forced to make simplified assumptions regarding the mechanical behavior of the fabric cushion. This incontrovertibly limits designers in taking advantage of the full potential of the fabric system. In order to optimize the airbag design, improve deployment simulations and overall dependability, a more sophisticated approach is needed. In this study, a simple unit cell model representing a single crossover of two orthogonal woven yarns is developed to simulate the in-plane mechanical behavior of both coated and uncoated plain weave airbag fabrics under multiple states of stress. Since the structural analysis of the deployment of the airbag is performed using the finite element method, the proposed mechanistic model is implemented as a User-Material-Model in the commercial code LS-DYNA. Here, the unit cell model represents the constitutive behavior of a continuum membrane. The approach results in capturing, in detail, the discrete nature of the fabric while retaining the computational efficiency of simple membrane formulation compared to explicitly modeling each yarn within the fabric. The procedure to calibrate the model inputs, namely the yarn geometric and mechanical properties for a given fabric is detailed. The sensitivity of the unit cell model and verification of the finite element implementation is discussed. A series of experiments were performed to validate the in-plane behavior of the model. The proposed model can be adopted by designers to better represent the nonlinear mechanical behavior of the fabric. It can also be used as a tool to design novel fabrics that are optimized for a particular application. Table of Contents iii Table of Contents Abstract ........................................................................................................................................... ii Table of Contents........................................................................................................................... iii List of Tables .................................................................................................................................. v List of Figures ................................................................................................................................ vi Nomenclature............................................................................................................................... viii Acknowledgements......................................................................................................................... x Chapter 1 – Introduction ................................................................................................................. 1 1.1 Motivation............................................................................................................................. 1 1.2 Goals and objectives ............................................................................................................. 3 1.3 Outline .................................................................................................................................. 4 Chapter 2 – Background ................................................................................................................. 6 2.1 Overview of airbag fabric technology .................................................................................. 6 2.2 Airbag fabric research........................................................................................................... 9 2.2.1 Summary...................................................................................................................... 14 2.3 Modeling of the mechanical behavior of fabrics ................................................................ 15 2.3.1 Representative mechanistic models ............................................................................. 15 2.3.2 Continuum approaches................................................................................................. 19 2.3.3 Current state-of-the-art................................................................................................. 20 2.3.4 Summary...................................................................................................................... 21 Chapter 3 - Development of a representative unit cell: theory and calibration ........................... 22 3.1 Background and approach .................................................................................................. 22 3.2 Unit cell definition .............................................................................................................. 24 3.3 Deformational mechanisms and constitutive relationship .................................................. 29 3.3.1 Yarn axial extension .................................................................................................... 29 3.3.2 Yarn bending................................................................................................................ 30 3.3.3 Coating extension......................................................................................................... 33 3.3.4 Unit cell in-plane shear behavior ................................................................................. 34 3.4 Numerical procedure........................................................................................................... 37 3.5 Characterization of constituent properties .......................................................................... 39 3.5.1 Microscopy .................................................................................................................. 39 3.5.2 Yarn extension test....................................................................................................... 42 3.5.3 Coating characterization .............................................................................................. 47 3.5.4 In-plane shear calibration............................................................................................. 48 3.6 Results................................................................................................................................. 50 3.7 Sensitivity analysis ............................................................................................................. 51 3.8 Summary............................................................................................................................. 54 Chapter 4 – Implementation and verification ............................................................................... 56 4.1 Background and approach .................................................................................................. 56 4.2 Explicit dynamic finite element analysis ............................................................................ 60 4.3 Implementation of unit cell based UMAT for FE shells .................................................... 61 4.3.1 Continuum formulation................................................................................................ 61 4.3.2 Special considerations – yarn separation, failure and element erosion ....................... 65 4.4 Verification ......................................................................................................................... 66 4.5 Summary............................................................................................................................. 68 Chapter 5 – Validation .................................................................................................................. 70 Table of Contents iv 5.1 Background and approach .................................................................................................. 70 5.2 Experimental evaluation ..................................................................................................... 70 5.2.1 Uniaxial extension ....................................................................................................... 70 5.2.2 Biaxial extension.......................................................................................................... 72 5.2.3 Bias extension .............................................................................................................. 78 5.3 Comparison of UC, FE-UC and experiments ..................................................................... 85 5.3.1 Uniaxial results ............................................................................................................ 85 5.3.2 Biaxial results............................................................................................................... 88 5.3.3 Bias results ................................................................................................................... 94 5.4 Summary............................................................................................................................. 99 Chapter 6 - Conclusion and recommendations ........................................................................... 100 6.1 Conclusions....................................................................................................................... 100 6.2 Recommendations and future work .................................................................................. 101 6.3 Summary........................................................................................................................... 102 References................................................................................................................................... 103 Appendix A – UMAT *MAT card documentation..................................................................... 109 Appendix B – UMAT pseudo code ............................................................................................ 117 Appendix C – Justification of the selection of a linear unit cell over sinusoidal geometry ....... 120 Appendix D – Conversion of membrane stress into specific stress............................................ 123 List of Tables v List of Tables Table 3-1: Existing center-line unit cell models vs. proposed model........................................... 23 Table 3-2: Unit cell configuration conditions............................................................................... 28 Table 3-3: Geometric properties of coated and uncoated airbag fabric as determined from microscopy.................................................................................................................................... 42 Table 5-1: Test methods in the literature for evaluating the biaxial behavior of fabrics............. 73 Table 5-2: Biaxial extension of coated fabric – experiment vs. simulation.................................. 91 Table 5-3: Biaxial extension of uncoated fabric – experiment vs. simulation.............................. 92 Table 5-4: Comparison of bias deformation of coated sample: experiment vs. simulation ......... 97 Table 5-5: Comparison of bias deformation of uncoated sample: experiment vs. simulation ..... 98 List of Figures vi List of Figures Figure 1-1: Biaxial extension of airbag fabric vs. current linear elastic assumption...................... 3 Figure 1-2: Overview of the scope of this study............................................................................. 4 Figure 2-1: Pierce's unit cell (1937).............................................................................................. 16 Figure 2-2: Kawabata's unit cell (1973)....................................................................................... 18 Figure 3-1: Simple unit cell geometry in initial and deformed states.......................................... 24 Figure 3-2: Procedure for generating stress-strain behavior of fabric under multiple states of stress using the unit cell approach ................................................................................................ 27 Figure 3-3: Bending geometry of an elastica................................................................................ 30 Figure 3-4: Typical shear stress-strain curve for single-coated and uncoated fabric .................. 35 Figure 3-5: Shear model behavior a) secant shear modulus as a function of strain b) regions in the shear stress-strain curve .......................................................................................................... 36 Figure 3-6: Plain weave of a) single-coated fabric and b) uncoated fabric ................................. 40 Figure 3-7: Cross section of 350dtex coated airbag fabric a) warp direction and b) fill direction ....................................................................................................................................................... 41 Figure 3-8: Cross section of 350dtex uncoated airbag fabric a) warp direction and b) fill direction ........................................................................................................................................ 41 Figure 3-9: Yarn specimen preparation procedure ...................................................................... 43 Figure 3-10: KES-G1 microtensile tester with loaded yarn sample ............................................. 44 Figure 3-11: Procedure of obtaining "pure" yarn load-elongation response ................................ 45 Figure 3-12: Average force-elongation curve for 350dtex nylon 6,6 airbag yarn ........................ 46 Figure 3-13: KES-FB1 textile shear tester shown in a) and b) shear deformation adopted in the KES-FB1 testing system............................................................................................................... 48 Figure 3-14: Shear stress-strain behavior for 350dtex nylon airbag fabric ................................. 49 Figure 3-15: Membrane stress-strain curves for 350dtex fabric produced by the unit cell model ....................................................................................................................................................... 51 Figure 3-16: Crimp parameter sensitivity for a) 1:1 biaxial extension and b) uniaxial extension52 Figure 3-17: Yarn bending rigidity parameter sensitivity for a) 1:1 biaxial extension and b) uniaxial extension ......................................................................................................................... 53 Figure 3-18: Coating thickness parameter sensitivity for a) 1:1 biaxial extension and b) uniaxial extension ....................................................................................................................................... 54 Figure 4-1: Outline of the basic concept of the unit-cell-membrane ........................................... 58 Figure 4-2: Effect of curvature on the internal structure of woven fabric ................................... 59 Figure 4-3: Numerical procedure of LS-DYNA with user material model option...................... 60 Figure 4-4: Verifying modes of shear: a) pure shear and b) rail shear ........................................ 67 List of Figures vii Figure 4-5: Verifying shear deformation behaviors of a single element with varying shear stiffnesses...................................................................................................................................... 68 Figure 5-1: Average uniaxial stress-strain of 350dtex fabric........................................................ 71 Figure 5-2: Biaxial tester at Drexel University............................................................................. 74 Figure 5-3: Biaxial specimen dimensions..................................................................................... 75 Figure 5-4: Biaxial testing specimen configuration...................................................................... 76 Figure 5-5: Average biaxial stress-strain curve for coated and uncoated samples ...................... 77 Figure 5-6: Photographs of biaxial extension at approximately 12% strain for a) uncoated and b) coated fabrics ................................................................................................................................ 78 Figure 5-7: Heterogeneous deformation of fabric bias extension................................................. 79 Figure 5-8: Bias extension-apparent shear angle recording set-up............................................... 80 Figure 5-9: Load-elongation of bias coated and uncoated airbag fabric ..................................... 81 Figure 5-10: Bias shear -- applied specimen end stress vs. center specimen shear angle ........... 82 Figure 5-11: Typical bias shear deformation sequence - 50mm x 100mm coated sample.......... 83 Figure 5-12: Typical bias shear deformation sequence - 50mm x 100mm uncoated sample...... 83 Figure 5-13: Detailed low shear angle plot of applied stress vs. center shear of bias sample..... 84 Figure 5-14: Uniaxial extension of coated airbag fabric - model vs. experiments ....................... 86 Figure 5-15: Uniaxial extension of uncoated airbag fabric - model vs. experiment..................... 86 Figure 5-16: Macroscopic strains of simulated uniaxial loaded coated airbag fabric at 30% extension – a) warp direction and b) fill direction........................................................................ 87 Figure 5-17: Yarn strains of simulated uniaxial loaded coated airbag fabric at 30% extension – a) warp direction and b) fill direction ........................................................................................... 88 Figure 5-18: Biaxial extension of coated airbag fabric - model vs. ex periments ........................ 89 Figure 5-19: Biaxial extension of uncoated airbag fabric - model vs. ex periments .................... 89 Figure 5-20: Simulated yarn strains at approximately 12% equal biaxial extension in the- a) warp direction and b) fill direction ........................................................................................................ 93 Figure 5-21: Bias load-elongation for coated and uncoated fabric - simulation vs. experiment .. 95 Figure 5-22: Applied shear stress vs. measured shear angle - simulation and experiments......... 95 Figure 5-23: Detailed low level applied shear stress vs. measured shear angle - simulation and experiments ................................................................................................................................... 96 Figure A1: Shear model behavior regions in the shear stress-strain curve ............................... 113 Figure A2: Secant shear modulus as a function of strain........................................................... 114 Figure C1: Approximation of yarn height by linear and sinusoidal unit cell ............................. 120 Figure C2: Approximation of sin theta by linear and sinusoidal unit cell .................................. 121 Figure C3: Approximation of cos theta by linear and sinusoidal unit cell ................................. 121 Figure C4: Approximation of warp fabric membrane stress by linear and sinusoidal unit cell . 122 Nomenclature viii Nomenclature ( )i Index referencing warp yarn (i=1) and fill yarn (i=2) Ai Specific area of yarn in the i material direction c Speed of sound in the fiber material c * Speed of sound across a crimped yarn cri Yarn crimp in the i material direction dfiber Diameter of a single fiber Ec Elastic modulus of coating (EI)yarn Yarn bending rigidity the i material direction Fb,i Vertical force component contributed to yarn bending Fc,i Vertical force component contributed to yarn extension Fct,i In-plane force generated at the end of the yarn in the i direction by the coating Fend,i Total in-plane force in-plane force generated at the end of the unit cell in the i material direction Ff,i In-plane force generated at the end of the yarn in the i direction by the yarn Fy,i Yarn axial tension force G Shear modulus G1 Unit cell shear modulus that is contributed by coating shear G2 Unit cell shear modulus that is contributed by yarn rotation G3 Unit cell shear modulus that is contributed to locking of yarns H0i Initial yarn centerline height at the center of the unit cell Hi Current yarn centerline height at the center of the unit cell K Bulk modulus L0i Initial yarn length Ld,i Linear density of the yarn Li Current yarn length Nfiber Number of fibers per yarn ni Yarns per inch Ni Normal in-plane membrane stress p Packing factor of the yarn rc Radius of curvature S12 In-plane unit cell shear membrane stress tc Coating thickness wc Areal density of coated fabric wu Areal density of uncoated fabric y0i Initial horizontal spacing of yarn yi Current horizontal spacing of yarn ε∆ Strain increment tensor iε Strain applied to the unit cell in the i material direction iyarn,ε Yarn strain in the i material direction ult iyarn,ε Ultimate yarn strain in the i material direction 0 iθ Original angle between material vector i and the local material axes t iθ Angle between material vector i and the local material axes at time step t Nomenclature ix γ1 Shear strain corresponding to the end of coating denominated shear stiffness γ2 Shear strain corresponding to the beginning of yarn rotation dominated shear stiffness γ3 Shear strain corresponding to the onset of contact between parallel yarns γ4 Shear strain at which shear locking occurs λi Total stretch in the i material direction νc Poisson’s ratio of the coating ρ Density of unit cell-shell Acknowledgements x Acknowledgements I wish first to thank my supervisors Dr. Frank Ko and Dr. Reza Vaziri for all their mentoring, guidance, ideas and assistance through the duration of this work. I am grateful for their willingness to share with me the worlds of textile and computational mechanics. I would like to thank the financial support from TRW Automotive on this project and the technical support from Dr. Chuan Lee from TRW. I also wish to thank Dr. Joseph Wartman and Mr. David Harmanos of the Civil, Architectural and Environmental Engineering Department at Drexel University for sharing and their aid in helping me operate the biaxial tester which was vital to validating the model developed in this work. Additionally, I would like to thank Dr. John Gosline and Dr. Ken Savage from UBC Zoology for sharing their tensile tester which was used to generate the uniaxial and bias data used for validation. I also offer much thanks to the past and present members of the UBC Composites Group and the Advanced Fibrous Materials Laboratory for many fruitful discussions and advice. Finally, I would like to thank my friends and family back east for their support and understanding while I pursue to advance my education and career. I especially want to thank my parents for their love and encouragement throughout the years. Chapter 1– Introduction 1 Chapter 1 – Introduction 1.1 Motivation The first recorded automobile fatality occurred in Birr, Ireland in 1869 (Fallon & O’Neill, 2005) -- an event that arguably marked the dawn of the study of automobile safety. Over the past two decades, the emergence of the airbag has established itself as an integral part in vehicle safety for passengers. The United States National Highway Traffic Safety Administration (NHTSA) estimates that as of 2009 more than 28,000 lives have been saved in the U.S. because of frontal airbags (National Highway Traffic Safety Administration, 2009). While the airbag has seen great success, the sobering statistic of 24,474 vehicle occupant fatalities in the U.S. during 2009 documented by the NHTSA illustrates that the airbag as well as other safety technologies need continuing improvement (National Highway Traffic Safety Administration, 2010). The concept of the airbag is fairly simple: upon collision, a charge is sent to ignite a gas explosion that rapidly inflates the airbag. The airbag provides a cushion in which the impact energy is dissipated, forces that act upon the passenger are distributed over a large area and excessive rotations of the passenger are limited. When the forces and rotation that act on the passenger are kept to a minimum, the likelihood of injury dramatically decreases. Considerable efforts of the airbag designers and manufacturers have been focused on producing airbag systems that are reliable and have predictable performance. Of fundamental importance to the airbag performance is the mechanical properties of the airbag fabric. Airbag fabrics, which are typically constructed of a simple plain weave of nylon yarns, exhibit unique characteristics that differ from the traditional engineering materials. More specifically, airbag fabrics are heterogeneous, anisotropic, have the ability to undergo large deformations and exhibit nonlinear mechanical behavior. Heterogeneity and anisotropy lends itself to the geometric assembly of discrete constituents while nonlinearity is due to both geometric deformations and material nonlinearity of the constituents. Chapter 1– Introduction 2 The applied pressure in the airbag is a follower-type loading, meaning the temporal and spatial distribution of the load depends on the structural response of the fabric. Structural response is governed by material behavior of the fabric as well as the operative boundary conditions. A great amount of effort has been devoted by the computational mechanics community to develop sophisticated gas models and fluid-structure interaction algorithms which have improved the accuracy of simulating the operative boundary conditions. Still, simplified assumptions are made regarding the mechanical behavior of the fabric during the design, analysis and simulation of the airbag structure. Of particular interest to the airbag industry is improving the structural analysis and design through a better understanding of the mechanical behavior as well as the failure behavior of the fabrics under deployment conditions. The most common simplified assumption in the industry’s structural analysis is to approximate the fabric as an orthotropic continuum with linear elastic mechanical behavior (Dörnhoff et al., 2008; Hirth, Haufe, & Olovsson, 2007; Wawa, Chandra, & Verma, 1993). The assumed elastic modulus is taken from the initial modulus measured by experimental data. This type of approach obviously neglects the internal changes of the fabric structure and the material nonlinearity of the yarn (Wawa et al., 1993). Figure 1-1 illustrates the shortcomings of the current linear elastic analysis capabilities to capture what is physically observed in biaxially strained fabrics from this study. From the figure, it can be observed that after 4% strain there are large portions of the fabric stress-strain curve where the assumed stiffness is first over estimated and then at strains exceed approximately 15%, the assumed mechanical behavior under predicts the stiffness of the physical fabric. The erroneous estimations can therefore lead to false predictions regarding energy absorption of the system and incorrect simulations of the airbag deployment and impact process. Additionally, there are no clear criteria to describe the failure of the airbag fabric which is a concern for the industry. Refinements in the formulation of the mechanical behavior that take into account the nonlinear stress-strain behavior up to failure can result in better predictions regarding both the dynamics and failure of the airbag structure. Chapter 1– Introduction 3 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 0.00 0.05 0.10 0.15 0.20 0.25 strain m em br an e st re ss (N /m m ) Test Test Assumption Assumption X Y Figure 1-1: Biaxial extension of airbag fabric vs. current linear elastic assumption 1.2 Goals and objectives The objective of this thesis is to model the nonlinear stress-strain behavior of airbag fabrics up to failure under deployment-like conditions. The aim of this work is to develop a constitutive model that accurately represents the anisotropy and nonlinearity of the airbag fabric material under states of stress seen in the deployment of the airbag using simple inputs based on the fabric’s constituents. Since the industry utilizes airbag simulations that are performed using dynamic explicit finite element codes, the material model is intended to be implemented into such codes keeping in mind computational efficiency. Experimental evaluation of the fabric is performed to generate input data for the material model as well as reference for validation. Figure 1-2 illustrates the general strategy in this study. Successful modeling of the stress-strain behavior can aid engineers in improving the safety and dependability of the airbag system. Chapter 1– Introduction 4 Figure 1-2: Overview of the scope of this study It should be made clear that the purpose of this work is to improve predictions of the elastic behavior of the airbag fabric up to failure under multi-axial states of stress. While a more sophisticated representation of the constitutive behavior of the airbag fabric may improve the structural analysis and simulated deployment kinematics, any conclusions drawn about improvements regarding post-impact kinematics or passenger energy dissipation should be made cautiously as the work does not consider visco-plasticity (i.e. permanent and rate-dependant deformations of the fabric). 1.3 Outline This section presents a general outline of the thesis. Chapter 2 begins with a brief overview of the airbag system along with the historical evolution of the fabric system. A literature review presents the work that has been performed in either evaluating the mechanical behavior of airbag Chapter 1– Introduction 5 fabrics experimentally or by modeling. Then a historical review of the classical literature in textile mechanics pertinent to modeling the elastic behavior of woven fabrics and the current state-of-art is presented. Chapter 3 presents a representative unit cell model based on a simple linear approximation of the yarn crossover geometry for woven fabrics. The structural mechanisms and constitutive relations that empower the mechanical response of the unit cell are discussed in detail. The procedures for characterization of constituent properties for a current generation airbag fabric are presented. A sensitivity analysis of the unit cell inputs is performed using properties of the airbag fabric as the nominal baseline. Chapter 4 describes the continuum formulation for a finite element shell using the unit cell model as the basis for the constitutive relationship. A brief overview of the importance of finite element analysis in airbag and crashworthiness simulations is provided. The unit cell based continuum formulation is implemented as a user material subroutine (UMAT) into the commercial dynamic finite element code, LS-DYNA. The derivation of the continuum formulation that transforms element strains to the unit cell and transforms the unit cell stresses back to the element under large deformations is presented. A few verification case studies are performed to confirm the successful implementation of the theory. Chapter 5 discuses the validation of the modeling techniques derived in Chapters 3 and 4. The fabric was tested under uniaxial, biaxial and bias extension to validate the model under different states of stress. The experimental procedure is carefully laid out and the correlation between the simple unit cell model predictions and the experimental results is discussed in detail. Chapter 6 presents the conclusions drawn from the current study and recommends possible future work in the area of modeling woven fabrics and their failure predictions under multiple stress states leading to airbag simulations. Chapter 2– Background 6 Chapter 2 – Background 2.1 Overview of airbag fabric technology During an automobile collision, the vehicle can undergo a rapid change in velocity while the passengers continue to move until an opposing force occurs. Highly concentrated opposing forces arising from the vehicle interior or seatbelt can result in serious injury to passengers. An airbag is designed to minimize these concentrated forces and reduce excessive motion (mostly upper body rotation) of a belted passenger. Upon collision, a charge is initiated which inflates the airbag that forms a cushion between the passenger and vehicle interior. The deployment sequence occurs rapidly between 25 to 30 milliseconds from the time of sensing the crash to full inflation of the airbag where the speed of the deploying airbag can reach up to 160 km/hr (100 mph) (Crouch, 1994; Mukhopadhyay, 2008). At which time gas pressures can reach 70 kPa and temperatures close to 600° C for a few milliseconds (Gon, 2010; Mukhopadhyay, 2008). The bag’s internal pressure is uniformly applied to the occupant and the bag gently dissipates the passenger’s kinetic energy while distributing the contact force over a large area and minimizing rotations of the passengers articulated segments. The mechanisms that contribute to airbag’s absorption and dissipation of energy are the mechanical biaxial stretching of the fabric and escaping gases. While historical references of the airbag can be traced back to the 1920s, the first patent in the United States for an automotive safety cushion assembly was issued in 1953 (Crouch, 1994; Mukhopadhyay, 2008). The concept was not executed in a commercial vehicle until the early 1970s when General Motors started to produce automobiles with the safety devices. Unfortunately, the feature garnered little public support due to expensive and complicated technology. With additional research and development of an effective and more economical airbag system, the technology started to be implemented into vehicles extensively in the late 1980s. Starting in 1998, all new vehicles in the United States were required to have driver and Chapter 2– Background 7 passenger airbags. Today, vehicles are outfitted with side curtain and thorax airbags for side and rollover impacts (Mukhopadhyay, 2008). The technical requirements for airbags are defined as: good foldability, high softness, resilient, heat resistant, low air permeability, high dynamic stress resistance, low fabric weight, high fabric strength, good abrasion resistance and stability upon aging (Gon, 2010; Mukhopadhyay, 2008; Schwark & Muller, 1996; Crouch, 1994). To meet these requirements, the airbag is constructed of woven textile membranes that are mostly constructed from multi-filament polyamid 6,6 (nylon) yarns. The woven fabric structure used in the airbag application is the plain weave; where the warp and fill yarns are interlaced in a regular sequence of one under and one over. The term ‘warp’ refers to the direction of yarns that run continuously through the weaving machine and are continuous for the entire length of the fabric roll. The ‘fill’ yarns (sometimes referred to as weft or woof) are the yarns that run transverse to the warp yarns. The mechanical behavior of woven fabrics is anisotropic and highly nonlinear. The nonlinearity is a result of internal structural changes during deformations where the deformations are finite but can also be attributed to material nonlinearity of the yarn constituents. Woven fabrics possess high flexibility which allows them to fold and conform to a variety of shapes. The low bending stiffness is attributed to the interlacing of yarn without rigid bonding of the overlapping points as well as thin parallel system of fibers which are only restricted by friction. The yarn material system, polyamide (or better known by its trade name – nylon) is the generic term for any long-chain synthetic polymeric amide which has recurring amide groups as an integral part of the main polymer chain (Harris, 1954). While this definition can cover a wide range of structures, the most important group from which commercial fibers are constructed include condensation polymers of an α-ω straight-chained aliphatic diamine with an α-ω straight- chain aliphatic dicarboxylic acid. Specifically, nylon 6,6 which is most commonly used in airbag fabrics is the condensation polymer of hexamethylene diamine with adipic acid (Harris, 1954). It should also be mentioned that polyester (PET) and nylon 4,6 yarns are used in small quantity for airbag fabrics and sewing threads in some parts of world. Nonetheless, nylon 6,6 is the fiber Chapter 2– Background 8 of choice compared to other synthetic or natural fibers because it has the highest strength-to- weight ratio at an economical price and other properties that are desirable in airbag applications. Nylon 4,6 has similar mechanical and thermal behavior to nylon 6,6 but is more expensive (Gon, 2010; Mukhopadhyay, 2008). Polyester is a cheaper fiber that is used extensively in seatbelts and other automotive textiles. However, properties that make polyester a suitable material candidate for seatbelts does not translate into airbag applications. Polyester undergoes less dimensional changes compared to nylon under moisture and temperature fluctuations which allows for smooth uptake and pull out of the seatbelt (Crouch, 1993). Additionally, polyester is more rigid than nylon which prevents excessive stretching of the belt during high loading that occurs as a result of an impact. The high elongation of nylon is advantageous in airbags as it promotes wider allocation of forces throughout the airbag as well as uniform stress distribution along perimeter seams. Additionally, nylon has a relatively higher melting point, high heat of fusion and absorbs 2-4% water by weight that provides quenching properties which aids in the prevention of burn through from hot particulates that could possibly break free from the inflator and travel into the bag (Crouch, 1993). The airbag fabric can also be coated with an elastomer which lowers the permeability of the fabric and provides an ablative shield to the fabric from the hot gases (Crouch, 1993; Crouch, 1994; Gon, 2010; Schwark & Muller, 1996). The first generation of airbags used a neoprene coating while today’s airbags are almost exclusively coated with silicone. Neoprene coatings can release HCl over time which can degrade the nylon yarns. Additionally, neoprene has a tendency to self-adhere which requires the airbag to be coated with talc while silicone does not exhibit adhering so no talc is used. Silicone rubbers are more stable and therefore do not degrade the fabric even after high temperature aging. This means the amount of coating required is significantly less if silicone is used instead of neoprene, resulting in a thinner, lighter and more compliant air bag (Gon, 2010). Silicone exhibits better wear and abrasion resistance compared to neoprene coatings. From a manufacturing point of view, the silicone used in the airbag application is a Pt-cured, low viscosity liquid rubber applied using knife coating equipment to apply a thin coat upon one side of the fabric (Mukhopadhyay, 2008; Schwark & Muller, 1996). After the coating is applied to the fabric, the fabric is pulled through an oven to induce polymerization and adhesion between the fabric and coating. Chapter 2– Background 9 There has been much debate by the airbag industry regarding the value of coated and uncoated fabrics (Crouch, 1993; Gon, 2010; Mukhopadhyay, 2008; Schwark & Muller, 1996). The coated fabrics offer better resistance to heat conductivity, tear performance, and low permeability. On the other hand, the process to coat the fabrics is environmentally unfriendly. The fabric becomes hard to handle due to the permeation of the solvent of the coating liquid. Uncoated fabrics are more environmental friendly and are easier to fold and pack into small spaces. Aside from the evolution from neoprene to silicone coatings, the fabric construction has changed throughout the twenty years of airbag use (Crouch, 1993; Mukhopadhyay, 2008). The first generation of fabric design consisted of 940 decitex (abbreviated as dtex) yarns. Decitex is a measure of the linear density of the yarn and is defined as the weight in grams per 10,000 meters of yarn. Another popular unit of linear density used in textile is denier, which is the weight in grams of 9000 meters of yarn. The higher the decitex or denier, the thicker the yarn and coarser the fabric weave. The first generations of fabrics were coarse, heavy and difficult to pack. The second generation of fabrics used during the mid-1990s were made of high tenacity 470 dtex nylon yarns. At this time, the transition from neoprene to silicone coatings was also seen. The new fabrics were gentler on the passenger’s skin during impact than its predecessors and had better packability. The fabrics were also lighter and had more controlled permeability. Manufactures are now evaluating fabrics that are constructed of high to super high tenacity yarn with linear densities of 235 to 350 dtex. The fabrics have improvements in weight reduction, packability and softness. In the future, the trend of low density, high strength yarns will continue that will result in bags becoming lighter, more robust and more compact (Gon, 2010; Mukhopadhyay, 2008). As the fabric system continues to evolve, the analysis and design procedures must as well. 2.2 Airbag fabric research This section details the research related to the evaluation of airbag fabric and modeling of the mechanical properties of airbag fabric chronologically. While there has been great effort in modeling the deployment and impact kinematics of airbags, the literature related to experimental Chapter 2– Background 10 evaluation and modeling of the mechanical properties of airbag fabrics is very limited. Keshavaraj et al (Keshavaraj, Tock, & Nusholtz, 1995; Keshavaraj, Tock, & Nusholtz, 1996) studied the biaxial properties of nylon 6, nylon 6,6 and polyester fabrics using a blister-inflation device. The fabrics were of a balanced construction with the same amount of yarns in the warp and fill direction although there has been no mention of the value of crimp in the yarns. The blister-inflation technique used in the study is a quasi-steady-state measurement in which a flat sheet of fabric is deformed into a semi-spherical blister via pressure drop across the fabric using compressed air. The biaxial stretching of the fabric and changing permeability as the fabric structure is inflated is recorded. A pressure gauge measures the internal pressure of the blister while the blister height is recorded manually. The temperature of the inflating gas was collected using a temperature sensor while volumetric flow rate is measured with an anemometer. The fabric thickness is measured before the sample is loaded into the rig. The biaxial stress-strain is then determined by a relationship previously derived for solid plastic films under blister inflation and is dependent on the internal pressure, diameter of pressurized sphere, blister height and fabric thickness. The biaxial stress-strain blister relationship is based on the assumption that a constant volume of fabric sheet deforms (uniformly) from a flat configuration into a spherical segment during the experiment. Permeability behavior of 630 denier and 420 denier fabrics made of nylon 6,6 and nylon 6 were compared under both low pressure drop and high pressure drop over a variety of temperatures. The nylon 6 fibers exhibited higher permabilities than the nylon 6,6 fibers. The biaxial stress- strain behavior was also evaluated for the same specimen matrix. The specimens were pressurized at set intervals and held so that the blister height could be recorded before increasing the pressure, hence the quasi-steady-state definition. During this test, the fabric was not extended to rupture. A ball-burst rupture test was also performed to test the fabrics to failure under both static and dynamic loading. The authors claim that the ball-burst experiments provided a more realistic view of the performance of the fabric under biaxial condition. The crosshead rates tested were 0.5 and 50 inches per minute. As a continuation of the previous work, Keshavaraj et al (Keshavaraj, Tock, & Haycook, 1996) developed an airbag fabric material model for nylon and polyester using a simple neural network architecture. The neural nets formulated were intended to be used as a design tool in Chapter 2– Background 11 determining permeability and biaxial stress-strain relationships for airbag fabrics. The authors claimed that the advantage that neural networks have is the computation efficiency in handling complex and nonlinear problems compared to the nonlinear finite element technology at the time the paper was written. The inputs of the model included the experimental data obtained from the blister-inflation experiments performed in a previous study (Keshavaraj et al., 1996) and the geometric properties of the fabric (i.e. yarn linear density, yarns per inch). The authors reported predictions that fitted well with the experimental data with extremely fast computation time. The ability of the artificial neural network to obtain the mechanical behavior at different state of stresses was not discussed. Hong (Hong, 2003) used a mix of dynamic experiments with finite element simulation- optimization techniques to back calculate the elastic properties of 315 denier (60 yarns x 60 yarns) silicone coated airbag fabric under deployment conditions. Hong discussed the anisotropy seen in uniaxial tests taken in the warp, fill and bias (yarns oriented 45 degrees) directions in which Hong argued that this can be problematic in deciding the proper values to use for the mechanical properties in simulations. To find the elastic constants that govern the airbag fabric response under deployment, Hong carried out an optimization simulation. In the optimization procedure, the material properties of the airbag fabric are optimized to minimize the difference in a drop tower test on a deploying airbag and finite element simulation results. The variables that are minimized between the test and simulation are the acceleration, velocity, displacement and force. Hong was able to find values for modulus of elasticity, shear modulus and Poisson’s ratio, however, the results were influenced by the type of element formulation used. The mechanical properties obtained were not compared to other mechanical tests. Rohr et al (Rohr, Harwick, & Nahme, 2004) performed a series of experiments to determine the strength and failure behavior of fabric under different loading rates and exposed temperatures. A series of uniaxial tests on nylon 6,6 airbag fabric in the warp and fill directions was performed at loading rates 0.08 mm/s and 500 mm/s using an Instron Type 8033 mechanical tester at temperatures between -35 ºC and 85 ºC. Additionally, the fabric was tested at a loading rate of 9000 mm/s using a drop weight tower at temperatures of -35 ºC and 20 ºC. The fabric geometry such as yarn linear density, areal density, yarns per inch in both the warp and fill directions was not reported in the study. The force was normalized by assuming the fabric as a continuum with Chapter 2– Background 12 constant cross-sectional area. The results of the uniaxial tests at room temperature, quasi-static conditions found no differences in mechanical behavior between the warp and fill directions however, the sample size was not mentioned nor any indication of the repeatability of the test. The investigation of the strain-rate parameter found that as strain rate increased, the ultimate strain decreased while failure stress increased but no empirical equations are derived based on their experimental findings. When varying the exposed temperature, as the testing temperature increased, the trend of the data showed that the failure stress decreased and failure strain increased for all three strain rates. However, the authors only tested the fabrics at three temperatures over a relatively small temperature range with respect to the possible range of temperatures that the airbag structure can undergo during deployment. Additionally, the authors carried out burst pressure tests on the fabrics. Little detail is given regarding the experimental set up and procedure of the test, regardless, the failure pressure was reported to be between 4-5 bar. In a subsequent study, Rohr et al (Rohr, Harwick, & Nahme, 2005) carried out biaxial cruciform tests to determine the material behavior under 1:1 biaxial loading. The experimental rig used consisted of four lever arms where one end of a lever is connected to a traditional mechanical tester cross head while the other ends traveled on a track equipped with piezoelectric sensors to record the pulling load from the fabric specimen. Different lever lengths can introduce different ratios of biaxiality, however this study only used a 1:1 loading ratio. Like the preceding study (Rohr et al., 2005), there is no reference to the fabric geometry or the size of the sample population. It is also unclear how stresses and strains were normalized but it is assumed the same procedures of the preceding paper were used. The results found little difference between the warp and fill directions for the particular fabric system tested. The authors indicated that the sample failed near the grips and at the corners of the specimen. A group of German automotive researchers noted the importance of fabric modeling, particularly in folding and deployment simulations (Dörnhoff et al., 2008). The fabric models used in their studies are orthotropic elasticity definitions which are implemented into the major dynamic finite element codes (LS-DYNA, PAM-CRASH, MADYMO FE). Biaxial extension and picture frame tests were performed to generate the stress-strain response of the fabric under extension and pure shear. For the biaxial test, the fabrics were not loaded to failure, rather loaded to a given point Chapter 2– Background 13 and the recovery of the fabric was monitored. The procedure outlining the picture frame test for evaluation of shear properties of the fabric was discussed, although no results are given. The authors claimed that using the generated stress-strain curves rather than mechanical data from the fabric supplier, it was possible to get a better representation of the real life airbag kinematics. While not a main focus in the paper, it discussed the need and challenge for accurate geometrical description of the airbag structural components such as patches, straps and vents to improve virtual airbag designs. Behera and Goyal (Behera & Goyal, 2009) used an artificial neural network system to predict performance parameter for five different airbag fabric systems coated with silicone and polyurethane. The inputs of the model consisted mostly of construction parameters such as fabric areal density, yarns per inch, warp/fill yarn linear density, yarn strength, yarn ultimate elongation, fabric cover, warp/fill crimp, thickness and yarn flexural rigidity. To validate the model output experimental testing followed ASTM standards for breaking load, tear strength, bursting strength, air permeability and specific packability. After the network was calibrated, the model was able to determine breaking loads, final elongations, permeability, tear strength, burst strength and packability. The model was able to predict failure strength and elongation with minimal error while air permeability and tearing strength outputs were found to have high percentages of error. The authors state that the prediction performance of the model is purely based on the amount of “training” or the amount of data available. Brueggert and Tanov (Brueggert & Tanov, 2002; Tanov & Brueggert, 2003) proposed a user defined material model for loosely woven airbag fabric in the finite element code, LS-DYNA, which takes into account the non-orthogonal orientation as the fabric shears. The model was based on a representative unit cell constructed of four bars to from a trellis that is diagonally braced by two springs. The linear elastic bars represent the extensional behavior of the fabric while the diagonal springs model captures the effect of the yarns locking upon excessive shear deformation. The authors demonstrated that the internal pressure and kinematics of a side- cylindrical shaped airbag recorded in experimental tests were in good agreement with a simulation using the generated material model. Chapter 2– Background 14 2.2.1 Summary The literature regarding evaluation of airbag fabric and modeling the constitutive behavior is quite limited. On the experimental evaluation side, airbag fabrics have been tested in uniaxial tension under a variety of strain rates and temperatures, and under biaxial tension using the inflation and cruciform techniques. The blister inflation technique assumes the stresses and strains are equal for the warp and fill direction which could be invalid if the fabric is unbalanced either with regard to fabric geometry or yarn mechanical behavior. The cruciform technique is susceptible to stress concentrations but can be used to mimic different states of stress on balanced and unbalanced fabrics. In the studies discussed, stresses are normalized by cross sectional area under the assumption that the fabric is a continuum. While the procedure of picture frame shear tests has been discussed, shear stress-strain results have not been published for airbag fabrics. Additionally, other modes of shear deformations like rail shear or shear under biaxially pre-stressed fabrics have not been investigated for airbag fabrics to date. Also missing from the current literature is an investigation of the mechanical properties of the airbag yarns as well as a direct comparison between coated and uncoated fabrics. The current avant-garde in terms of modeling elastic behavior of airbag fabrics has been dominated by finite element orthotropic elastic definitions and to a lesser degree artificial neural networks. The orthotropic elastic material models common in the commercial codes are as useful as their inputs making them only accurate for the stress state at which the inputs are given. The artificial neural networks have been more of an academic exercise that requires a wide range of experiments with large sample sizes to be of use. Additionally, the ability of artificial network to predict mechanical behavior and failure under multiple states of stress has not been studied. The current cutting edge methods, a mix of unit cell constitutive relationships with finite element continuum formulations have been proposed and demonstrated for use in airbags although further study and validation of the material behavior is needed. Chapter 2– Background 15 2.3 Modeling of the mechanical behavior of fabrics 2.3.1 Representative mechanistic models Perhaps the grandfather of the study of mechanics of woven fabrics is Dietzius Haas (Haas, 1918). Haas was interested in studying the deformations of biaxially stressed textile fabrics for use in airship design. Haas’ analysis considered the effects of crimp interchange, thread straightening and thread shear on the forces seen in the principal axes of a fabric system. Mathematical relationships to describe the equilibrium of forces were derived based on the basic geometry of the fabrics. The consideration of fabric shear was based on the assumption that the fabric would shear in a frictionless trellis like manner under a biaxial stress field until the resultant force at the intersection of the warp and fill yarns aligned with the physical orientation of the warp and fill yarn. The analysis was practical and significant as it improved the stability of airship, particularly when attempting to make turns (Hearle, 1969). A 1937 paper by F.T. Pierce is a classical paper in woven fabric mechanics that is highly cited in the literature (Peirce, 1937). In the work, Pierce derived the geometry for a representative unit cell model of woven fabric based on yarns with solid circular cross sections as seen in Figure 2-1. The unit cell model is able to capture the finite internal structural changes that occur within the fabric under the condition that the force exerted by the warp yarn on the filling yarn equals the force exerted by the filling on the warp. Thereby under the conditions of equilibrium and continuity, the stress-strain curve of the fabric at a macro scale can be determined. However, solving the geometric and kinematic equations has to be performed using numerical methods as the relations are complex and nonlinear. Later, a study by Freeston developed a theoretical analysis of the Pierce unit cell under biaxial loading to provide simpler analytical expressions (Freeston, Platt, & Schoppee, 1967). The circular cross section and the assumptions of perfectly flexible, incompressible yarns in the Pierce model limits the analysis to a small range of fabric systems. Future improvements of this technique were made by Kemp (Kemp, 1958) for race- track shaped yarns and Shanahan and Hearle (Shanahan & Hearle, 1978) for lenticular shaped yarns that were compressible. Chapter 2– Background 16 Actual Fabric Structure Idealized Unit Cell Structure df d w pf H hf D/2 Figure 2-1: Pierce's unit cell (1937) Grosberg was among the first to present a mechanistic model of shear for plain weave fabrics (Grosberg & Park, 1966; Grosberg, Leaf, & Park, 1968). Grosberg recognized the factors that influence shear rigidity, namely the resistance against change of the interlacing angle caused by friction and elastic restriction. The model was based on a unit cell of intersecting warp and fill yarns and considered elastic lateral bending of the yarns, slippage of the yarns at the intersection and subsequent elastic rotation of the yarn crossovers. The frictional contact between the yarns was assumed to be a line contact however the length of the contact had to be back calculated from experiments for the slippage portion of the model due to simplifications made in the geometry. In the elastic portion of the model, the line contact is computed with geometrical relationships. Additionally, Grosberg recognized that the geometry of the fabric structure and contact force changes when the fabric specimen is put under tension and provided a correction within the model to account for the presence of a pre-tension. However, the correction cannot account for changing tensions as the structural configuration of the unit cell is updated throughout the analysis. While the model presented in the papers had good agreement with the experimental results, greater deviations were seen in extremely tight and extremely loose fabrics. The model has not been able to replicate the shear behavior of different fabric systems observed by other researchers in the recent literature (Sun & Pan, 2005). Chapter 2– Background 17 A simplified model of fabric shear and a discussion of the geometrical limits of fabric were developed by Skelton (Skelton, 1976). In many ways, the simplified model is similar to the Grosberg analysis in that it considered the frictional force required to rotate the yarns at the intersection for the same unit cell geometry but neglecting the consideration of lateral bending of the yarns. The theoretical maximum shear angle with respect to fabric tightness was determined by taking into account the geometric construction of the fabrics under several assumptions: yarns are thin interwoven strips, yarns are interwoven cylinders and yarns are in side-by-side contact. Additionally, Skelton concluded that for fabrics that are tightly woven, the occurrence of shear would not be contributed to yarn rotation but by yarn distortion, in which case, the fabric shear behavior is more like a laminar sheet material. The next significant contribution to a mechanistic approach of woven fabrics was made by Kawabata (Kawabata, Niwa, & Kawai, 1973a; Kawabata, Niwa, & Kawai, 1973b; Kawabata, 1989) who considered both extensional and shear responses. The fabric is represented by simple a unit cell constructed of straight bars connected by pins at the yarn crossover as seen in Figure 2-2. The structural parameters required to construct the unit cell are the warp and fill yarn density and crimp. Similar to the equilibrium condition in the Pierce model, under biaxial deformation, the contact force between the warp and fill yarn is balanced and the axial force in the yarn is broken down into its components to compute the force seen at the fabric ends. Adequate predictions were found comparing the experimental performance on cotton, wool and polyester fabric systems under biaxial loads. However, the equilibrium condition cannot be applied to solve uniaxial extension behavior because no tension is applied to the yarn in the non- loaded transverse direction (Kawabata, Niwa, & Kawai, 1973b). Therefore, no resistance force is preventing the straightening of the load yarn (geometric stiffening). It was found that in order to capture the geometric stiffening a resistance force from the transverse yarn acts upon the loaded yarn direction. By considering the bending rigidity and intra-fiber shear in the resistance force of the transverse yarn, the straight line unit cell model could capture the uniaxial behavior seen in experiments with reasonable accuracy. Chapter 2– Background 18 X3 X2 X1 H01 H02 2y01 2y02 y01 y02 Actual Fabric Structure Idealized Unit Cell Structure X1 X2 Figure 2-2: Kawabata's unit cell (1973) The same unit cell model was also used to describe the shear deformation of fabrics using some empirical relationships of the yarns in rotation (Kawabata, Niwa, & Kawai, 1973c). The shear model is capable of capturing the coupled behavior extension has on the shear resistance of the fabric. In the Kawabata model, as the unit cell structure is deformed by extension and rotation, the shear force is found by balancing the torque force required to rotate the intersecting yarns, which is a function of contact force and friction. To calibrate the shear portion of the model, specialized equipment was developed to determine the torque required to rotate a crimped yarn under several magnitude of contact forces. This method was able to obtain reasonable results of fabric under simple shear (rail shear) as well as shear combined with constant extension. Overall, the Kawabata unit cell approach used to determine the extensional behavior is very popular in present day research due to its simplicity and accuracy to describe the complex nonlinear behavior of fabrics. The method has been modified to include coating extension and inelastic effects (Stubbs & Thomas, 1984); sinusoidal profile of yarns instead of linear elements (A. Shahkarami & Vaziri, 2006; A. Shahkarami, 2006; A. Shahkarami, 2006); and incorporated as the constitutive relationship in some finite element analysis (Ivanov & Tabiei, 2004; King, Jearanaisilawong, & Socrate, 2005; A. Shahkarami & Vaziri, 2007). Chapter 2– Background 19 2.3.2 Continuum approaches There have been considerable efforts by textile engineers to assume the fabric system as a continuous sheet in order to use plate theory in the analysis and design of fabric structures. Kilby (Kilby, 1963) developed planar stress-strain relationships of a simple trellis in which linear elements pivoted together at the yarn intersection, although the passing over and under of the yarns is not considered. The analysis found that the fabric can be treated as an elastic lamina and yielded a single expression for the general modulus in any direction using the modulus of elasticity in the warp and fill directions, shear modulus and Poisson’s ratio and angle between the direction of extension and the warp yarn as input. Due to the nonlinearity of fabrics, the analysis is limited to small strains. Alley and Fasion (Alley & Faison, 1972) attempted to analyze the fabric response using the generalized form of Hooke’s Law for continuum analysis of membrane structures, more specifically parawings. The authors described that nine coefficients are needed to construct the plane stress anisotropic constitutive relations, four of which are coefficients of interaction of first and second kinds. The coefficients of interaction are analogous to the shear coupling parameters seen lamina constitutive relationships. Unlike lamina, the resulting Hookian relationship for fabric is asymmetric and a function of both axial and shear loads. A series of experimental tests were performed on polyurethane coated nylon fabric to obtain the nonlinear coefficients for that particular fabric system. Shanahan, Lloyd and Hearle (Shanahan & Hearle, 1978) examined the uses of plate and shell continuum formulations to describe the complex deformations of fabrics under the assumption that fabrics behave as a sheet. Due to the orthogonal nature of the woven fabric geometry, it is convenient to make use of structural axes. Therefore, Shanahan and co-workers were able to construct the linear elastic stiffness matrix that had 13 independent stiffnesses that accounts for extension, bending and coupling. Again, due to the complexity of fabric mechanical behavior, the analysis is limited to small strain, linear problems though the authors state the analysis provides the framework for solving for nonlinear situations. Aside from the computational limitations of the analysis are the difficulties in evaluating the fabrics experimentally to obtain the elastic constants. The authors argue that the large displacements and finite areas to obtain Chapter 2– Background 20 measurable effects in addition to lack of rigidity cause impediments using traditional measurement techniques for anisotropic engineering materials. 2.3.3 Current state-of-the-art The current methods in fabric analysis rely heavily on the finite element method. Shockey et. al (Shockey, Erlich, & Simons, 2000) were one of the earliest researchers to discretely model the individual yarns as solid continuum orientated in a plain weave structure. A small patch of fabric was modeled to capture the dynamic response of the fabric subjected to aircraft engine fragmentations. A similar approach was preformed by Duan et. al. (Duan, Keefe, Bogetti, & Cheeseman, 2005) for ballistic impact of fabrics under different boundary conditions. Shahkarami (A. Shahkarami, 2006) proposed a less intricate model, where a unit cell of two crossover yarns is analyzed to capture the detailed behavior of fabric under biaxial loading. Lomov and Verpoest have developed sophisticated algorithms to generate finite element models for a wide range of fabric structures ranging from plain weaves, twill weaves and 3D woven structures (Lomov, Gusakov, Huysmans, Prodromou, & Verpoest, 2000; Lomov et al., 2001). While these fully 3D analyses provide a wealth of detail regarding the mechanical and dynamical behavior of the yarns and fabric, a significant amount of computational power and time are required. Therefore, only small portions of the structures are modeled. To remedy the computational requirements but retaining important details of the fabric deformation, shell elements whose constitutive relationships are based on a mechanistic representative unit cell have been developed by a number of researchers for a variety of applications. Brueggert and Tanov (Tanov & Brueggert, 2003) developed a user material model (UMAT) for shell element in the commercial code LS-DYNA where the constitutive relationship was based on a square arrangement of linear springs capturing the extensional response of the fabric while a set of diagonal springs represent the shear behavior of the fabric. Ivanov and Tabiei (Ivanov & Tabiei, 2004) developed a micromechanical model for Kevlar fabric based on the Kawabata unit cell geometry with a viscoelastic stress-strain yarn model to capture strain rate dependency. The model was implemented as a UMAT for the LS-DYNA to simulate ballistic impact of Kevlar fabric. King (King et al., 2005) developed a material model for the commercial code ABAQUS/Standard to predict the behavior of Kevlar fabric under quasi-static uniaxial Chapter 2– Background 21 extension, bias extension and picture frame shear. Shahkarami (A. Shahkarami & Vaziri, 2006; A. Shahkarami, 2006) developed a UMAT for LS-DYNA intended for shell formulations based on a unit cell with a sinusoidal yarn profile for ballistic impact simulations of Kevlar fabric panels. Overall, this approach of using a respective micromechanical model as the constitutive relationship of a shell finite formulation increases the computational efficiency while still accurately capturing the internal structural changes that affect the macroscopic mechanical behavior of the fabric. 2.3.4 Summary The study of the mechanics of woven fabrics is nearly a century old and has been applied to the study of everything from airships to bullet proof armor. The unit cell method has proven to be effective in generating the extensional stress-strain behavior of fabrics while predicting shear behavior has been less successful. The continuum approaches are easy to implement for the analysis of structures but require a great deal of testing to capture the anisotropic and nonlinear nature of the fabric. Fully 3D finite element models provide the most detail about the fabric’s mechanical behavior, but are too computationally demanding to model large structures. The current state-of-the-art techniques discussed offer a favorable compromise of mechanistic unit cell methods with finite element analysis that provides accuracy and detail without extensive calibration of elastic constants. Chapter 3- Development of a representative unit cell: theory and calibration 22 Chapter 3 - Development of a representative unit cell: theory and calibration 3.1 Background and approach The first step in developing a modeling approach is to establish a geometry that best represents the fabric structure that is simple yet does not sacrifice accuracy or neglect reality. A unit cell based on the center-line positioning of the yarn which is approximated by straight lines originally proposed by Kawabata (Kawabata, Niwa, & Kawai, 1973a; Kawabata, Niwa, & Kawai, 1973b; Kawabata, 1989) was chosen for this study. This simple center-line approach has been adopted by several researchers over the years for a variety of applications as shown in Table 3-1. While the center-line unit cell geometry has been repeated throughout the literature, the structural mechanisms that empower the unit cell vary greatly depending on the fabric system and application. For instance, some textile composite preforms have lower interlacing density and lower yarn crimp which permits more fiber mobility in the yarn that in turn allows the yarn to exhibit a more compressible behavior. (Lomov & Verpoest, 2000) However, tighter woven fabric structures like those used in airbags tend to inhibit fiber mobility which results in an incompressible yarn (Lomov, 2000). Likewise, fabric systems of fiber-glass and Kevlar can be approximated to have linear elastic behavior while other fabrics are composed of highly nonlinear materials such as nylon or polyester. Therefore, when implementing this type of unit cell approach, it is important for the user to consider the application and material system in order to incorporate the proper structural mechanisms to reflect the physical nuisances of the fabric. Thus as novel applications and material systems are considered for this particular unit cell theory, new mechanisms will be developed that will add to the library of knowledge that the fabric designer or analyst can use. Chapter 3- Development of a representative unit cell: theory and calibration 23 A goal in the proposed model is to implement mechanisms whose inputs are simple constitutive properties of the fabric, more specifically fabric geometry and yarn stiffness. In that way, the model can be used to analyze the in-plane behavior of a fabric without excessive or specialized testing. Additionally, it can be utilized as a design tool to predict certain mechanical properties of a virtual fabric. The way the proposed unit cell model adds to the current body of knowledge is by including: nonlinear yarn extension behavior for nylon, yarn bending rigidity, and coating extension. All of which are based on structural mechanics using simple geometry and material properties (Young’s Modulus, Poisson’s ratio). In Table 3-1, the proposed models features are compared to the current models in the literature. Table 3-1: Existing center-line unit cell models vs. proposed model Biaxial, Uniaxial, Bias Shear YesYes(EC)Nonlinear Extension, Bending Rigidity, Incompressible LinearAutomotive Airbag Nylon 6,6; Silicone- coated Nylon 6,6 PROPOSED MODEL BiaxialYesNoLinear Ext, Perfectly Flexible, Incompressible Linear & Sinusoidal Architectural Fabrics Teflon- coated fiberglass; PVC- polyester Bridgens (2008) Biaxial* , Ballistic Impact n/aYes(NC)Linear Ext, Perfectly Flexible, Compressible(EC) SinusoidalBallistic Protection KevlarShahkarami (2005) Uniaxial, Biasn/aYes(EC)Linear Ext, Bending Rigidity(EC), Compressible(EC) LinearBallistic Protection KevlarKing (2004) Nonen/aYes(EC)Linear Ext w/ viscoelasticity, Perfectly Flexible LinearBallistic Protection KevlarTabiei & Inanov (2004) Biaxial, Uniaxialn/aNoLinear Ext, Perfectly Flexible, Compressible(EC) LinearComposite Preform Forming FiberglassBoisse (2001) Biaxial, Picture Frame Shear Yes(EC)Yes(EC)Linear Ext, Perfectly Flexible, Compressible(EC) LinearArchitectural Fabrics Teflon- coated fiberglass Kato (1999) BiaxialYesNoLinear Ext, Perfectly Flexible, Compressible(EC) LinearArchitectural Fabrics Teflon- coated fiberglass Stubbs (1980) Biaxial, uniaxial, rail shear n/aYes(EC)Nonlinear Extension, Bending Rigidity, Incompressible & Compressible(EC) LinearApparelCotton; polyester Kawabata (1973) Experimental Validation Methods Coating Included Shear Included Yarn BehaviorUC ShapeApplicationFabric System Researcher * Basic UC material behavior verified using data from Boisse EC= Experimentally Calibrated NC = Numerically Calibrated Chapter 3- Development of a representative unit cell: theory and calibration 24 3.2 Unit cell definition The simple unit cell geometry used in this study is shown in Figure 3-1. The coordinate system origin is defined as the intersection of the warp and fill yarns such that the coordinate axis X1 is along the neutral line of the warp direction, axis X2 in the fill direction and axis X3 is the through- thickness direction of the fabric. The assumptions established in this study’s formulation of the unit cell are the following: • The yarns are elastic • Displacements are equal for each yarn end • No slippage occurs at the yarn crossovers • Environmental effects such as temperature or moisture changes are not considered; i.e. the behavior of the fabric is the same as it is at room temperature • Strain rate effects are neglected for the time being due to lack of experimental procedure and data FEnd,2 FEnd,2 FEnd,1 FEnd,1 Initial State Deformed State x3 x2 x1 x3 x2 x1 L01 H02 2y01 2y02 y01 y02 θ01 θ02H01 L02 tc Fc,1 Fb,1 Fc,2Fb,2 Figure 3-1: Simple unit cell geometry in initial and deformed states Chapter 3- Development of a representative unit cell: theory and calibration 25 The construction of the straight line unit cell geometry requires two geometric inputs. The first being the yarns per unit length in the warp and fill directions (in textiles, the common unit is yarns/inch). From this parameter, the horizontal half spacing of the unit cell, y, can be calculated as: ( )201 2 1 n y = (3-1) ( )102 2 1 n y = (3-2) where n is the number of yarns per unit length. From here forward, the 1 and 2 indices will indicate the warp and fill yarn, respectively. The 0 index represents the original undeformed structure. The second geometry input parameter is the percent crimp of the warp and fill yarns. The crimp of the yarn is the quantification of the amount of undulation caused by weaving and can be found by measuring a horizontal distance, Xi, between two points that are parallel to the yarn and the actual length, l0i, of the yarn between the same two points. The crimp can be quantified as: i ii i X Xl cr − = 0 (3-3) where i = 1,2. Therefore the length of the yarn can be determined using the expression: ( )iii cryL += 100 (3-4) The final geometric parameter to assemble the unit cell structure is the yarn height which can be found simply by: 2 0 2 00 iii yLH −= (3-5) To this point, the yarn geometry has been defined in the unit cell, now the coating geometric parameters needs to be defined. The length and width of the coating are equal to the fabric unit Chapter 3- Development of a representative unit cell: theory and calibration 26 cell spacing, y01 and y02, as shown in Figure 3-1. The thickness of the coating, tc, can be found using the areal density fraction between the uncoated and coated fabric and the total height (thickness) of the unit cell. ( ) −+= 122 0201 u c c w wHHt (3-6) where wu is the areal density of the uncoated fabric and wc is the areal density of the coated fabric. The equation (3-6) is only applicable to coated fabrics that have the same geometric construction as the uncoated fabric. Additionally it is assumed that the coating maintains a constant thickness throughout the unit cell. When a displacement is applied to the unit cell, the coating and fabric are assumed to stretch by the same amount. The tensile force developed at the end of the unit cell, denoted as Fend,i in Figure 3-1, is the sum of forces generated by the fabric and coating and is expressed as: ictifiend FFF ,,, += (3-7) where Ff,i is the fabric end force and Fct,i is the end force from coating extension. Finally, the unit cell end forces can be translated into the membrane stress (force per unit width of fabric) by: iiendi nFN ,= (3-8) The membrane stresses developed depend on the reconfiguration of the unit cell structure which is governed by constitutive relations and mechanisms of the yarn and coating. To determine the new structural configuration of the perturbed fabric structure requires solving a set of highly nonlinear equations for which a numerical procedure is needed. A computer code was developed to solve these equations for a particular strain or can be iterated to generate stress-strain curves of the fabric system for uniaxial and multiple ratios of biaxial extension. Figure 3-2 shows a flowchart of this code. Chapter 3- Development of a representative unit cell: theory and calibration 27 Input n, c, yarn f-d, EI, εult Specify Stress State Biaxial Specify Biaxial Ratio, Strain increment Specify Warp or Fill Displacement, Strain increment Uniaxial Check ε< εult End Save Ni N-R Procedure ε1 =ε1i-1 + ∆ε ε< εult ε> εult ε1 =ε1 i-1 + ∆ε1 ε2 =ε2 i-1 + ∆ε2 Figure 3-2: Procedure for generating stress-strain behavior of fabric under multiple states of stress using the unit cell approach Depending on the stress state, the solution procedure varies slightly due to conditional geometric behavior of the unit cell. To be clear, under biaxial load the change of yarn spacing in both the warp and fill direction can be determined from the strains while the change in the yarn length is a function of the yarn height. Under uniaxial load the length of the yarn transverse to the applied load is equal to its original length while the yarn spacing is a function of the yarn height. Under these conditions, the yarn is not allowed to compress axially to support negative strains hence compressive resistance is provided by the bending rigidity of the yarn. Table 3-2 summarizes these conditional geometric properties of the fabric under different modes of extension. Finally, a yarn failure criterion was established in which the code runs until either the warp and fill yarn surpasses the specified ultimate yarn strain. Chapter 3- Development of a representative unit cell: theory and calibration 28 Table 3-2: Unit cell configuration conditions Variable Biaxial Uniaxial – Warp Uniaxial - Fill y1 = y01 + d1 = y01 + d1 f(H1, H2) y2 = y02 + d2 f(H1, H2) = y02 + d2 L1 f(H1, H2) f(H1, H2) = L01 L2 f(H1, H2) = L02 f(H1, H2) H1 Unknown Unknown Unknown H2 Unknown Unknown Unknown The center-line unit cell method has two main equations that need to be evaluated in order to solve for the end forces that arise from the perturbed fabric structure: continuity and equilibrium. The continuity equation for an incompressible yarn where the cross-section of the yarns remain unchanged during stretching, regardless of the profile shape is given as: 210201 HHHH +=+ (3-9) where H01 and H02 are the original height of the warp and fill yarn, respectively. H1 and H2 are the height of the warp and fill yarns in the perturbed structure. The equilibrium condition considers the sum of the vertical force components that arise between the warp and fill yarns at the cross-over. The forces that are considered in this study are axial extension whose vertical component exerts a contact force at the cross-over and the forces that arise from bending the yarn. The equilibrium condition as shown in Figure 3-1 is expressed as: ( ) ( ) ( ) ( )22221111 HFHFHFHF bcbc −=− (3-10) where the magnitude of the vertical cross-over forces are a function of the yarn height, Fc is the contact force that arise from yarn extensions and Fb is the reaction force that arises from bending. Chapter 3- Development of a representative unit cell: theory and calibration 29 3.3 Deformational mechanisms and constitutive relationship Before solving equations (3-9) and (3-10), the deformational mechanisms of the fabric’s constituents need to be derived and discussed in detail. Some of these mechanisms dictate the equilibrium between the warp and fill yarns while others simply contribute to the end force through super-position. Furthermore, the shear stress-strain behavior of the fabric needs to be established. 3.3.1 Yarn axial extension The yarn axial force is determined simply by the relation ( ) ( ) i ii iiy L LLAEF 0 0 , − = ε (3-11) where ( )εE i is the axial stiffness of the yarn, the magnitude of which is a function of strain. The nonlinear relationship of ( )εE is determined experimentally and will be discussed in greater detail in Section 3.5. The variable Ai is the specific area of the yarn which is determined through the following relationship ρ9000 , id i L A = (3-12) where Ld,i is the linear density of the yarn (denier) and ρ is the density of the fiber material (g/cm3) resulting in Ai being measured in (mm2). The total contact force or reaction force at the yarn crossover that arises due to the extension can be shown to be: i i iyic L HFF ,, 2= (3-13) Chapter 3- Development of a representative unit cell: theory and calibration 30 3.3.2 Yarn bending While yarns are known for their high flexibility, the small bending rigidity is important to consider capturing low level stress behavior, particularly under the uniaxial case where the only force developed by the unloaded yarn to maintain equilibrium at the crossover is due to the bending rigidity of the yarn. For determining the bending reaction force due to the changing angle of the yarn, first consider the case of bending a straight linear elastica into a crimped form as shown in Figure 3-3. With respect to the unit cell, the elastica shown in the figure is composed of two unit cell halves from two crossovers. Hence, the bending force, Fb,i is divided by a factor of two. The bending property of the yarn can be characterized by the linear equation: ( ) c yarn r EI M = (3-14) where (EI)yarn is the bending rigidity of the yarn and rc is the radius of curvature. Fb,i/2 dS x+ yi θi φ φ rc Fb,i/2 yi dx dφ S Figure 3-3: Bending geometry of an elastica Chapter 3- Development of a representative unit cell: theory and calibration 31 There has been a great deal of research performed concerning the mechanics of the bending rigidity of yarn particularly on twisted yarns or cordage materials (Freeston & Schoppee, 1975; Hearle, 1969; Platt, Klein, & Hamburger, 1959). Fortunately, the airbag fabric is constructed of low-twist continuous filaments with a circular cross-section, therefore a simple treatment is defensible. The theoretical bending rigidity of low-twist, non-blended yarns is bounded by two values (Platt et al., 1959). The lower bound assumes each individual fiber has complete freedom of motion in a frictionless manner and is defined as: ( ) 64 4 fiber fiberfiber LB yarn d ENEI pi = (3-15) where Nfiber is the number of fibers in the yarn, Efiber is the modulus of elasticity of the fiber and dfiber is the diameter of the fiber. The upper bound assumes the fibers have no freedom to move and act like a complete cluster bonded by friction. Therefore the yarn bending rigidity can be computed using the number of fibers in the yarn divided by the yarn packing factor times the rigidity of the lower bound. This yields the expression: ( ) yarnfiberUByarn IEEI = (3-16) where Efiber is the modulus of elasticity of the fiber and Iyarn is the second moment of inertia using the cross-sectional geometry of the yarn. To determine the force equilibrium between the reaction force and bending moment, from Figure 3-3, it can be shown that ϕd ds r c = (3-17) and ϕcos dxds = (3-18) Chapter 3- Development of a representative unit cell: theory and calibration 32 Substituting (3-17) into (3-14) and representing the moment by the reaction force and the bending moment gives: ( ) xF ds dEIM byarn 2 −== ϕ (3-19) Substituting (3-18) into (3-19) and integrating ( ) xdxFdEI byarn ∫∫ −= 2cos ϕϕ (3-20) Applying the boundary conditions seen in Figure 3-3 where at x = 0, φ = θ yields: ( ) ( )ϕθ sinsin2 2 −= x EI F yarnb (3-21) Therefore at the yarn intersection where x = y and φ = 0, and adding the second half of the bending force, the total bending force at the crossover can be expressed as: ( ) i i i iyarn ib L H y EI F 2, 8 = (3-22) This expression is identical to the one proposed by Grosberg (Grosberg, 1966; Hearle, 1969) in his analysis of the bending of fabrics. However, we are interested in the vertical force that arises when the yarn is bent into a new configuration from its original woven structure. If we reference the yarn in its woven state as datum (original configuration), the vertical force from bending out of this state can be expressed as: ( ) −= i i i i i iyarn ib L H L H y EI F 0 0 2, 8 (3-23) Once the vertical force components are known and balanced, the end horizontal force from the yarns which are of interest can be found simply by: i i iyif L yFF ,, = (3-24) Chapter 3- Development of a representative unit cell: theory and calibration 33 3.3.3 Coating extension In the case of the coated fabric, the end force from the coating must be taken into account. Assuming the coating is therein isotropic elastic continuum, the stresses under biaxial load using Hooke’s law can be expressed as: ( )2121 1 ενενσ c c c E − − = (3-25) ( )1222 1 ενενσ c c c E − − = (3-26) where Ec is the modulus of elasticity for the coating and cν is the Poisson’s ratio of the coating. Resolving these stresses into an end force associated with the fabric unit cell geometry, the force- strain relationship for the coating is: − − − − = 02 022 01 011 2 2 1, 1 y yy y yytyEF c c cc ct νν (3-27) − − − − = 01 011 02 022 2 1 2, 1 y yy y yytyEF c c cc ct νν (3-28) The fabric and coating end force are superimposed to obtain the total end force per unit cell ictifiend FFF ,,, += (3-29) Finally, the unit cell end forces can be translated into the membrane stress (force per unit width of fabric) by: iiendi nFN ,= (3-30) Chapter 3- Development of a representative unit cell: theory and calibration 34 3.3.4 Unit cell in-plane shear behavior Up to this point the constitutive behavior discussed pertains to the extensional behavior of the fabric. The fabric can undergo shear deformations characterized by rotation of the yarns at the crossover. At first glance the shear problem seems trivial -- the yarns bend laterally until overcoming static friction at the crossover then rotating that mimicks the rotation of a trellis structure. Several researchers have proposed simple mechanisms based on elasticity to describe the shear forces that cause this behavior with varying degrees of success depending on the fabric system and geometry. However, the culprits that contribute to the shear behavior are more complicated than one’s first instinctive analysis of the problem. Intra-fiber friction, yarn torque, yarn sliding, intra-yarn shear of the fibers and the rate dependency of friction are issues that are difficult to capture using simple unit cells. The problem of shear becomes progressively more complicated when considering coated fabrics. To the author’s best knowledge, Farboodmanesh has probably conducted the most extensive studies on the shear behavior of coated fabric (Farboodmanesh, 2003; Farboodmanesh et al., 2005). The shear behavior of a typical single-coated fabric versus uncoated is illustrated in Figure 3-4. Essentially, at low level shear stress the shear behavior is governed by a coating- fabric interaction marked by a high shear modulus. As the shear strains grow, the shear stiffness of the system decreases and transitions to a behavior that resembles the uncoated fabric. Farboodmanesh (Farboodmanesh, 2003) attempted to model this behavior borrowing from micromechanical models used in traditional fiber reinforced composites but had limited success. Developing a robust analytical-constituent-based shear model for coated or uncoated fabric is an enormous and difficult task which is the reason why many of the unit cell models that include shear require experimental calibration. Chapter 3- Development of a representative unit cell: theory and calibration 35 Shear Strain Single-Coated Fabric Sh e a r St re ss Uncoated Fabric Sh e a r St re ss Sh e a r St re ss Figure 3-4: Typical shear stress-strain curve for single-coated and uncoated fabric Due to the difficulties of evaluating fabric shear using the fabric’s constituents, empirical data from the fabric is warranted to empower the shear behavior of the proposed unit cell method. This empirical data can be obtained either through picture frame test where the fabric undergoes pure shear; rail shear test using the Kawabata Evaluation System (KES) (Kawabata, 1989) for fabrics; or could potentially be numerically calibrated through a full 3D finite element model of the fabric and its constituents. Each method has advantages and disadvantages. Picture frame requires special fixture and the design of the fixture must be carefully done to ensure reproducible results. Kawabata Evaluation System requires specialized equipment and some post-processing to neglect the effects of tension on the specimen. The use of 3D finite element models can be used to evaluate the pure shear behavior of virtually designed fabrics but capturing the correct boundary conditions can be challenging. Regardless of the approach used to obtain the shear data, the following method can be used to represent the shear behavior of the unit cell. A similar approach was used by Shahkarami (A. Shahkarami & Vaziri, 2006; A. Shahkarami, 2006) to describe the shear behavior of uncoated fabrics and is modified for the current model to include coated fabrics. A spline fit of the shear modulus as a function of shear strain illustrated in Figure 3-5a can be used to describe the shear stress-strain behavior based on typical behavior seen in Figure 3-5b. Chapter 3- Development of a representative unit cell: theory and calibration 36 G1coated G2 G3 G1uncoated Sh e a r M o du lu s Shear Strain γ1 γ2 γ3 γ4 Sh e a r M o du lu s (a) Coated Transition Zone Shear Strain Sh e a r St re ss γ2γ1 γ3 γ4 Locking Transition Zone Single-Coated Fabric Uncoated Fabric Sh e a r St re ss (b) Figure 3-5: Shear model behavior a) secant shear modulus as a function of strain b) regions in the shear stress-strain curve For coated fabrics, the fabric-coating shear interaction modulus, G1 needs to be defined as well as the extent of the transition zone which is bounded by γ1 and γ2. For uncoated fabrics, these values can be set equal to zero. The pre-locking shear stiffness of the fabric, G2 is the same for both uncoated and coated fabrics of the same construction. As the fabric continues to deform in shear, the amount of rotation at the crossovers reach a geometric limit and begin to lock. This is Chapter 3- Development of a representative unit cell: theory and calibration 37 a gradual process controlled by friction, the current packing state of the fibers within the yarn and geometric features of the fabric. Within the spline model, this process is bounded by γ3 and γ4 which are the same for both the coated and uncoated fabric. Likewise the locking shear modulus, G3, is the same for uncoated and coated fabrics. In the end, the spline fit can be summarized as shown in equation (3-31). ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ≥−−+−−+ <≤−−+− − − + <≤−−+ <≤− − − + <≤ = 4121234233 431212 2 3 34 23 2 3212122 21 2 1 12 12 1 11 12 2 1 2 1 2 1 2 1 2 1 2 1 0 γγγγγγγ γγγγγγγ γγ γ γγγγγγ γγγγγ γγ γ γγγ GGGGG GGGGG GGG GGG G S (3-31) It is important to note that this representation of the shear is purely elastic. In reality, the overcoming of friction is a plastic behavior in the fabric which is exhibited by hysteresis in the loading-unloading behavior of the fabric. Since the airbag is a one-time use structure subject to high strain rates, the effect of friction and plasticity are assumed to be negligible at this time. 3.4 Numerical procedure With the continuity equation and the deformational mechanisms that influence equilibrium defined, a technique to solve the nonlinear equations to determine the fabric stress at an associated strain can be discussed. While there are many methods in the literature to solve multi- variable, nonlinear equations such as Brent’s method, the Newton-Raphson method was used for its efficiency and speed at arriving at the roots of a system of nonlinear equations (Press, 1992). First the functional equations are defined as ( ) ( ) ( )020121211 , HHHHHHf +−+= (3-32) ( ) ( ) ( )2211212 , bcbc FFFFHHf −−−= (3-33) Chapter 3- Development of a representative unit cell: theory and calibration 38 The essential part of the multivariable Newton-Raphson method is assembling the Jacobean matrix for which partial derivatives of the functional equations are taken with respect to the yarn height as shown in equation (3-34). [ ] ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = 2 2 1 2 2 1 1 1 H f H f H f H f J (3-34) where 1 1 H f ∂ ∂ = 2 1 H f ∂ ∂ =1 (3-35) and 1 1 1 1 1 2 H F H F H f bc ∂ ∂ − ∂ ∂ = ∂ ∂ (3-36) and 2 2 2 2 2 2 H F H F H f bc ∂ ∂ + ∂ ∂ −= ∂ ∂ (3-37) The derivative of the contact force with respect to the yarn height is given as: ( ) ( ) + +− = ∂ ∂ 2 322 0 2 0 0 12 ii ii i i i i i ci yH LH L L L AE H F ε (3-38) The derivative of the yarn bending force with respect to the yarn height is given as: ( ) ( ) + −= ∂ ∂ 2 322 2 0 0 2 8 ii i i i i i i bi yH H L H y EI H F (3-39) With the elements of the Jacobean matrix known, the iterative procedure can be performed as shown below. Chapter 3- Development of a representative unit cell: theory and calibration 39 [ ] − = 2 1 2 1 2 1 f f J H H H H guess guess new new if k=0 i guess i HH 0= else 1− = k i guess i HH (3-40) The amount of iterations can be reduced between strain increments if the values of Hi from the previous step are used as the guess for current step. 3.5 Characterization of constituent properties To develop the stress-strain behavior of the fabric using the unit cell method described thus far, a series of experiments were performed to characterize the constituent properties. The fabrics investigated for this study are constructed of high tenacity nylon 6,6 multifilament yarns with a linear density of 350dtex which was provided by TRW Automotive. One system has a single side silicone coating which does not penetrate through the full thickness of the fabric while the other system remains uncoated. Both fabric systems have identical geometrical features (i.e. yarns/in, % crimp in warp and weft directions). The areal density of the coated fabric is 218.93 g/m2 and 171.56 g/m2 for the uncoated fabric as determined by ASTM D3776-95. 3.5.1 Microscopy Due to the dense weave and small yarn width, it is difficult to measure features of the fabric visually; therefore the use of optical microscopy was warranted. To obtain the number of yarns per inch in the warp and fill directions, micrographs were taken in plan view with respect to the fabric as shown in Figure 3-6a for a coated sample and Figure 3-6b for an uncoated sample. Images were taken randomly throughout the fabric samples to collect a global set of data. Chapter 3- Development of a representative unit cell: theory and calibration 40 (a) (b) Figure 3-6: Plain weave of a) single-coated fabric and b) uncoated fabric The other geometric parameter of interest is the amount of crimp in each yarn. Using microscopy, the fabric was set in epoxy so that the edge of the fabric was perpendicular to the mold face. Upon hardening, the sample is sanded and polished until the fabric edge is exposed. Images are taken of the sample and then using the software package ImageJ (Rasband, 1997- 2009) the yarn length and spacing are measured to calculate crimp according to equation (3-3). Figure 3-7 shows images of the cross section of the fabric in the warp and fill directions for coated and uncoated specimens, respectively. Additional parameters obtained from the sectioned samples were the filament diameter and count for both warp and fill directions. Chapter 3- Development of a representative unit cell: theory and calibration 41 (a) (b) Figure 3-7: Cross section of 350dtex coated airbag fabric a) warp direction and b) fill direction (a) (b) Figure 3-8: Cross section of 350dtex uncoated airbag fabric a) warp direction and b) fill direction Chapter 3- Development of a representative unit cell: theory and calibration 42 Table 3-3 summarizes the geometric parameters for both the coated and uncoated systems as measured from the micrographs. From these parameters, the geometry of the unit cell and some constitutive properties can be determined. Table 3-3: Geometric properties of coated and uncoated airbag fabric as determined from microscopy Coated Uncoated Warp Fill Warp Fill Yarns/in 60.43 55.27 60.87 55.09 % Crimp 6.30% 8.90% 6.28% 8.79% #Filaments/yarn 140 140 140 140 Filament Diameter (µm) 16.95 16.95 16.95 16.95 Coating Thickness (mm) 0.082 0.082 N/A N/A Overall, the Table 3-3 confirms that the coated and uncoated fabric samples used in the study have identical geometric construction. From the figures, the plain weave geometry can be observed and it can also be seen that the yarns have virtually no twist and are composed of many continuous filaments. The silicone coating is very thin and does not penetrate through the entire thickness of the fabric. 3.5.2 Yarn extension test Besides geometric structure, the mechanical properties of the yarn are essential inputs to a potential model of the fabric. To evaluate the mechanical properties of the yarns, namely failure strain and force-elongation behavior, extension tests were performed. The magnitude of crimp can also be quantified using this method and compared to the values obtained through microscopy by recording the percent elongation at which the yarn becomes completely straight. Warp and fill yarns were carefully extracted from the uncoated fabric sample by gently pulling away neighboring yarns in the weave structure. Yarns were sampled from different locations of Chapter 3- Development of a representative unit cell: theory and calibration 43 the fabric roll to generate a global population and to avoid any possible localized effects. The sample population was 10 yarns in each of the warp and fill direction. A length of 150 mm (3 in) was marked off using a felt tip marker on the fabric and then the yarns were withdrawn as shown in Figure 3-9. This procedure of marking the length on the fabric as opposed to measuring on an extracted yarn ensures that the gauge length is based off of the yarn in its crimped condition as seen in the fabric structure. A condition that is important to keep true if one wants to back calculate the percent crimp from the extension test. Figure 3-9: Yarn specimen preparation procedure It should be noted that yarns were not taken from the coated fabric due to the degree of difficulty of peeling away the coating layer to get the yarns without causing significant damage to the yarns. It was also found that extracting lengths greater than 25 mm was near impossible due to the hindrance of the coating. Extension tests were attempted on a few samples that were suitable for testing, however the reproducibility of the tests was not acceptable so they are not included in the study. The other potential test considered was testing a coated yarn, but since the coating does not penetrate the thickness of the fabric combined with the plain weave structure, the yarn is not continuously coated for lengths required for testing. The yarns were mounted onto paper frames as seen in Figure 3-9c using epoxy. The paper frame ensures the proper gauge length distance between grips preventing additional slack or causing a Chapter 3- Development of a representative unit cell: theory and calibration 44 pre-stress in the yarn. Additional slack or smaller gauge length than the yarn extracted can cause erroneous results in the magnitude of crimp if one wanted to subtract the uncrimping stiffening from the load-elongation curve, a procedure that will be discussed later on. A larger gauge length can cause the yarn to stretch, causing a pre-stress in the yarn as well as providing incorrect results of the magnitude of crimp. A Kato Tech KES-G1 microtensile tester (Figure 3-10) with a 5kg load cell located at the Advanced Fibrous Materials Laboratory at The University of British Columbia was used to obtain force-displacement information of the yarn up to failure. Upon loading the sample into the grips, the paper frame is cut before extension is applied. The elongation rate was 2mm/min at ambient conditions according to ASTM D 3883-04 (ASTM D3883-04, 2008). Controller Load Cell Sample Data Acquisition Figure 3-10: KES-G1 microtensile tester with loaded yarn sample As the yarn is extended, the crimp in the yarn begins to straighten before the yarn undergoes stretching, a process commonly referred to as uncrimping. Obviously, this uncrimping of the yarn is a form of geometric stiffening that needs to be removed in order to obtain the “pure” mechanical response of the yarn. Figure 3-11 illustrates the technique of determining the Chapter 3- Development of a representative unit cell: theory and calibration 45 geometric stiffening region of the force-elongation curve as specified by Option C of ASTM D 3883-04 (ASTM D3883-04, 2008). The straight line portion of force elongation curve is extrapolated by line AB. The point A represents the magnitude of the crimp in the yarn and the elongation where the crimp is fully removed from the yarn. The Point C is obtained by constructing a line parallel to the Force axis from Point A. Point C corresponds to the tensile force required to remove crimp without stretching the yarn. The curve is graphically shifted so that Point C becomes the origin, therefore obtaining the pure force-elongation behavior of the yarn as seen in Figure 3-12. Figure 3-11: Procedure of obtaining "pure" yarn load-elongation response Chapter 3- Development of a representative unit cell: theory and calibration 46 0.00 5.00 10.00 15.00 20.00 25.00 0.00 0.05 0.10 0.15 0.20 0.25 strain Fo rc e (N ) Warp Fill Figure 3-12: Average force-elongation curve for 350dtex nylon 6,6 airbag yarn Figure 3-12 shows the average force-elongation curves. Overall, the tests were reproducible which suggests the yarn responses are uniform throughout the fabric and there are no regional effects. All yarns failed within the middle of the specimen, away from the grips. The average failure strains were 21.35% and 20.81% for the warp and fill yarn, respectively. The average failure load was 21.12 N and 20.18 N. The average crimp for the warp and fill yarns obtained from the procedure specified by ASTM D 3883 was found to be 6.33% and 8.74%, respectively. It can be seen from Figure 3-12 that the yarn exhibits a hyperelastic-like behavior which includes a high initial modulus, a transition period where the stiffness decreases and then as undergoes higher stiffening at higher strains until failure. This behavior arises from the semicrystalline structure of nylon polymer fibers. As a tensile load is applied to the yarns, crystalline block segments separate from the lamellae in the polymer and both block and tie chains become oriented in the direction of the tensile axis which results in stiffening of the polymer. Chapter 3- Development of a representative unit cell: theory and calibration 47 One may argue that it may be more beneficial to test non-crimp yarns (i.e. yarns that have not been processed into fabric) to reduce the geometric stiffening effect from decrimping. While this would reduce the amount of work associated with sample preparation and post-processing of data, this approach would not capture any additional drawing that the polymeric yarns may go under during the warping process. From Figure 3-12, it can be seen that the warp yarn is stiffer and fails at a higher load. Considering the warp and fill yarns are identical in terms of linear density and filament count, additional drawing of the warp yarn from manufacturing is a plausible explanation for the difference in mechanical behavior of the two yarn directions. In the end, the obtained response can be considered ex situ as we have obtained the response of the yarn from a processed fabric. Of course there may be some behaviors, namely failure stress and strain, which are affected by the in situ environment of the fabric. Pan (Pan, 1996) and Shahpurwala (Shahpurwala & Schwartz, 1989) have discussed some of the in situ variables that can impact yarn such as pressure-independent adhesion, a frictional component dependent on the confinement pressure of the yarn and statistical distribution of fiber strength. The values of crimp for the warp and fill yarns found using the extension tests are close to the values obtained using the sectioning method. The extension test advantages in testing crimp that it can be used to test a large sample population of yarns with relative ease regarding sample preparation, testing and processing. However, the method is sensitive to keeping the proper gauge length. The sectioning method can be time consuming for large sample populations considering the effort of cutting samples, casting them in epoxy, polishing and imaging. However, the images obtained can be used to measure crimp with a high level of assurance assuming the fabric sample does not undergo dimensional changes from handling or the epoxy environment. 3.5.3 Coating characterization The silicone coating used on the fabric was not available for mechanical testing, although airbag grade silicone mechanical properties are readily available in the literature (Crouch, 1994; Schwark & Muller, 1996). The modulus of elasticity of the coating system is 2.75 MPa (Schwark & Muller, 1996) and Poisson’s ratio is assumed to be 0.35. The thickness of the Chapter 3- Development of a representative unit cell: theory and calibration 48 coating was found to be approximately 0.082 mm based on equation (3-6) and was also confirmed using microscopy. 3.5.4 In-plane shear calibration The properties of the fabric were evaluated in a previous study (Ko, December 2006) using a KES-FB1 shear tester at Philadelphia College of Textiles. A fabric sample is held by two parallel chucks on both edges of the fabric as shown in Figure 3-13a. The one edge moves parallel to the fixed grip which applies a shear force to the fabric. A constant normal force is applied to prevent buckling or wrinkling of the specimen. Figure 3-13b better illustrates the boundary conditions of the test. The test was performed only on coated fabric but as previously discussed, the low level shear behavior of the uncoated fabric was observed to be the same as the coated fabric after the influence of the coating is overcame. Figure 3-14 is a reproduction of the results in the study and it can be seen that the tests had a high reproducibility. For characterization of the shear relationship described in equation (3-31), the loading portion of the curve is used. λ1 ,Λ1 Λ2λ2γ Fs W2 (a) (b) Figure 3-13: KES-FB1 textile shear tester shown in a) and b) shear deformation adopted in the KES-FB1 testing system Chapter 3- Development of a representative unit cell: theory and calibration 49 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 0.05 0.1 0.15 0.2 Shear Strain (radians) M em br an e Sh ea r St re ss (N /m m ) Loa ding Unl oa din g M em br an e Sh ea r St re ss (N /m m ) Figure 3-14: Shear stress-strain behavior for 350dtex nylon airbag fabric Since the fabric undergoes a shear deformation under a constant load, the fabric sample is subject to both shear and tensile forces. The model depends on the pure shear response of the fabric so the tensile response measured in the fabric needs to be removed. The extension of the unit cell can be used considering the boundary conditions that arises from KESF testing system as shown in is as follows. 111 =Λ=λ (3-41) γ λ cos 2 2 Λ = (3-42) where λi is the yarn stretch, Λi is the orthogonal stretch with respect to the initial fabric configuration, and γ is the shear angle as shown in Figure 3-13b. Now that the stretches applied to the unit are expressed as a function shear strain, Kawabata demonstrated that the pure membrane shear stress-strain behavior that arises from yarn rotation, Chapter 3- Development of a representative unit cell: theory and calibration 50 S12, can be obtained by removing the tensile forces that contribute to the recorded shearing force, Fs as shown below (Kawabata, Niwa, & Kawai, 1973c; Kawabata, 1989): ( )γ γ tan cos 2 2 12 WF nS S −= (3-43) where n2 is the fill yarns per inch, W2 is the constant normal load and Fs is the applied shear load. The drawback to using the KES-FB1 for calibrating shear is that it can only give information about low level shear behavior – in the current shear model the tests can only help find G1 and G2. At high shear stresses, the shear behavior is no longer governed by yarn rotation rather than the shear stiffness of the yarn. Therefore, the jamming stiffness of the fabric needs to be defined. Here it is proposed that the jamming shear stiffness of the yarn is related to the shear stiffness of the fiber material knocked down by the fiber packing factor of the yarn: ( ) p EG υ+ = 123 (3-44) where E is longitudinal tensile modulus of the yarn, ν is the Poisson’s ratio of the yarn and p is the packing factor of the yarn. 3.6 Results The simple unit cell model was run to evaluate the unit cell model for the cases of uniaxial stress applied to the warp and fill directions and biaxial state of stress. Figure 3-15 shows the model generated membrane stress-strain curves under equal biaxial and uniaxial state of stress for the warp and fill directions. Chapter 3- Development of a representative unit cell: theory and calibration 51 0.00 10.00 20.00 30.00 40.00 50.00 60.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 strain m e m br a n e s tr e s s (N /m m ) Model-Warp Model-Fill 1:1 m e m br a n e s tr e s s (N /m m ) (a) 0.00 10.00 20.00 30.00 40.00 50.00 60.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 strain m e m br a n e s tr es s (N //m m ) Model-Warp Model-Fill m e m br a n e s tr es s (N //m m ) (b) Figure 3-15: Membrane stress-strain curves for 350dtex fabric produced by the unit cell model 3.7 Sensitivity analysis A sensitivity analysis was performed to investigate the influence of the essential yarn input parameters. For clarity of presentation, only the warp stress is shown but the same trends witnessed in the warp direction are applicable to the fill direction. For the sensitivity study, the nominal input values are those obtained through characterization and therefore are identical to Figure 3-15. Figure 3-16 shows the sensitivity of the unit cell model to yarn crimp for equal biaxial extension and uniaxial load. Under uniaxial stress, increasing the yarn crimp increases the failure strain of the fabric. The increase in crimp requires more elongation to uncrimp or geometrically deform the yarn before material straining can occur. Decreasing crimp eliminates the amount of geometric deformation and therefore requires less elongation to reach the fabric’s failure point. However, the change in crimp does not change the stiffness or failure load under uniaxial Chapter 3- Development of a representative unit cell: theory and calibration 52 extension. For biaxial extension, the trend of increasing failure strain with crimp can be seen but there is also a decrease in stiffness and the ultimate stress. 0.00 10.00 20.00 30.00 40.00 50.00 60.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 strain w a rp - m em br a n e s tr es s (N /m m ) +25%Cr +10%Cr Nominal -10%Cr +25%Cr 0.00 10.00 20.00 30.00 40.00 50.00 60.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 strain w ar p - m em br an e st re ss (N /m m ) +25%Cr +10%Cr Nominal -10%Cr -25%Cr +Crimp -Crimp +Crimp -Crimp 1:1 w a rp - m em br a n e s tr es s (N /m m ) w ar p - m em br an e st re ss (N /m m ) (a) (b) Figure 3-16: Crimp parameter sensitivity for a) 1:1 biaxial extension and b) uniaxial extension Figure 3-17 shows the sensitivity to the bending rigidity of the yarn. If the yarn is considered to be perfectly flexible, the unit cell is not able to capture the stiffening effect as the fabric uncrimps. The yarn simply elongates until the crimp is removed and then forces are generated as the straightened yarn elongates. Outside the low level stress behavior during the uniaxial extension, variations up to two times the nominal value of the bending rigidity have little effect on the mechanical behavior of the fabric regardless of the stress state. Chapter 3- Development of a representative unit cell: theory and calibration 53 0.00 10.00 20.00 30.00 40.00 50.00 60.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 strain w a rp - m e m br an e s tr e s s (N /m m ) No Bending Nominal 2x EI 10X EI 0.00 10.00 20.00 30.00 40.00 50.00 60.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 strain w a rp - m e m br a n e s tr e s s (N /m m ) No Bending Nominal 2x EI 10X EI -EI +EI EI=0 1:1 -EI +EI w a rp - m e m br an e s tr e s s (N /m m ) w a rp - m e m br a n e s tr e s s (N /m m ) (a) (b) Figure 3-17: Yarn bending rigidity parameter sensitivity for a) 1:1 biaxial extension and b) uniaxial extension Figure 3-18 shows the sensitivity to the thickness of the coating in the unit cell model. There is virtually no effect on the extensional behavior between an uncoated fabric, the nominal values and coating that penetrates fully through the thickness of the fabric. Intuitively, one would expect this result since the stiffness of the coating is much less than that of the fabric. Chapter 3- Development of a representative unit cell: theory and calibration 54 0.00 10.00 20.00 30.00 40.00 50.00 60.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 strain w a rp - m e m br a n e s tr e s s (N /m m ) No Coating Nominal Full Coating 0.00 10.00 20.00 30.00 40.00 50.00 60.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 strain w a rp - m e m br a n e s tr e s s (N /m m ) No Coating Nominal Full Coating 1:1 w a rp - m e m br a n e s tr e s s (N /m m ) w a rp - m e m br a n e s tr e s s (N /m m ) (a) (b) Figure 3-18: Coating thickness parameter sensitivity for a) 1:1 biaxial extension and b) uniaxial extension 3.8 Summary A unit cell model based on a simple linear geometry was proposed and the governing deformational mechanisms were derived. The characterization techniques to obtain inputs for the model were carefully outlined and the geometric and mechanical properties of a 350dtex airbag fabric are obtained. The results of the unit cell model represent the behavior of an ideal fabric where deformation is uniform throughout. The next step would logically be to perform a validation study comparing the results of the simple unit cell model to experimental results. However, stress heterogeneity is seen in many of the standard tests to evaluate the in-plane behavior of the fabric. Stress heterogeneity arises due to specimen geometry, yarn orientation as well as the sample’s boundary conditions. This Chapter 3- Development of a representative unit cell: theory and calibration 55 warrants further computation to accurately model the physical conditions of certain tests. Additionally, besides simulating physical experimental specimens, we would like to use mechanical response of the developed unit cell as the basis for structural analysis of the airbag in the crash simulation. More specifically, the technique developed needs to be formulated in a way that it can be of use with the current design tools. Therefore in the next chapter, a finite element continuum approach using the unit cell as the constitutive behavior is presented that can be utilized in modeling stress heterogeneity in test samples. Chapter 4– Implementation and verification 56 Chapter 4 – Implementation and verification 4.1 Background and approach As established to this point, the airbag contributes significantly to the overall crashworthiness of an automobile. The analysis and simulation of vehicles for crashworthiness is still fairly young field and has evolved greatly the past 30 years (Khalil & Du Bois, 2004). Starting as simple lumped parameter model, today models are based on 3D continuum mechanics with intricate and complex geometry that often require the use of supercomputers for simulation. In the past decade, the simulation of the airbag and occupant/airbag interaction has become a standard option in many commercial simulation codes (Dörnhoff et al., 2008; Hirth et al., 2007). The high costs conducting and difficulty measuring the airbag deployment and impact makes simulations desirable to evaluate a large number of parameters (Khan & Moatamedi, 2008). The airbag deployment and impact, like most crashworthiness analysis of vehicles, involves highly nonlinear structural mechanics due to large deformations, rotations and material nonlinearity. Also considering the equations of state to model gas flow from the inflator, the governing equations pose a great challenge in the analysis, warranting the use of a numerical method like the finite element method. The finite element method of structural dynamics solves the set of nonlinear partial differential equations of motion for body or structure in a space-time domain along with the material stress strain relationship within defined boundary conditions. The body or structure is descritized into a finite number of components (elements) that are connected by points (most commonly referred as nodes). The temporal and spatial discretization results in a solution of the equations in the forms of the values of the field parameters at common nodes of the elements. Within the domain of an element, the values of all the parameters are approximated through the use of the Chapter 4– Implementation and verification 57 appropriate shape functions. The state of stress and strain in the material corresponds to the minimum energy of the system which is an approximate representation of the overall equilibrium of the system. The finite element simulation of airbags is based on the assumption that the fabric can be approximated using membrane shell elements with an orthotropic, linear stress-strain material relationship (Bedewi, Marzougui, & Motevalli, 1996; Hirth et al., 2007; Khan & Moatamedi, 2008). The membrane assumption is valid since the out-of-plane mechanical properties of the fabric are very low compared to the in-plane properties. However, the orthotropic material assumption as stated numerous times before, is incorrect as the material is highly anisotropic as the yarns can rotate to a non-orthotropic state. The current state-of-the-art fabric constitutive model in LS-DYNA (*MAT_FABRIC) utilizes a nonlinear orthotropic stress-strain formulation with non-orthogonal material axes (Hallquist, 2006). However, to calibrate the inputs for the model requires cruciform biaxial test data for extensional responses and picture frame shear tests for the shear response. This necessitates special equipment and multiple tests for different shaped airbags which can be time consuming particularly when evaluating novel fabric systems. Additionally, at stress ratios that deviate from the tested configuration, the constitutive behavior may not be correct due to rearrangement of the fabric’s internal structure. The proposed alternative is to incorporate the unit cell model described in the previous chapter as the basis for the constitutive behavior for shell-membrane elements. Figure 4-1 illustrates the basic concept of the proposed implementation of the unit cell model as the basis for representing the in-plane mechanical behavior of the fabric. The fabric structure is replaced by a continuum of a homogeneous anisotropic material. Stretches and rotations are applied to the continuum membrane which are identical to that occurring in the unit cell. One key aspect of the unit cell- continuum formulation is to translate the continuum strains which are taken with to respect the element coordinate system and compute the strains that occur in the fabric with respect to the yarn configuration which may not coincide with the element coordinate system upon deformation. Once the strains are transformed to the yarn orientation, the forces from the unit cell can be computed. Then, the forces are normalized and translated back into the element local coordinate system. Chapter 4– Implementation and verification 58 X3 X2X1 H01 H02y01 y02 Coating Original Unloaded Structure n2 n3 n4 n1 q10 q20 x y= = LS-DYNA applies strain to element UMAT finds “yarn strains” from element coordinate system UMAT transforms unit cell stress back to element coordinate system Deformed Loaded Structure UMAT applies yarn strain to unit cell algorithm x y n1 n2 n3 n4 Unit cell forces and normalized stress are calculated x y n1 n2 n4 n3 q20 q10 q2 q1 Figure 4-1: Outline of the basic concept of the unit-cell-membrane Therefore, the outlined approach can determine the fabric state of stress with consideration of the internal changes the yarns undergo without explicit modeling of each yarn. This approach is therefore very computationally efficient compared to other techniques that attempt to simulate the mechanical response and contact behavior of every yarn within the fabric. The increase in efficiency can lead to faster solution times or allow more resources to be dedicated to more computationally intensive processes in the airbag simulation. Additionally, the history of the evolution of certain fabric parameters such as the yarn tension, yarn failure, contact forces and yarn orientation can be tracked for each element for the duration of the simulation. While there are many advantages using this method, the assumptions made in the development can impose limits on the use of the model. For instance, the continuum assumption is only valid if the length scale of the element is larger than the yarns and empty spaces within the physical fabric. More specifically, the element dimensions cannot be smaller than the dimensions of a single crossover. This means more dense fabrics can use smaller elements in the simulation Chapter 4– Implementation and verification 59 while still retaining the physical conditions. Since the airbag fabric is a very dense weave and the evolution of new airbag fabrics are increasingly becoming denser, this approach has plenty warrant in the structural analysis of the airbag deployment. If this method was applied to small structures made of coarse woven fabrics, the limiting element size may be larger than the required mesh size needed to obtain the correct results. The other main assumption that can limit the capabilities of the model is the no-slip condition between the yarns. During the failure of fabric structures, the yarns can slip and slide near the location of the failure or tear to accommodate the equilibrium within the fabric. This obviously changes the internal geometry of the fabric and the level of tension can vary throughout the yarn length along a single crossover. Under the no-slip assumption, yarns crossover if affined to the midpoint of both yarns and the yarn tension developed is constant through the length of the yarn within the unit cell. The final main assumption is that the unit cell remains in a ‘flat’ planar configuration during out- of-plane deformations. More clearly, the curvature of the structure is not reflected in the unit cell geometry. This effect of curvature is significant as it can strain the yarn as well as change the internal structure of the fabric as seen in Figure 4-2. This assumption is appropriate for structures with large radius of curvatures with fabrics that have small yarn spacing shown where the out-of-plane deformation is negligible. In the airbag application, the curvature of the structure is typically fairly large and the density of the fabric is very high. However, if this method were to be applied to small diameter tubes like those used in biomedical or petroleum applications, the effects of the curvature on the unit cell geometry maybe more pronounced and may affect the mechanical response of the fabric. Figure 4-2: Effect of curvature on the internal structure of woven fabric Chapter 4– Implementation and verification 60 4.2 Explicit dynamic finite element analysis The numerical testbed for this study is the commercial nonlinear dynamic finite element code, LS-DYNA. The code is used extensively in the automotive industry as it contains sophisticated contact algorithms for modeling impact problems in addition to specialized capabilities for modeling airbags, seatbelts and anthropomorphic test devices. Most importantly, LS-DYNA allows users to code their own subroutines defining material models, more commonly referred to as User Material Model (UMAT). Figure 4-2 shows the typical analysis process used by LS- DYNA with the UMAT option invoked. Update Velocities Update Displacements Apply Force Boundary Conditions Process Elements Process Contacts Update Accelerations User Material Model Start Figure 4-3: Numerical procedure of LS-DYNA with user material model option The user-defined subroutine (UMAT) allows for formulation of constitutive relationships for applications where the standard library of material definitions fail to accurately represent the physical material behavior. The UMAT is coded in FORTRAN and contains a common block to access the strain increments, material constants, element failure flags and load curves from the main program. At each time step, LS-DYNA passes the element strain increments evaluated at the shell element integration point to the subroutine. For this study, the integration points are located at the center of the element for the constant stress membrane. The UMAT computes the Chapter 4– Implementation and verification 61 stresses within the element and passes the values back to the main program to continue the analysis. 4.3 Implementation of unit cell based UMAT for FE shells 4.3.1 Continuum formulation The challenge in the continuum formulation of the constitutive behavior of fabric is that the configuration of the fabric is continuously changing. The first issue to be addressed is the measuring the kinematics of the changing configuration. From continuum mechanics, the fundamental measure of a deformed body at a point in time can be given by the deformation gradient which is defined for a two-dimensional body as ( ) ( )0 j i ij x tx tF ∂ ∂ = (4-1) The deformation gradient can be used to find the stretch and rotations that a body undergoes in two points in time. To do so, the right Cauchy-Green deformation tensor needs to be constructed as per: FFC T= (4-2) The Cauchy-Green deformation tensor is symmetric and positive-definite. The Green deformation tensor can now be used to find the stretch and rotation in each yarn. Within the constraints of the UMAT, the displacement gradient can be constructed at a time t based on the increments of strain that are passed by the main program ttt ∆εxx += −1 (4-3) where xt-1 is the displacement gradient of the previous time step and ∆∆ ∆∆ = t y t xy t xy t xt εε εε ∆ε (4-4) Chapter 4– Implementation and verification 62 where t x ε∆ tyε∆ t xyε∆ are the current strain increments in the local element coordinate system xyz. The local axes system used in LS-DYNA is based on the shell node numbering where the x-axis is defined parallel to the n1 - n2 edge as shown in Figure 4-1 (Hallquist, 2006). The z-axis is therefore taken to be normal to the element mid-plane and the y-axis is determined by the cross product: xzy ×= (4-5) The material deformation gradient can therefore be found using IxF += tt (4-6) where I is the identity matrix. To track the configuration of the material axis, two vectors q1 and q2 are assumed to be parallel to the original orientation of the warp and fill yarns respectively. Therefore, the vectors are assigned based on the angle of the yarn [ ]Tiiiq 000 sincos θθ= (4-7) where 0iθ is the initial angle of the yarn taken with respect to the local x axis. According to Bathe (Bathe, 1982), the stretch of each material yarn axis can be found using equation (4-8) 020 qq T i t i C=λ (4-8) Therefore, with the stretch in each yarn vector calculated, the displacement along each yarn within the unit cell at the current time step is determined below: Chapter 4– Implementation and verification 63 ( )10 −= tiiti yd λ (4-9) With the stretch of the warp and fill material yarn axis known along with the Cauchy-Green deformation tensors, the angle between the yarn axes at the current time step can be found using equation (4-10) (Bathe, 1982): 21 0201cos λλφ qqT C = (4-10) With the updated angle between the two yarn vectors known, the vector coordinates need to be updated. Assuming that a positive value of strain results in the yarn vectors becoming acute, the change in the yarn angle is determined by equation below ( ) 2 00 φθθθ −−=∆ wf (4-11) A positive change in yarn angle follows the rotation convention shown below for each yarn vector: θθθ ∆−= 011t θθθ ∆−= 022t (4-12) (4-13) Therefore the newly rotated yarn vectors for the warp and fill directions can be expressed according to equation (4-14): ( ) ( )[ ]Ttitiiq θθ sincos= (4-14) With the unit cell displacement and current angle between the yarns now known, the yarn strain can be found and the resulting forces can be computed using the procedure outlined in the previous chapter. The obtained stresses now must be transformed from the yarn coordinate system back into the local element coordinate system. Keeping with the tensor notation, the Chapter 4– Implementation and verification 64 local stress tensor can be computed back into the local element xyz system using the scalar magnitude of the internal yarn forces and the dyadic product of the unit yarn vectors at the current time step according to Peng and separately by King (King et al., 2005; Peng & Cao, 2005). ( ) ( ) ( )122112222111 qqqqh Sqq h Nqq h N ⊗+⊗+⊗+⊗=σ (4-15) where N1 is the magnitude of the force per unit length from the warp yarn end, N2 is the magnitude of the force per unit length from the fill yarn end, S12 is the magnitude of the shear force per unit length and h is the thickness of the shell element. Specifically, the components of the stress tensor within the element coordinate system can be shown to be: 21 12 2 22 1 21 coscos2coscos θθθθσ h S h N h N x ++= (4-16) 21 12 2 22 1 21 sinsin2sinsin θθθθσ h S h N h N y ++= (4-17) ( )122112222111 sincossincossincossincos θθθθθθθθτ +++= h S h N h N xy (4-18) This form differs from the work of Shahkarami (A. Shahkarami & Vaziri, 2007) as the shear component is included in the stress tensor rather than treating shear separately. Additionally, the treatment of the material axes is consistent with the procedures for finite deformation, non- orthogonal constitutive behavior in continuum mechanics (Bathe, 1982; Holzapfel, 2000). Within the explicit finite element framework, the determination of the critical time step is crucial for analysis. LS-DYNA requires the bulk and shear moduli of the material to be defined in order to determine the critical time step as well as calculating transmitting boundary, contact stiffness etc. Within the context of the proposed continuum model, the bulk and shear moduli are based on published values of the speed of sound in bulk material the yarn is composed of and the density of the fabric structure and can be computed using equations (4-19) and (4-20) respectively. Chapter 4– Implementation and verification 65 ( )ν ρ 213 *2 − = cK (4-19) ( )ν ρ + = 12 *2 cG (4-20) where ρ is density of the fiber material , ν is the Poisson’s ratio of the yarn material. The values of c* can be found using the expression: icr c c + = 1 * (4-21) where c is the speed of sound in the fiber material and cr is the crimp of the yarn. With respect to best practices, for unbalanced fabrics, the yarn crimp that results in the larger values of K and G governs. 4.3.2 Special considerations – yarn separation, failure and element erosion Within the user material model framework, the displacements in the warp and fill direction will always be given so that the yarn heights are unknown and the length of the yarns is a function of the height. This is different from the previous chapter regarding uniaxial extension of the fabric where the transverse displacement was unknown and was solved for assuming the transverse yarn did not compress. In that case, it was necessary to establish certain unknown unit cell conditions in order to generate a solution for uniaxial stress condition. In the user material model framework, the axially incompressible yarn condition is still established however the yarn spacing will always be determined from the continuum stretches. Therefore, under these imposed conditions, during large negative displacements the algorithm will not find a solution since continuity of the unit cell structure is broken. Here the yarns separate and are no longer in contact. In this case, the yarns are decoupled so that the deformed geometry and forces of each yarn can easily be computed. Chapter 4– Implementation and verification 66 An instantaneous failure criterion based on the ultimate yarn strain is used to initiate failure of the individual warp and fill yarns. Once the yarn surpasses the specified ultimate strain, the yarn is flagged by the subroutine and the yarn is permanently eliminated from the unit cell. When a yarn is removed from the unit cell, the shell element is no longer capable of carrying extensional load in that direction or any shear loading. When both of the unit cell yarns have exceeded the ultimate yarn strain, the shell element is then eroded from the finite element mesh. The erosion results in the element being removed from the calculation while retaining its mass properties on the nodes attached to the element. Nodes that become completely unconstrained due to the erosion of elements are than also eliminated from the computation. Under certain conditions that arise from single yarn failure, the elements can undergo an excessive and unrealistic deformation which promotes numerical instabilities. To remedy possible instabilities, a user-defined criterion was established in the user material model to erode unhealthy elements. One last consideration of the UMAT formulation is the need to refer to the local x axis of the element. This requires the 1-2 edge of each element to be parallel throughout the mesh to ensure the yarn orientation is consistent throughout the simulated part. Therefore, the current UMAT definition cannot be used in arbitrarily meshed structures. In the driver side airbag structure, the bag is constructed of two circular pieces of fabric so the meshing within the constraints of the UMAT is feasible for the majority of the bag. Possible problematic areas are near edges and vent hole. Future work will need to address this limitation. 4.4 Verification Code verification checks that the algorithms are working properly and that the solution of the mathematical model is accurate. The addition of the unit cell algorithm with the continuum formulation along with interactions between the subroutine and main program presents several nuances that can cause problems. To verify the successful implementation of the code a series of simple tests were performed. The first of these test simply check that the extensional behavior of a single element to the simple stand-alone computer algorithm discussed in the previous chapter. Prescribed displacements are applied to the element to confirm the biaxial and uniaxial response of the implemented unit cell agree with the results from the simple model developed in the previous chapter. The second set of simple tests consisted of applying displacements to a square Chapter 4– Implementation and verification 67 patch of 5x5 elements to investigate the elements whose displacements are controlled by the connectivity of the neighboring elements. This provides us with an opportunity to find arbitrary displacement ratios where the algorithm has difficulties converging. The final set of verification studies investigated the rotation-nonorthogonal behavior of the yarn vectors as illustrated in Figure 4-4 and Figure 4-5. Three single element tests were performed: 1) pure shear, 2) rail shear and 3) a balanced trellis like shear varying the shear stiffness of the yarn crossover. Under pure shear, one would expect that the only deformation that arises is shear, therefore no stretch is generated in the yarns. From Figure 4-4a, the model correctly exhibits the proper pure shear behavior. Under rail shear test there is positive stretch generated in direction of the deformed edge in addition to shear stresses generated. In Figure 4-4b, the element is sheared so that the deformed edge from shear is parallel to the fill yarns. Again, the calculated response of the model mimics the expected behavior. 1.00 1.05 1.10 1.15 0.00 0.10 0.20 0.30 0.40 shear strain Ya rn St re tc h Warp Fill Ya rn St re tc h 1.00 1.05 1.10 1.15 0.00 0.10 0.20 0.30 0.40 shear strain Ya rn St re tc h Warp Fill Ya rn St re tc h (a) (b) Figure 4-4: Verifying modes of shear: a) pure shear and b) rail shear Finally, in Figure 4-5, an element with balanced yarn properties (meaning that the properties for each yarn direction are identical) has two opposing corners with assigned displacements while the remaining two corners are free. The purpose of this example is to examine the impact of shear stiffness on the non-predefined rotation of the yarn. Assigning no shear stiffness or resistance at the crossover, the crossover should behave like a pin. Thus, the crossover should Chapter 4– Implementation and verification 68 act like a trellis where any deformation is attributed to rigid body rotation of the yarns. It can be seen from Figure 4-5 that shear strain of the crossover results in no stretch of the yarns. If very high or infinite shear stiffness is assigned, the crossovers should act like they are fixed and remain in the orthogonal state. Therefore, any deformation of the element should be attributed to stretch as seen in the calculations. Finally, an assigned realistic value of shear stiffness is applied and it can be observed that the element experiences a mix of rotation and extension. Overall, this confirms that the model matches deformational behaviors that one would expect under a multitude of conditions. 1.00 1.05 1.10 1.15 1.20 1.25 1.30 0.00 0.10 0.20 0.30 0.40 shear strain Ya rn St re tc h Infinite Shear Stiffness No Shear Stiffness Realistic Shear Stiffness Ya rn St re tc h Figure 4-5: Verifying shear deformation behaviors of a single element with varying shear stiffnesses 4.5 Summary The structural analysis of airbag and fabric structures requires a constitutive model that can consider the complex and often heterogeneous deformations of the fabric structure, but at the same time, the model needs to be simple enough to keep computational costs low. Here, the mechanistic unit cell model is implemented as the material behavior for an efficient membrane element in the finite element code LS-DYNA. Using the developed user material model, a wide variety of fabric deformational behaviors can be obtained without explicitly modeling each yarn. Chapter 4– Implementation and verification 69 The user material model has been carefully verified to ensure the in-plane extensional and shear behaviors are computed correctly and are passed to LS-DYNA without any error. Now that the unit cell model has been successfully implemented in terms of computational correctness, the next step is to check the model for physical correctness. In the next chapter, a validation study is performed demonstrating the models accuracy under a variety of deformations. Chapter 5– Validation 70 Chapter 5 – Validation 5.1 Background and approach While the verification and sensitivity of the model has been performed, neither gauges the competence of the model for representing the physical system. Validation of a computational model is the process of determining the degree to which a model is accurate representation of the real world from the perspective of the intended uses of the model (Schwer, 2002). The intent of this chapter is to confirm correctness of the models quasi-static in-plane mechanical properties to experimental data. Since published, publically available experimental data on the mechanical properties of airbag fabric is very limited, especially with regards to the current state-of-art fabrics, a series of experiments were performed to rigorously confirm the results of the models. Of the experiments performed, the biaxial extension of the fabric is believed to be the closest to mimicking the conditions that occur in the airbag. During the simulation of the quasi-static tests using the user material model within the dynamic explicit finite element framework, the total kinetic energy was kept to a minimum so that any influence from inertial effects is negligible. 5.2 Experimental evaluation 5.2.1 Uniaxial extension The uniaxial extension of fabric is often performed due to its simplicity and can be used to compare the stiffness of the warp and fill directions. The uniaxial tensile properties of the coated and uncoated fabrics were measured according to the modified strip tensile test method as specified by ASTM D5035-95. Five samples taken in each of the warp and fill directions for both coated and uncoated fabrics were tested. The ends of the fabric were wrapped with layers of masking tape to protect that fabric from crushing at the grips. The specimen width was 25 Chapter 5– Validation 71 mm with a gauge length of 100mm. An Instron 1122 testing machine located in the Department of Zoology at The University of British Columbia was used for testing. The crosshead speed was 100 mm/min. The stress-strain curves in the warp and fill directions for the coated and uncoated samples are shown Figure 5-1. The stress is normalized based on a membrane assumption, simply corresponding to the force divided by unit length. The vertical error bar show the maximum and minimum stress values at selected data points. The horizontal error bar at the failure point illustrates the minimum and maximum failure strains recorded. Failure occurred within the middle of the gauge length indicating uniform stress distribution. Overall the tests were very reproducible. The sample population for the uncoated sample was reduced to four since one of the specimens had broken near the grip. Figure 5-1: Average uniaxial stress-strain of 350dtex fabric From Figure 5-1, it can be seen that there are no discernable difference between the deformational behavior of coated and uncoated samples as observed in the biaxial extension 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 strain m em br an e st re ss (N //m m ) Exp-Coated-warp Exp-Coated-fill Exp-uncoated-warp Exp-uncoated-fill Chapter 5– Validation 72 tests. The stress-strain behavior is very different from the biaxial behavior of the fabric. The uniaxial response is characterized by a sigmoidal shape which consists of a low initial modulus followed by gradual stiffening into high linear modulus. Compared to the biaxial results, fabrics under uniaxial loading have higher failure stresses but lower apparent modulus. 5.2.2 Biaxial extension As discussed previously, the biaxial loading is the fundamental loading condition seen in an airbag. To evaluate the biaxial behavior, there are three methods discussed in the literature: burst test, inflated cylinder test, and cruciform plane biaxial extension test (Bassett, Postle, & Pan, 1999). These methods are better illustrated in Table 5-1 showing the original fabric configuration and the deformed fabric during operation. In the burst test, stress and strains are quantified based on the internal pressure within the fabric and the fabric shape and requires impermeable fabrics. In the inflated cylindrical test, the results can be affected by the presence of seams unless circularly woven samples are tested and also requires fabrics that are impermeable. The cruciform biaxial extension test can evaluate both permeable and impermeable fabrics at a wide range of stress ratios and can directly measure stresses and strains. For this study, biaxial response of the fabric is evaluated using the cruciform biaxial extension method. Chapter 5– Validation 73 Table 5-1: Test methods in the literature for evaluating the biaxial behavior of fabrics Cruciform Inflated Cylinder Burst DeformedOriginal Pressure Source Sample (Flat) Pressure Source Sample (Cylindrical) Sample (Flat) Test Method Sample (Spherical) Biaxial extension experiments of coated and uncoated fabric samples were performed using a custom-made large load capacity biaxial tester (see Figure 5-2) located at the Hess Research Laboratories at Drexel University. The tester was designed with fibrous materials in mind, more specifically geotextiles, but is suitable for a wide range of textiles and composites that require large applied loads (Wartman, Harmanos, & Ibanez, 2005). The tester consists of a servo- hydraulic test frame with two pairs of grip carriages oriented 180 degrees apart that move in equal and opposite directions. The carriages are driven by a 67 kN hydraulic actuator powered by a hydraulic manifold with high-pressure and low-pressure accumulators. Each actuator drives the grip carriages via connection arms. Each grip carriage has a dynamic range of motion of ±10.2 cm and peak velocity of 30 cm/s. A 2.0 cm threaded rod extends from each carriage to be used as the grip attachment point. The grip widths are approximately 20.23 cm (8 inches) wide and have pyramid-teethed attachment plates to provide sufficient grip. A computer system consisting of a digital controller running ANCO Engineer’s ANIPC-400 software controls the Chapter 5– Validation 74 tester through a 4 channel data acquisition system. A 32 channel analog-to-digital converter is used to obtain the data. Figure 5-2: Biaxial tester at Drexel University Specimens were cut into a cruciform shape as shown in Figure 5-3 leaving a 20.32 cm x 20.32 cm (8” x 8”) test area. Great care was taken while cutting the sample to ensure the fabric remained orthogonal and true to the desired dimensions in order to avoid unwanted bias. Masking tape placed in layers was used as a grip tab to protect the fabric from being damaged by the grips. A sample size of 5 coated and 5 uncoated samples were prepared. Chapter 5– Validation 75 Figure 5-3: Biaxial specimen dimensions The alignment and calibration of the loading actuators was first performed. Using a straight edge and a laser level, each extender arm from the actuator was measured and adjusted to ensure each grip carriage and arm was equidistant. Upon confirmation of the actuator extensions calibration, the grip assemblies were attached to each arm and were checked to ensure that the grip slots were leveled. An overhead digital camera was placed over the sample to qualitatively measure fabric displacements within the gauge length in addition to the recorded cross-head displacements. An extension rate of 5mm/sec (2.5mm/sec per actuator) and a sample rate of 50Hz were inputted into the digital controller. On a separate digital controller, the sampling rate of the overhead digital camera was defaulted at 2Hz. Samples were loaded into the grip assemblies with the 5.08cm grip tab fully inserted into the grip slot and secured. The orientation of the fabric (warp or fill direction) with respect to the load cells (A,B,C,D) was recorded. To ensure there was no bias or unbalance in the load cells, samples were rotated, as demonstrated in Figure 5-4 into different configurations so that the load cells had the opportunity to measure warp and fill Chapter 5– Validation 76 directions during the duration of testing. Results from using this procedure show that there was no noticeable asymmetry between the load cells. Figure 5-4: Biaxial testing specimen configuration A pretension of 44.5 N (10 lbs) was introduced to each actuator which is equivalent to 2.2 N/cm (1.25 lbs/in) of stress imposed on the sample. The pretension is an ad-hoc procedure to eliminate excessive experimental noise obtained from the load cells early on in the testing sequence. The samples were loaded until final fracture. Upon completion of the test, the failed sample was removed and the alignments of the actuators were checked after the actuators had returned to their original positions. A new sample was placed into the grip fixtures and the loading procedure was repeated. The load data recorded had some inherent noise which was eliminated using a moving average filter. The average membrane stress-strain curve is shown in Figure 5-5. The stress is normalized based on a membrane assumption, simply force divided by unit length. The vertical error bars show the maximum and minimum stress values at selected data points. The horizontal error bars at the failure point illustrates the minimum and maximum failure strains recorded. Overall, the biaxial response for the coated and uncoated samples was shown to be reproducible. The failure mode of the fabric was tearing which initiated at the corners of the samples. Chapter 5– Validation 77 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 0.00 0.05 0.10 0.15 0.20 0.25 strain m em br an e st re s s (N /m m ) Coated-warp Coated-fill Uncoated-warp Uncoated-fill Uncoated Warp Uncoated Fill Coated Fill Coated Warp m em br an e st re s s (N /m m ) Figure 5-5: Average biaxial stress-strain curve for coated and uncoated samples The fabric under biaxial extension exhibits a hyper-elastic like behavior very similar to the yarn behavior exemplified by a high initial modulus, some softening before stiffening into a high linear modulus until failure as seen in Figure 5-5. There is no drastic change in extensional properties between the coated and uncoated samples. This is to be expected since the silicone coating has a considerably lower stiffness compared to the yarn system. However, it was also observed during the experiments that the coated fabric fractured at a lower strain than the uncoated fabric. The probable cause of the discrepancy of the failure points between the coated and uncoated sample is due to the difference in the stress concentrations seen in the corners of the samples. The difference lies in the prohibitive nature of the coating to resist in-plane yarn rotation and sliding. Qualitatively, yarn sliding at the edges of the sample apparent by fraying of the edges occurs in the uncoated fabric which can be seen from the photographs taken and shown in Figure 5-6a. This sliding and rotation allows the yarns to conform to the load path putting less stress on the yarns. On the other hand, in Figure 5-6b, the coated fabric’s edges have no fraying. Additionally, the edges have a slight curvature caused by crimp interchange between the warp and fill yarns due to no slip condition. Chapter 5– Validation 78 (a) (b) Figure 5-6: Photographs of biaxial extension at approximately 12% strain for a) uncoated and b) coated fabrics 5.2.3 Bias extension The bias extension test refers to the uniaxial extension of a rectangular fabric specimen that has been cut at 45 degrees to the yarn directions. The test is fairly simple to conduct compared to other methods used for shearing a fabric such as rail shear or picture frame test and can provide reasonably reproducible results. One drawback is that the strain is not homogeneous throughout Chapter 5– Validation 79 the sample in which three distinctive shearing zones can be identified. Referring to Figure 5-7, Zone A undergoes pure shear deformation until the yarns reach a jamming point. Zone B is mix of yarn extension and shear while Zone C remains undeformed. Zone A Zone B Zone B Zone B Zone B Zone C Zone C Zone A Zone B Zone B Zone B Zone B Zone C Zone C a) b) L ½ L Figure 5-7: Heterogeneous deformation of fabric bias extension Five samples each of the uncoated and coated fabric specimens were tested. The specimens’ dimensions were 50mm wide by 100mm long gauge length which has been used in other research (Harrison, Clifford, & Long, 2004; Page & Wang, 2000; Spivak & Treloar, 1968). The samples were cut vigilantly to ensure the 45 degree orientation and optical microscopy was used to confirm the fabric angle. Masking tape was applied to the gripped portion of the fabric to protect it from grip induced damaged. A 1cm x 1cm orthogonal grid was drawn on each sample using a felt-tip permanent marker as a reference to measure rotation at the center of the specimen as well as monitor the deformations in other regions in the fabric. Additionally, the grid square in the center of the sample will be used as a Point of Reference to quantify shear strain verses the applied stress to the specimen in the upcoming discussion. The bias tests were performed on the Instron 1122 tester at room temperature with a digital camera mounted onto a tripod to record the shear angle at the center of the specimen as shown in Chapter 5– Validation 80 Figure 5-8. The camera recorded video at 30 frames per second at a resolution of 640x460 pixels. The video started to record before the extension was performed and a clicking noise which acts as an audio marker was made simultaneously when the Instron loading was engaged. Using the audio marker, the video images can be correlated to an exact data point recorded by the Instron. Data acquisition from the Instron was performed at 100 Hz. A 500 kg load cell was installed and an extension rate of 60 mm/min was used. After jamming, the samples started to slip out of the grips, therefore they were not tested to failure. Since the low level shear behavior is of interest, the range of data obtained when the specimen slips at higher shear strains is ignored. Instron Tester Camera & Tripod Bias Specimen Figure 5-8: Bias extension-apparent shear angle recording set-up Post-processing the video to obtain the shear strain can be a daunting task. The duration of each bias test was about one minute and the video was recorded at 30 fps, which results in 1800 images available for measurement for one sample. Even using one frame per second, it would require 60 images to manually measure for shear angles. Therefore, to reduce the amount of images while retaining accuracy, images were measured at 1 fps for the first 10 seconds where the majority of the shear deformation occurs and the one frame every five seconds is measured to Chapter 5– Validation 81 until 50 seconds has past for the test. It was witnessed that after a minute slippage of the specimens in the grips begins to occur in the sample, therefore any data points beyond that time are not considered. The angle of the lower corner of the center square marked off on the specimen is measured using the software ImageJ. Figure 5-9 shows the average load-elongation behavior of the coated and uncoated bias fabric samples. At low levels of extensions, the coated samples exhibit a stiffer behavior than the uncoated fabric. After about 20mm, the stiffnesses of the coated and uncoated fabric are almost identical. 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 Extension (mm) A pp lie d St re ss (N /m m ) Exp - Coated Exp - Uncoated Figure 5-9: Load-elongation of bias coated and uncoated airbag fabric Figure 5-10 shows the average applied specimen stress and shear angle results for the coated and uncoated specimens. To be clearer, the y-axis (applied shear stress) is the load recorded by the Instron normalized by the specimen width and the x-axis (shear angle) is the measured change in angle of the center square. The alphabetic markers signify the correlated video snapshots shown in Figure 5-11 for the coated fabric and Figure 5-12 for the uncoated fabric so that the stress- strain curve can be referenced with respect to the observed deformed configuration of the Chapter 5– Validation 82 sample. The Point of Reference (P.O.R.) where the center shear angle was measured is indicated in the first frame. Since the camera position moved during the testing, it is not possible to superimpose semi-transparent images to illustrate the average deformed configuration of the sample. Therefore in the construction of Figure 5-11 and Figure 5-12, the sample whose applied stress-strain response was closest to the average is referenced. A B C D E F G 0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 Center Square Shear Angle (degrees) A pp lie d sh ea r st re ss (N /m m ) Uncoated Coated A pp lie d sh ea r st re ss (N /m m ) Figure 5-10: Bias shear -- applied specimen end stress vs. center specimen shear angle The horizontal error bars at each data point represent the deviations that occur mostly due to human error in measuring shear angle and effects that result in yarn slippage. The error in the shear angle for both specimens is acceptable, although the uncoated samples exhibit higher strain error. Overall, Figure 5-10 attempts to quantify the local shear deformation of the pure shear deformation zone as opposed to the total elongation of the sample which is mainly a global response. Chapter 5– Validation 83 Figure 5-11: Typical bias shear deformation sequence - 50mm x 100mm coated sample Figure 5-12: Typical bias shear deformation sequence - 50mm x 100mm uncoated sample Chapter 5– Validation 84 It can be observed that at lower levels of shear strain there is a significant difference in the deformational behavior of the coated and uncoated samples. Figure 5-13 better illustrates this low shear strain behavior for both of the fabric sample sizes tested. It can be observed that the coated fabric has a much higher initial shear stiffness compared to the uncoated fabric. While the coating has a low Young’s Modulus and is very thin, it still seems to inhibit yarn rotation during low shear strains. At higher shear strains, the coated fabric starts to behave similarly to the uncoated sample having a region of low shear stiffness before dramatically stiffening when the yarns have rotated into a jammed (locked) condition. 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 Center Square Shear Angle (degrees) A pp lie d St re ss (N /m m ) Uncoated Coated Figure 5-13: Detailed low shear angle plot of applied stress vs. center shear of bias sample Chapter 5– Validation 85 5.3 Comparison of UC, FE-UC and experiments 5.3.1 Uniaxial results Figure 5-14 and Figure 5-15 show the stress-strain results for the coated and uncoated uniaxial extension of the simple single unit cell model, simulated sample using the finite element material model formulation and the experimental results. The simple single unit cell analytical model is an adequate approximation of the uniaxial behavior of the fabric assuming that the deformation is homogeneous. In reality, the deformation of the sample is not homogeneous due to the transverse contraction along the free edges of the sample from crimp interchange of the yarns. Figure 5-16a demonstrates the predicted strain heterogeneity in the coated fabric at 30% strain in the warp direction. Figure 5-16b shows the fabric strain in the fill direction. For brevity, only the warp-loaded coated simulation is shown but the following discussion is applicable to fill-loaded direction and uncoated fabric. As it can be seen by comparing the warp and fill strains, the simulation predicts an area of biaxial tension near the clamped portion of the sample caused by constriction of the transverse yarns against yarn crimp interchange. Along the center of the specimen, the strain is highest and uniform where the axial yarns have almost completely decrimped. Stepping away from the macroscopic strains, Figure 5-17a and Figure 5-17b shows the calculated warp and fill yarn strains. Since the failure criteria of the unit cell is based on the ultimate yarn strain, studying the yarn strains developed can indicate the location where possible fabric failure initiates. In the examples given in Figure 5-17, at approximately 30% fabric strain, the warp yarns near the middle of the sample approach the ultimate warp yarn strain which was the area in which failure had occurred in the experiments. The simulated sample using the finite continuum formulation has the ability to replicate both the macroscopic and yarn deformations that occur physically in the sample. Therefore, the predicted stiffness and failure from the finite element simulation more closely resemble the behavior of to the experiments compared to the calculated behavior from the single unit cell. Outside the deformation mechanisms, the calculated stress-strain behavior of the coated and uncoated fabric simulations are almost identical. This is to be expected as shown previously from the sensitivity analysis of the unit cell, experimental observations and the lack of shear deformation in the type of loading. Chapter 5– Validation 86 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 strain m e m br an e s tr e ss (N //m m ) Exp-Coated-warp Exp-Coated-fill Analytical UC-Coated-warp Analytical UC-Coated-fill FE UC-Coated-warp FE UC-Coated-fill Figure 5-14: Uniaxial extension of coated airbag fabric - model vs. experiments 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 strain m em br an e s tr es s (N //m m ) Exp-uncoated-warp Exp-uncoated-fill Analytical UC-uncoated-warp Analytical UC-uncoated-fill FE UC-Uncoated-warp FE UC-Uncoated-fill Figure 5-15: Uniaxial extension of uncoated airbag fabric - model vs. experiment Chapter 5– Validation 87 a) b) Figure 5-16: Macroscopic strains of simulated uniaxial loaded coated airbag fabric at 30% extension – a) warp direction and b) fill direction Chapter 5– Validation 88 a) b) Figure 5-17: Yarn strains of simulated uniaxial loaded coated airbag fabric at 30% extension – a) warp direction and b) fill direction 5.3.2 Biaxial results Figure 5-18 and Figure 5-19 show the stress-strain results of the simple unit cell model, simulated biaxial cruciform test and the experimental cruciform test. Both the simple model and finite element model were run until reaching their calculated ultimate load of the fabric. The simulated cruciform sample is a quarter model of the actual test to keep computational time low. Overall, the deformational behavior of the single unit cell and simulated model closely resemble the global stress-strain characteristics seen in the experiments. Chapter 5– Validation 89 0.00 10.00 20.00 30.00 40.00 50.00 60.00 0.00 0.05 0.10 0.15 0.20 0.25 strain m em br an e s tr es s (N /m m ) Exp-Coated-warp Exp-Coated-fill UC Model-warp UC Model-fill FE Model-warp FE Model-fill Figure 5-18: Biaxial extension of coated airbag fabric - model vs. ex periments 0.00 10.00 20.00 30.00 40.00 50.00 60.00 0.00 0.05 0.10 0.15 0.20 0.25 strain m em br an e st re ss (N /m m ) Exp-uncoated-warp Exp-uncoated-fill UC Model-warp UC Model-fill FE Model-Warp FE Model-Fill Figure 5-19: Biaxial extension of uncoated airbag fabric - model vs. ex periments Chapter 5– Validation 90 Table 5-2 shows the simulated deformation of the sample compared to photographs taken during the biaxial experiments. Likewise, Table 5-3 shows a similar comparison for the uncoated fabric. Qualitatively, the simulations capture the deformational nuances of the corners of the specimens, specifically the slight curvature that is developed. On the other hand, there are visible differences between the uncoated simulation and the experiment. The uncoated sample in the experiment exhibits sliding and pullout of the yarns; thereby giving the edges of the sample a straight rather than curved appearance seen in the simulation. However, this discrepancy does not greatly affect the global force-displacement relationship of the system as evident in the calculated stress-strain curves. While the single unit cell model and finite element material formulations have very good agreement on the stress-strain behavior, the failure predictions by the two methods vary. Notably, the finite element simulation of the coated fabric adequately predicts the point of failure seen experimentally for coated fabric (around 12%). The single unit cell model greatly over- predicts the failure point of the coated fabric, understandably as the edge effects and stress concentrations are not considered. The failure of the simple unit cell model presents the failure of a fabric under homogeneous biaxial displacement. Much like the uniaxial sample, the biaxial sample has edge effects that cause heterogeneous displacement condition. The initiation of failure in the simulation occurs at the corner of the specimen where yarns strains are the highest. Upon the first onset of yarn failure, propagation of failure to neighboring yarns occurs suddenly causing the simulation to become unstable. In the experiments, a similar phenomena is exhibited -- failure typically initiating at the corner of the sample and rapidly propagating across the cruciform arm. Figure 5-20 (a) and (b) shows the warp and fill yarn strains for the coated failure at 12% biaxial strain respectively. From the figures, it can be seen that the fill yarn near the corner of the specimen approaches its ultimate strain. It should be noted that modeling the cruciform shape used in this study within the finite element framework is quite difficult. While the global stress-strain behavior is believed to be correct, the local behavior of the cruciform corner is a singularity point where calculating the correct magnitude of stress becomes a very challenging task. Refining the mesh past dimensions that are smaller than the dimensions of the unit cell results in earlier failures at the corner as one would expect. One could bevel the corner to elevate the effect of the singularity, but this condition would also Chapter 5– Validation 91 neglect the physical system. Therefore, the failure predictions of the biaxial simulation may require further technique and investigation. Nonetheless, we will address possible weaknesses in the user material model in predicting failure of biaxial samples assuming the singularity problem is less evident in actual applications. Table 5-2: Biaxial extension of coated fabric – experiment vs. simulation Strain Experiment Simulation 0% ~5% ~12% Chapter 5– Validation 92 Table 5-3: Biaxial extension of uncoated fabric – experiment vs. simulation Strain Experiment Simulation 0% ~5% ~10% Chapter 5– Validation 93 (a (b Figure 5-20: Simulated yarn strains at approximately 12% equal biaxial extension in the- a) warp direction and b) fill direction Interestingly, the failure predictions of the uncoated fabric by the simple unit cell and finite element material model deviate greatly from what is seen in the experiments. The simple unit Chapter 5– Validation 94 cell model over-predicts the failure point while the continuum material model under-predicts the failure of the uncoated fabric. The shortcomings of predicting failure of the uncoated fabric samples are apparent when considering the assumptions that are the basis for each model. The simple unit cell model represents an ideal fabric. The failure stress predicted by the simple model represents the failure of an infinitely long fabric sample under biaxial extension where the yarns remain orthogonal with no edge effects or sliding and friction. On the other hand, the material model is based on continuum assumptions that allow for non-orthogonal yarn configurations and considers no sliding between the yarns. Henceforth, the actual fabric sample can be descritized to consider the stress concentration from edge corners. This continuum assumption seems to be valid for the coated fabric system where the coating reduces slip thus ensuring a more continuum like structure. Additionally, as mentioned previously the yarn failure strain criteria is based on the ex-situ response. Certainly the confining pressure of neighboring yarns and filament friction are aspects not considered within the model that could potentially affect the failure criteria. 5.3.3 Bias results The bias extension is a test to examine the shear behavior of the fabric. Due to the high deformational heterogeneity, it is not possible to model the test using the simple unit cell model. Figure 5-21 shows the simulated and average experimental results of the force-elongation relation for the bias sample. A good correlation is seen between simulation and experiments for both coated and uncoated samples. At higher extensions, around 35 mm, the response of the model for both coated and uncoated fabrics becomes slightly stiffer than the experiments. While the global force-displacement seems reasonable, examining shear deformation in detail suggests weaknesses in the model. Figure 5-22 shows the applied stress versus the measured shear strain at the center of the specimen. At low shear strains there is reasonable agreement between the simulation and experiments. Figure 5-23 gives a more detailed view of the low level stress- strain. However, at larger center shear strains the simulations of the coated and uncoated samples deviate from the experiment, even though the global stiffness of the simulation and experiments were found to be similar. Table 5-4 and Table 5-5 show the deformed sample against the simulation at various extensions for the coated and uncoated fabric, respectively. From the tables it can be seen that the simulation accurately captures the distinct deformation zones seen in the bias experiments. Chapter 5– Validation 95 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 Extension (mm) Fo rc e (kg f) Exp - Coated Exp - Uncoated FE - Coated FE - Uncoated Figure 5-21: Bias load-elongation for coated and uncoated fabric - simulation vs. experiment 0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 shear strain (degrees) Ap pl ie d sh ea r st re ss (N /m m ) Exp-Uncoated FE-Uncoated Exp-Coated FE-Coated Figure 5-22: Applied shear stress vs. measured shear angle - simulation and experiments Chapter 5– Validation 96 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 shear strain (degrees) A pp lie d sh ea r st re ss (N /m m ) Exp-Uncoated FE-Uncoated Exp-Coated FE-Coated Figure 5-23: Detailed low level applied shear stress vs. measured shear angle - simulation and experiments The agreement between the simulation and experiments at low shear levels shown in Figure 5-23 is encouraging since the shear model at those regions is calibrated using experimental data. At large shear deformations as shown in Figure 5-22, the approximation of the locking stress seems to have an effect on the shear deformation. The locking shear does not consider possible yarn deformations that have been documented in the literature such as inter-filament friction, yarn torque, rapping of the yarns at the crossover and changes in fiber volume fraction in the yarn (Grosberg et al., 1968; McBride & Chen, 1997; Skelton, 1976). Therefore, it can be argued that the shortcomings in the simulations do not lie in the shear model but in the calibration techniques. A possible and more suitable technique for calibrating the shear behavior of the unit cell that would include both low level shear and the full locking shear effect is by testing the fabric using a picture frame set-up as discussed in Chapter 3. Chapter 5– Validation 97 Table 5-4: Comparison of bias deformation of coated sample: experiment vs. simulation Extension Experiment Simulation 0 mm 10 mm 20 mm 30 mm Chapter 5– Validation 98 Table 5-5: Comparison of bias deformation of uncoated sample: experiment vs. simulation Extension Experiment Simulation 0 mm 10 mm 20 mm 30 mm Chapter 5– Validation 99 5.4 Summary Considering the lack of experimental data for the mechanical behavior of 350 dtex airbag fabric, a series of three experiments were conducted to validate the model: uniaxial extension, biaxial extension and bias extension. Of the three experiments, the biaxial extension most closely resembles the condition seen in the inflated airbag. Overall, the deformational behavior of the simulated fabric closely resembles that measured in the experiments. Additionally, the single unit cell model incorporated in MATLAB can be particularly useful to quickly evaluate different plain weave fabric parameters for their mechanical properties under simple deformations. Chapter 6- Conclusion and recommendations 100 Chapter 6 - Conclusion and recommendations 6.1 Conclusions The importance of accurately simulating the mechanical behavior of airbag fabric in the structural analysis of the airbag and the shortcomings of the current methods has been established in the previous literature. The proposed approach can closely predict the macroscopic nonlinear mechanical behavior of airbag fabric using the constituent properties under a variety of deformations. Additionally, not only does the approach successfully simulate the macroscopic behavior of the fabric, it can also approximate deformations of the constituents at the structural level such as crimp interchange and yarn strain – all at vary small computational cost. This is a significant improvement on the current constitutive definitions for airbag fabric. The basis of the constituent based approach is the unit cell model – the simplified approximation of the plain weave fabric structure. The structural mechanisms that contribute to the macroscopic load-elongation behavior that arises from deformation of the constituents are carefully defined. The characterization of the constituent properties for the model is fairly simple: geometric spacing and crimp; yarn load-elongation behavior; and in-plane shear behavior of the fabric. After implementation into the explicit finite element code, LS-DYNA, the unit cell model was used to simulate three quasi-static tests. The first of the simulated test was a uniaxial extension of a fabric sample – a simple and popular test to quickly evaluate a fabric performance. Good correlation of deformational behavior and failure was seen between the model and experiments. To truly understand the models use in the airbag application, a biaxial extension test of the fabric was conducted using a customized cruciform test set-up. Again, there was good agreement between the simulated and recorded stress-strain behavior of the fabric. Finally, a bias extension test, where a sample is loaded with the yarns oriented at 45 degrees from the loaded direction Chapter 6- Conclusion and recommendations 101 was performed to evaluate the model’s ability to represent the shear behavior of the fabric. At low levels of shear deformation, the model results closely resemble the experiments. However, at higher shear strains while the global load-elongation of the model is consistent with experiments, the measured shear at the center of the sample deviates from what is calculated in the simulation. 6.2 Recommendations and future work There are many parameters that can potentially affect the airbags performance which have not been explored in the literature. Certain yarn structural mechanisms such as yarn bending are of importance in quasi-static simulations but could be less influential on the airbag dynamics during the rapid dynamic simulation. Outside of structural mechanisms, assumptions neglecting the out-of-plane effects of the fabric should also be evaluated, particularly regarding any impact of simulated unfolding of the airbag. The environments that the airbag fabric experience can vary greatly during its lifetime which can possibly affect its overall performance. Before deployment, the fabric and its constituents can undergo the degradation effects from the environment such as temperature and humidity changes. During deployment, the fabric undergoes high strain rates and can experience temperatures up to 600 ºC, all of which are certainly different than the conditions measured in the experiments. All these scenarios are not currently implemented in the current model. If further experimentation shows sensitivity to these effects, additional unit cell mechanisms can be added to the model to enhance its capabilities. The other potential improvement of the proposed model is that it is purely elastic. Due to the discrete and free movement nature of the constituents, fabrics exhibit hysteresis. Of course this hysteresis acts as an energy dissipation agent. In the occupant crash simulations, documenting the total energy dissipation between the impact of the occupant and bag are essential in evaluating the likelihood of injury. The exact or estimated percentage of the fabric hysteresis contribution to the total energy dissipation of the system is not documented in this study. Further experiments and simulations should be conducted regarding this subject. Outside of energy Chapter 6- Conclusion and recommendations 102 dissipation, quantifying yarn sliding seems to be important in predicting fabric failure under complex modes of deformation. 6.3 Summary Overall, considering the current methods used by airbag designers, the current approach can be a powerful tool to better understand how the nonlinear behavior of the fabric can affect designs and passenger safety. Not only is the nonlinear behavior of the fabric accurately represented, but details into crimp interchange and yarn strain are implicitly computed – providing a wealth of information at low computational costs. While the studied application was intended for automotive airbags, the developed approach and methodology can be applied to a variety of textile structures such as architectural fabric roofs, ballistic armor, inflatable structures, parachutes and the like. References 103 References Alley, V. L., & Faison, R. W. (1972). 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Unpublished PhD, The University of British Columbia, Shahkarami, A., & Vaziri, R. (2006). An efficient shell element based approach to modelling the impact response of fabrics. 9th Int. LS-DYNA Users Conference, Dearborn, Michigan. 12. Shahkarami, A., & Vaziri, R. (2007). A continuum shell finite element model for impact simulation of woven fabrics. International Journal of Impact Engineering, 34(1), 104-119. Shahpurwala, A. A., & Schwartz, P. (1989). Modeling woven fabric tensile strength using statistical bundle theory. Textile Research Journal, 59(1), 26-32. Shanahan, W. J., & Hearle, J. W. S. (1978). An energy method for calculations in fabric mechanics part II: Examples of application of the method to woven fabrics. Journal of the Textile Institute, 69(4), 92. Shockey, D. A., Erlich, D. C., & Simons, J. W. (2000). Improved barriers to turbine engine fragments No. U.S. Department of Transportation). Menlo Park, California: SRI International. Skelton, J. (1976). 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Implementation and validation of a finite element approach to simulate occupant crashes with airbags: Part I - airbag model. ASME Applied Mechanics Division -Publications- AMD, 169, 269. Appendix A – UMAT *MAT card documentation 109 Appendix A – UMAT *MAT card documentation *MAT_USER_DEFINED_MATERIAL_MODELS User defined material 48. This model is the Continuum Plain Weave Fabric Model for shell elements with one through-thickness integration point (membrane) developed at The University of British Columbia. This model employs a mechanistic unit cell approach and non-orthogonal formulation capable of representing the constitutive behavior of plain weave fabrics. The unit cell formulation is based on simple yarn constituent properties and can employ both linear and nonlinear yarn extensional behavior. Additional structural mechanisms include transverse yarn compression, yarn bending rigidity and coating extension. A four region shear stress-shear input parameter is capable of fitting the shear behavior of both uncoated and coated fabrics. The unit geometry can be approximated by simple linear or sinusoidal approximation of the yarn crimp. The routine includes an instantaneous ultimate strain based yarn failure criterion which removes failed yarns from the unit cell and invokes element erosion when both warp and fill yarns have surpassed their capacity. The current implementation is a scalar routine. Card 1 1 2 3 4 5 6 7 8 Variable MID RO MT LMC NHV IOTHRO IBULK IG Type I F I I I I I I Default none none 48* 38* 40* 0* 1* 2* Card 2 1 2 3 4 5 6 7 8 Variable IVECT IFAIL ITHERM IHYPER IEOS Type I I I I I Default 0* 1 0 0 0 *Note: These values are not necessarily defaults. These variables must set as indicated to ensure proper operation of this user material model. Card 3 1 2 3 4 5 6 7 8 Variable K G SHTH thw0 thf0 niter TOL geoflag Type F F F F F I F I Default none none none 0 90 none none 0 Appendix A – UMAT *MAT card documentation 110 Card 4 1 2 3 4 5 6 7 8 Variable H01 n1 cr1 ey1u EIy1 den1 yro1 lcd1 Type F F F F F F F I Default 0 none none none none none none none Card 5 1 2 3 4 5 6 7 8 Variable H02 n2 cr2 ey2u EIy2 den2 yro2 lcd2 Type F F F F F F F I Default 0 none none none none none none none Card 6 1 2 3 4 5 6 7 8 Variable gm1 G1 gm2 G2 gm3 G3 gm4 ETOL Type F F F F F F F F Default none none none none none none none 1.1 Card 7 1 2 3 4 5 6 7 8 Variable fda fdb stfnr Ec Nu_c tc Type F F F F F F Default none none none none none none VARIABLE DESCRIPTION MID Material Identification. A unique number must be chosen. RO Mass density. MT User material type. To specify the fabric model, ‘48’ must be specified. LMC Length of the material constant array. For the fabric model, ‘38’ must be specified. NHV Number of history variables. For the fabric model, ‘40’ must be specified. Appendix A – UMAT *MAT card documentation 111 VARIABLE DESCRIPTION IORTHO Set to 0 for the fabric model IBULK Address of the bulk modulus in the material constants array. Set to ‘1’. IG Address of the shear modulus in the material constants array. Set to ‘2’. IVECT Vectorization flag. For the fabric code, set to ‘0’. IFAIL To allow element erosion, set IFAIL=1. ITHERM For the fabric code, set to ‘0’. IHYPER For the fabric code, set to ‘0’. IEOS For the fabric code, set to ‘0’. K Bulk Modulus. See Remark 1. G Shear Modulus. See Remark 2. SHTH Shell Thickness. thw0 Initial Warp Angle. thf0 Initial Fill Angle. niter Maximum number iterations for the unit cell calculation. TOL Tolerance of the unit cell solution. See Remark3. geoflag Unit cell geometry flag: EQ 0.0: Linear EQ 1.0: Sinusoidal H01 Initial amplitude of the sine function representing the warp yarn. Set to 0.0 for linear as the code internally determines the initial yarn height. n1 Warp yarns per inch. cr1 Warp yarn crimp. ey1u Ultimate strain of warp yarn. EIy1 Warp yarn bending rigidity. See Remark 4. den1 Warp yarn linear density in denier. yro1 Warp yarn material density. lcd1 Load curve ID to specify warp yarn stress-strain curve. H02 Initial amplitude of the sine function representing the fill yarn. Set to 0.0 for linear as the code internally determines the initial yarn height. n2 Fill yarns per inch. cr2 Fill yarn crimp. ey2u Ultimate strain of fill yarn. EIy2 Fill yarn bending rigidity. See Remark 4. den2 Fill yarn linear density in denier. yro2 Fill yarn material density. lcd2 Load curve ID to specify fill yarn stress-strain curve. gm1 Shear strain designating a transition zone. See Remark 5. G1 Initial Shear modulus. gm2 Shear strain designating a transition zone. See note 5. G2 Intermediate Shear modulus gm3 Shear strain designating a transition zone. See Remark 5. G3 Maximum Shear modulus. gm4 Shear strain designating a transition zone. See Remark 5. ETOL Value of erosion shear for highly deformed elements to maintain stability. Recommended value of 2.00. stnfr Factor in calculating minimum contact force. Typically set to 1.1. fda Maximum transverse compression parameter (mm). fdb Intra-ply contact stiffness ((N-mm)(1/3) ). Ec Modulus of elasticity of coating Appendix A – UMAT *MAT card documentation 112 VARIABLE DESCRIPTION Nu_c Poisson’s ratio of coating tc Coating thickness Remarks: 1. The suggested units for analysis are g , mm, ms, N, MPa. Internally, the conversion of the yarns per inch will formulate the unit cell geometry in mm. 2. Bulk Modulus is used to determine stable time step and is required for determining transmitting boundaries, contact interfaces, rigid body constraints. Within mechanistic unit cell empowered model, the following equations can be used to calculate the correct bulk modulus that accounts for the crimp of the yarn. ( )ν ρ 213 *2 − = cK where ρ is density of the fiber, ν is the Poisson’s ratio of the yarn material and cr c c + = 1 * where c is the speed of sound in the fiber material and cr is the crimp of the yarn. With respect to best practices, for unbalanced fabrics, the yarn crimp that results in the larger values of K governs. 3. Shear Modulus is used to determine stable time step and is required for determining transmitting boundaries, contact interfaces, rigid body constraints. Within mechanistic unit cell empowered model, the following expression can be used to calculate the correct shear modulus that accounts for the crimp of the yarn: ( )ν ρ + = 12 *2 cG where ρ. ν and c* are defined in remark 1. 4. Tolerance of the unit cell Newton-Raphson scheme to reach a solution. A smaller tolerance has a trade off of longer computational time. Appendix A – UMAT *MAT card documentation 113 5. The theoretical bending rigidity of low-twist, non-blended yarns is bounded by two values. The lower bound assumes each individual fiber has complete freedom of motion frictionless matter and is defined as: fiberyarn nEIEI = where n is the number of fiber within the yarn and EIfiber is the bending rigidity of a single fiber. The upper bound assumes the fibers have no freedom to move and act like a complete cluster bonded by friction. Therefore the yarn bending rigidity can computed using the number of fibers in the yarn divided by the yarn packing factor times the rigidity of the lower bound. This yields the expression: ( ) yarnfiberyarn IEEI = where Nfiber is the number of fibers in the yarn and Iyarn is the second moment of inertia using the cross- sectional geometry of the yarn. For greater discussion on the theoretical bending rigidity of yarn, determination of values within the bounds, determination of bending rigidity of blended yarns and determination of bending rigidity of highly twisted yarns the paper by Platt, Klein and Hamburgaer (1959) is highly suggested. 6. Due to the difficulties of evaluating fabric shear using the fabric’s constituents, empirical data from the fabric is warranted to empower the shear behavior of the proposed unit cell method. This empirical data can be obtained either through picture frame test where the fabric under goes pure shear; rail shear test using the Kawabata Evaluation System (KES) for fabrics; or could potentially be numerically calibrated through a full 3D finite element model of the fabric and its constituents. Each method has advantages and disadvantages. Picture frame requires special fixture and the design of the fixture must be carefully done to ensure reproducible results. Kawabata Evaluation System requires specialized equipment and some additional calculation to remove the effects of tension and to obtain pure shear behavior. The use of 3D finite element models can be used to evaluate the pure shear behavior of virtually design fabrics but capturing the correct boundary conditions can be challenging. Regardless the approach to obtain the shear data, the following method can be used to represent the shear behavior of the unit cell. A spline fit of the shear modulus as a function of shear strain illustrated in Figure 2A can be used to describe the shear stress-strain behavior based of typical behavior seen in Figure 1A. Coated Transition Zone Shear Strain Sh e ar St re ss γ2γ1 γ3 γ4 Locking Transition Zone Single-Coated Fabric Uncoated Fabric Sh e ar St re ss Figure A1: Shear model behavior regions in the shear stress-strain curve Appendix A – UMAT *MAT card documentation 114 G1coated G2 G3 G1uncoated Sh ea r M o du lu s Shear Strain γ1 γ2 γ3 γ4 Sh ea r M o du lu s Figure A2: Secant shear modulus as a function of strain For coated fabrics, the fabric-coating shear interaction modulus, G1 needs to be defined as well as the extent of the transition zone which is bound by γ1 and γ2. For uncoated fabrics, these values can be set equal to zero. The pre-locking shear stiffness of the fabric, G2 is the same for both uncoated and coated fabrics of the same construction. As the fabric continues to deform in shear, the amount of rotation at the crossovers reach a geometric limit and begin to lock. This is a gradual process controlled my friction, the current packing state of the fibers within the yarn and geometric features of the fabric. Within the spline model, this process is bound by γ3 and γ4 which are the same for both the coated and uncoated fabric. Likewise the locking shear modulus, G3, is the same for uncoated and coated fabrics. History Variables: The fabric continuum model will return many more variables than “normal” material models. These variables include the stretch of each yarn vector, yarn strain, unit cell geometric parameters, yarn failure flags, number of iterations, solution error etc. To access these history variables, be sure to set the NEIPS (shells) variables in the *DATABASE_EXTENT_ BINARY card to 40. The following table shows the history variables available and their location in the history variable list: History Variable Model Variable Description 1 strchw Stretch of warp yarn vector 2 strchf Stretch of fill yarn vector 3 thw Angle of the warp yarn vector 4 thf Angle of the fill yarn vector 5 fail1 Warp yarn failure flag (0=no fail, 1=failed) 6 fail2 Fill yarn failure flag (0=no fail, 1=failed) 7 ey1 Warp yarn strain 8 ey2 Fill yarn strain Appendix A – UMAT *MAT card documentation 115 History Variable Model Variable Description 9 sigw Un-rotated fabric stress - warp 10 sigf Un-rotated fabric stress - fill 11 gmw Shear strain - warp 12 gmf Shear strain - fill 13 R Newton-Raphson scheme residual error 14 iter Number of iterations of the unit cell N-R scheme 15 iterMXd Counter for number of times the maximum of iterations was exceeded throughout the analysis 16 Ten1 Warp tensile force 17 Ten2 Fill tensile force 18 Fc1 Contact force from warp yarn 19 Fc2 Contact force from fill yarn 20 Fb1 Bending force from warp yarn 21 Fb2 Bending force from fill yarn 22 H1 Height of warp yarn in the unit cell 23 H2 Height of fill yarn in the unit cell 24 d1 Warp yarn displacement 25 d2 Fill yarn displacement 26 dcomp Total transverse compression within the unit cell 27-30 blank - 31 g11 Component of the displacement tensor 32 g12 Component of the displacement tensor 33 g21 Component of the displacement tensor 34 g22 Component of the displacement tensor 35 sy1 Warp yarn stress (from load curve) 36 sy2 Fill yarn stress (from load curve) 37 sig(1) Calculated stress in 1 dir of the element reported back to DYNA 38 sig(2) Calculated stress in 2 dir of the element reported back to DYNA 39 sig(4) Calculated stress in 4 dir of the element reported back to DYNA 40 blank - Appendix A – UMAT *MAT card documentation 116 References: Shahkarami, A., & Vaziri, R. (2007). A continuum shell finite element model for impact simulation of woven fabrics. International Journal of Impact Engineering, 34(1), 104-119. Shahkarami, A. (2006). An efficient unit cell based numerical model for continuum representation of fabric systems. PhD thesis, The University of British Columbia Kawabata, S., Niwa, M., & Kawai, H. (1973). The Finite-deformation Theory of Plain-Weave Fabrics Part I: The Biaxial Deformation Theory. Journal of the Textile Institute, 64(1), 21. Kawabata, S., Niwa, M., & Kawai, H. (1973). The Finite-deformation Theory of Plain-Weave Fabrics Part II: The Uniaxial Deformation Theory. Journal of the Textile Institute, 64(2), 47. Platt, M. M., Klein, W. G., & Hamburger, W. J. (1959). Mechanics of elastic performance of textile materials. Textile Research Journal, 29(8), 611-627. Appendix B – UMAT pseudo code 117 Appendix B – UMAT pseudo code 1. Read the current strain increments, t∆ε 2. Calculate displacement gradient from ttt ∆εx += −1x 3. Calculate deformation gradient using I+= tt xF 4. Calculate the Cauchy-Green deformation tensor using FFC T= 4. Read yarn angle orientations from input deck, Calculate yarn vectors using [ ]Tiiiq 000 sincos θθ= 5. Calculate the current yarn stretch using 020 qq T i t i C=λ 6. Calculate current angle between warp and fill yarn using 21 0201cos λλφ qqT C = 7. Calculate change of angle 8. Calculate new yarn orientation θθθ ∆−= 011t and θθθ ∆−= 022t 9. Update yarn vectors to the current yarn orientation ( ) ( )[ ]Ttitiiq θθ sincos= 10. Calculate the current yarn displacement using ( )10 −= tiiti yd λ 11. Read yarn failure flag from history variables. If either of the warp or fill yarns have failed go to 33 12. Set up Newton-Raphson scheme. Set initial values of Hi equal to H0i if t=0 else the initial value Hi is equal to Hi t-1 Appendix B – UMAT pseudo code 118 13. Calculate unit cell current spacing by yi = y0i + di 14. Calculate current yarn length 15. Calculate yarn strain 16. Read current yarn stiffness from inputted curve deck 17. Calculate yarn tension using ( ) ( ) i ii iiy L LLAEF 0 0 , − = ε 18. Calculate vertical component of yarn tension (contact force) i i iyic L HFF ,, 2= 19. Calculate vertical component of yarn bending ( ) −= i i i i i iyarn ib L H L H y EI F 0 0 2, 8 20. Calculate the continuity functional equation ( ) ( ) ( )210201211 , HHHHHHf +−+= 21. Calculate the equilibrium functional equation ( ) )()(, 2211212 bcvc FFFFHHf −−−= 22. Check convergence criteria -- If |f1 + f2| < Tol go to 29 23. Calculate derivatives and assemble Jacobean matrix 27. Calculate new Hi using [ ] − = 2 1 2 1 2 1 f f J H H H H guess guess new new 28. Go to 13 29. Calculate fabric end force i i iyif L yFF ,, = Appendix B – UMAT pseudo code 119 30. Calculate coating end force − − − − = 02 022 01 011 2 2 1, 1 y yy y yytyEF c c cc ct νν and − − − − = 01 011 02 022 2 1 2, 1 y yy y yytyEF c c cc ct νν 31. Calculate membrane stress from total end forces ( ) iictifi nFFN ,, += 32. Check for yarn failure -- If no yarn failure go to 35 33. If ult iyarniyarn ,, εε ≥ then Ni = 0 , S12 = 0 and yarn is flagged in history variable -- go to 36 34. If both yarns have failed then erode element from the mesh 35. Calculate shear component from yarn rotation using ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ≤−−+−−+ <≤−−+− − − + <≤−−+ <≤− − − + <≤ = 4121234233 431212 2 3 34 23 2 3212122 21 2 1 12 12 1 11 12 2 1 2 1 2 1 2 1 2 1 2 1 0 γγγγγγγ γγγγγγγ γγ γ γγγγγγ γγγγγ γγ γ γγγ GGGGG GGGGG GGG GGG G S 36. Transform yarn stress tensor into the element coordinate system using 21 12 2 22 1 21 coscos2coscos θθθθσ h S h N h N x ++= 21 12 2 22 1 21 sinsin2sinsin θθθθσ h S h N h N y ++= ( )122112222111 sincossincossincossincos θθθθθθθθτ +++= h S h N h N xy 37. Return updated stress tensor to LS-DYNA to continue analysis Appendix C – Justification of the selection of a linear unit cell over sinusoidal geometry 120 Appendix C – Justification of the selection of a linear unit cell over sinusoidal geometry Early in the course of the research, the linear unit cell was compared to a more complex sinusoidal geometry by (A. Shahkarami, 2006) that more closely resembles the geometry of the unit cell. The linear unit cell assumption results in a misrepresentation of the yarn height at the crossover. It should be noted that the initial lengths of unit cell yarns in both geometries are equal to the actual yarn length and the yarn spacing is also true to the fabric structure (based on the measured yarns/in and % crimp ). Referencing Figure 1C, the percent difference between the approximation of the yarn height for the linear and sinusoidal unit cell during biaxial deformation is about 9%. Referencing Figure 2C and Figure 3C, the trigonometric functions that resolve the yarn forces into horizontal and fabric components have a smaller percent difference. Finally, the fabric end stress computed by the linear and sinusoidal unit cell is shown in Figure 3D. The difference between the computed stress of the two geometries is between 1.5% and 2.0% under biaxial deformation. 0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 strain w a rp ya rn he ig ht (m m ) 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00 di ffe re n ce (% ) warp yarn height - sin warp yarn height - tri % difference Figure C1: Approximation of yarn height by linear and sinusoidal unit cell Appendix C – Justification of the selection of a linear unit cell over sinusoidal geometry 121 0.245 0.250 0.255 0.260 0.265 0.270 0.275 0.280 0.285 0.290 0.295 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 strain si n (th e ta ) 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00 di ffe re n c e (% ) sin(theta) - sin sin(theta) - tri % difference Figure C2: Approximation of sin theta by linear and sinusoidal unit cell 0.954 0.956 0.958 0.960 0.962 0.964 0.966 0.968 0.970 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 strain c o s (th e ta ) 0 10 20 30 40 50 60 70 80 90 100 di ffe re n ce (% ) cos(theta) - sin cos(theta) - tri % difference Figure C3: Approximation of cos theta by linear and sinusoidal unit cell Appendix C – Justification of the selection of a linear unit cell over sinusoidal geometry 122 0.000 5.000 10.000 15.000 20.000 25.000 30.000 35.000 40.000 45.000 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 strain w ar p s tr e s s N /m m 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00 di ffe re n c e (% ) warp-stress - sin warp-stress - tri % difference Figure C4: Approximation of warp fabric membrane stress by linear and sinusoidal unit cell Overall for the fabric used in this study, no dramatic difference on the fabric stresses was found between geometries was found. Therefore, the linear unit cell was chosen since it is less computationally intensive and requires less preprocessing to determine the initial unit cell geometry, Appendix D – Conversion of membrane stress into specific stress 123 Appendix D – Conversion of membrane stress into specific stress In this study in order to keep consistent with the thin membrane assumption, stresses are normalized based on force per unit width. Another popular stress normalization unit within the textile community is specific stress based on force per mass per unit length. Hearle (Hearle, 1989) shows that Specific stress on fabric in N/tex = 2g/min weight"" fabric N/mmin h force/widt A traditional engineering unit of stress (force per unit area) can then be determined from the specific area provided the density of the fabric is known as shown below: Specific stress of 1 N/tex = density in g cm-3 × stress of 1 GPa
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Nonlinear mechanical behavior of automotive airbag fabrics : an experimental and numerical investigation Zacharski, Steven Edward 2010
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Title | Nonlinear mechanical behavior of automotive airbag fabrics : an experimental and numerical investigation |
Creator |
Zacharski, Steven Edward |
Publisher | University of British Columbia |
Date Issued | 2010 |
Description | Over the past two decades, the airbag has become an essential safety device in automobiles. The airbag cushion is composed of a woven fabric which is rapidly inflated during a car crash. The airbag dissipates the passenger’s kinetic energy thereby reducing injury through biaxial stretching of the fabric bag and escaping gas through vents. Therefore, the performance of the airbag is greatly influenced by the mechanical properties of the fabric. Unlike traditional engineering materials, airbag fabrics are composed of discrete constituents and have highly nonlinear mechanical behavior that arises from both geometric deformations and material nonlinearity. Henceforth, airbag designers are forced to make simplified assumptions regarding the mechanical behavior of the fabric cushion. This incontrovertibly limits designers in taking advantage of the full potential of the fabric system. In order to optimize the airbag design, improve deployment simulations and overall dependability, a more sophisticated approach is needed. In this study, a simple unit cell model representing a single crossover of two orthogonal woven yarns is developed to simulate the in-plane mechanical behavior of both coated and uncoated plain weave airbag fabrics under multiple states of stress. Since the structural analysis of the deployment of the airbag is performed using the finite element method, the proposed mechanistic model is implemented as a User-Material-Model in the commercial code LS-DYNA. Here, the unit cell model represents the constitutive behavior of a continuum membrane. The approach results in capturing, in detail, the discrete nature of the fabric while retaining the computational efficiency of simple membrane formulation compared to explicitly modeling each yarn within the fabric. The procedure to calibrate the model inputs, namely the yarn geometric and mechanical properties for a given fabric is detailed. The sensitivity of the unit cell model and verification of the finite element implementation is discussed. A series of experiments were performed to validate the in-plane behavior of the model. The proposed model can be adopted by designers to better represent the nonlinear mechanical behavior of the fabric. It can also be used as a tool to design novel fabrics that are optimized for a particular application. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-12-16 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0071553 |
URI | http://hdl.handle.net/2429/30411 |
Degree |
Master of Applied Science - MASc |
Program |
Materials Engineering |
Affiliation |
Applied Science, Faculty of Materials Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2011-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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