UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Progressive damage modeling of composite materials under compressive loads Zobeiry, Navid 2004

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_2005-0145.pdf [ 9.06MB ]
Metadata
JSON: 831-1.0063388.json
JSON-LD: 831-1.0063388-ld.json
RDF/XML (Pretty): 831-1.0063388-rdf.xml
RDF/JSON: 831-1.0063388-rdf.json
Turtle: 831-1.0063388-turtle.txt
N-Triples: 831-1.0063388-rdf-ntriples.txt
Original Record: 831-1.0063388-source.json
Full Text
831-1.0063388-fulltext.txt
Citation
831-1.0063388.ris

Full Text

PROGRESSIVE D A M A G E MODELING OF COMPOSITE MATERIALS UNDER COMPRESSIVE LOADS by N a v i d Z o b e i r y B.Sc. (Civil Engineering), University of Tehran, 2002 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Applied Science in THE FACULTY OF GRADUATE STUDIES Department of Civil Engineering THE UNIVERSITY OF BRITISH COLUMBIA December 2004 © Navid Zobeiry, 2004 ABSTRACT The in-plane compressive strength of fibre reinforced composite materials is known to be less than their corresponding tensile strength. There are a multitude of compression damage mechanisms that occur in composites, the form of which depends on the properties of the constituents and fibre lay-up. These mechanisms primarily consist of a combination of matrix cracking (yielding), localized buckling of fibres or kinking, and delamination. Whether the interest is to assess the structural integrity of composite materials in-service or to quantify their energy absorption capability under axial crushing, it is crucial to have predictive analysis tools that capture the physics of the damage mechanisms and their propagation under compressive loads. In this study, a constitutive model is formulated for the complete in-plane response of composite materials within the framework of a previously developed continuum damage mechanics model CODAM (Williams, 1998; Williams et al., 2003; Floyd, 2004). While the previous CODAM formulation was limited to simulating the progression of damage under tensile loading, the current formulation accounts for the initiation and propagation of damage under compression, tension and load reversals in each mode of loading. The model is implemented in the commercial finite element code, LS-DYNA, and combined with a modified crack band model originally developed by Bazant (Bazant and Planas, 1998) to overcome the mesh sensitivity problems that plague all strain-softening type constitutive models. The new model is validated against two sets of experimental data available in the literature, namely, eccentric compression loading of notched sandwich panels of various sizes (Bayldon, 2003a, 2003b), and axial compression of composite panels with central open holes of various panel and hole sizes (Soutis et al., 1993, 2002). It is shown that for these loading applications the predictions of the compressive strengths and the degree of size effect are in good agreement with the measured experimental results. Since the ii Abstract formulation of the model and its calibration are based entirely on the fundamental physics of the damage mechanisms, these successful validations instil confidence in exercising the model for predicting the response of composite structures of various sizes under a variety of in-plane loading applications involving compression, tension and load reversals. iii TABLE OF CONTENTS ABSTRACT II TABLE OF CONTENTS IV LIST OF TABLES VIII LIST OF FIGURES IX NOMENCLATURE XV ACKNOWLEDGEMENTS XVIII CHAPTER 1 : INTRODUCTION , 1 1.1. Background 1 1.2. Motivation 4 1.3. An Outline of This Study 6 CHAPTER 2 : COMPRESSIVE FAILURE OF FIBRE COMPOSITES 7 2.1. Introduction 7 2.2. Failure Mechanisms in Unidirectional Composites 8 2.2.1. Elastic Microbuckling 10 2.2.2. Fibre Crushing 10 2.2.3. Splitting 10 2.2.4. Buckle Delamination 11 2.2.5. Shear Band Formation 11 2.2.6. Plastic Microbuckling or Kinking 12 Table of Contents 2.2.6.1. His tory o f K i n k B a n d A n a l y s i s 12 2.2.6.2. K i n k i n g Format ion 16 2.3. Failure Mechanisms in Angle-ply Composites 19 2.4. Summary 21 CHAPTER 3 : ANALOG MODEL 22 3.1. Introduction 22 3.2. Elements : 25 3.2.1. R i g i d L i n k 25 3.2.2. Spr ing and T / C Spr ing 25 3.2.3. S l ide r 27 3.2.4. Fuse and T / C Fuse 27 3.2.5. G a p 29 3.2.6. L o c k 29 3.3. Collection of Elements or Boxes 31 3.3.1. Laminate B o x 31 3.3.1.1. Laminate B o x Equat ions 33 3.3.2. Rubb le B o x 42 3.4. Analog Model 46 3.4.1. M o d e l Cons t ruc t ion 46 3.4.2. M o d e l Formula t ion 48 3.4.2.1. L o a d Reversa l : Compres s ion -Un load ing Before B a n d Broaden ing-Tens ion 49 3.4.2.2. L o a d Reversa l : C o m p r e s s i o n - U n l o a d i n g After B a n d Broaden ing-Tens ion 51 3.4.2.3. L o a d Reversa l form Tens ion to Compress ion 53 3.5. Further Remarks on the Analog Model 55 CHAPTER 4 : DAMAGE GROWTH, ENERGY CONCEPT AND SCALING LAW 56 4.1. Damage Growth 57 4.2. Compressive Fracture Energy 61 Table of Contents 4.2.1. Main Fracture Energy 62 4.2.2. Band Broadening Fracture Energy 63 4.3. Compressive Fracture Energy in the Damage Propagation Process 65 4.3.1. Compressive Damage Propagation 65 4.3.2. Compressive Fracture Energy 67 4.3.2.1. Linear Damage Height Function 68 4.3.2.2. Generalized Damage Height Function 69 4.3.3. Summary of Fracture Energy Functions 71 4.3.4. Comparison with Experimental Results 71 4.4. Scaling Law 73 4.4.1. Crack Band Method 73 4.4.2. Modified Crack Band Method 74 4.4.3. Scaling Factor in Compression 74 4.4.3.1. Scaling Factor for an Element with Peak Stress at the Damage Initiation Strain 75 4.4.3.2. Scaling Factor for an Element with Peak Stress not at the Damage Initiation Strain 76 4.4.4. Summary of Scaling Factor Equations 77 CHAPTER 5 : MODEL VERIFICATION AND VALIDATION 78 5.1. Model Verification 79 5.2. Simulations of Notched Sandwich Panels under Eccentric Compression 82 5.2.1. Sandwich Panel Failure 83 5.2.2. Test Geometry 85 5.2.3. Failure Mechanisms of Woven Glass Laminates 88 5.2.4. Material '• 91 5.2.5. Finite Element Model 96 5.2.5.1. Experimental Results .' 96 5.2.5.2. Damage Initiation Strain 97 5.2.5.3. Scaling Factor 101 5.2.6. Simulations 103 5.2.7. Summary 107 5.3. Simulation of Open Hole Plates under Compression 108 5.3.1. Introduction 108 5.3.2. Test Geometry 110 vi Table of Contents 5.3.3. Material 112 5.3.4. Stress-Strain Curve Scaling 116 5.3.4.1. Analytical Solution •. 116 5.3.4.2. Stress Contour 117 5.3.4.3. Scaling 118 5.3.5. Simulation Results 121 5.3.5.1. Notch Size Effect 122 5.3.5.2. Specimen Sire Effect 124 5.3.6. Summary of Observations 126 CHAPTER 6 : CONCLUSIONS AND FUTURE WORK . 127 6.1. Conclusions 127 6.2. Future Work 130 REFERENCES 131 APPENDIX A: TAYLOR SERIES FOR DAMAGE HEIGHT FUNCTION 139 LIST OF TABLES Table 4-1 Damage height and fracture energy functions for different values of n in Equation 4-26 ; 71 Table 4-2 Material parameters used to verify fracture energy functions 71 Table 5-1 Compressive damage parameters for AS4/PEEK unidirectional panels.. 79 Table 5-2 Tensile damage parameters for AS4/PEEK unidirectional panels 80 Table 5-3 Material parameters for AS4/PEEK unidirectional panels 80 Table 5-4 Test geometries of notch size effect experiments (ECC). All numbers are in mm 86 Table 5-5 Test geometries of specimen size effect experiments (CDC). All numbers are in mm 86 Table 5-6 Properties of the foam core. All properties are in the loading direction... 91 Table 5-7 Properties of the face sheets. Properties in both directions are the same.. 91 Table 5-8 Damage parameters for woven glass-epoxy laminate 95 Table 5-9 Damage initiation strains for different CDC experiments based on their elastic analysis 101 Table 5-10 Scaling factors for notched sandwich panel under eccentric compression... 102 Table 5-11 Damage parameters for woven glass-epoxy laminate which give the best result for CDC simulations 106 Table 5-12 Specimen sizes with different notch sizes for Soutis et al. experiment (2002) 111 Table 5-13 Material parameters for the carbon-epoxy laminate in the compressive load direction 112 Table 5-14 Damage parameters for [(±45/0/90) 3] s carbon-epoxy laminate 115 Table 5-15 Modified parameters for the simulation of 50*50 mm specimens 120 viii LIST OF FIGURES Figure 2-1 The main compressive failure mechanisms of unidirectional composites. (a) Elastic microbuckling, (b) Kinking, (c) Fibre crushing, (d) Splitting, (e) Buckle delamination, (f) Shear band formation (Fleck, 1997) 9 Figure 2-2 Kink band formation under compressive load for a unidirectional laminate. (a) Before failure, (b) After failure 13 Figure 2-3 Kink band Formation in a unidirectional laminate under compressive load. (a) Out-of-plane kink band, (b) In-plane kink band 15 Figure 2-4 Overall behaviour of unidirectional fibre composites under compressive loads in the fibre direction when kinking occurs (Moran et al., 1995) 16 Figure 2-5 Initial damage height in kinking process (Fleck et al., 1995). Fibres rotate under compressive load 17 Figure 2-6 Schematic showing the behaviour of a single fibre during the early stage of kink band formation (Moran et al., 1995) 18 Figure 2-7 (a) Overview of arrested damage zone, (b) Band broadening in progress. (c) A close-up view of the microbuckle band, (d) 45 degree off-axis microbuckle band linking the 0/45 degree delaminated interfaces, (e) Out-of-plane 0 degree microbuckle band extending into the 45 degree off axis layer. (Sivashanker, 2001) 20 Figure 3-1 The behaviour of a rigid link element under compressive and tensile loads 25 Figure 3-2 The symbol and behaviour of a spring element under compressive and tensile loads 26 Figure 3-3 The symbol and behaviour of a T/C spring element under compressive and tensile loads 26 Figure 3-4 The symbol and behaviour of a slider element under compressive and tensile loads '. 27 IX List of Figures Figure 3-5 The symbol and behaviour of a fuse element under compressive and tensile loads 28 Figure 3-6 The symbol and behaviour of a T/C fuse element under compressive and tensile loads 28 Figure 3-7 The symbol and behaviour of a gap element under compressive and tensile loads 29 Figure 3-8 The symbol and behaviour of a lock element under compressive and tensile loads 30 Figure 3-9 Laminate box and the arrangement of basic units. All the elements are T/C springs and T/C fuses 32 Figure 3-10 Basic unit as used in the laminate box. "m" represents the response of matrix and 'J/" represents the response of fibre in a representative volume element 32 Figure 3-11 Force-displacement response of a basic element in the laminate box 33 Figure 3-12 Grouping of matrix and fibre elements in the laminate box under tensile loads 34 Figure 3-13 The response of group "w" under tensile loads in which the group consists of four elements 35 Figure 3-14 The response of group "tn" under tensile loads in which group consists of an infinite number of elements 36 Figure 3-15 Force-displacement response of the laminate box under tensile loads, m and/represent the response of matrix and fibre respectively 37 Figure 3-16 Grouping of elements in the laminate box under tensile loads..: : 38 Figure 3-17 Individual responses of Group m and Group / of elements in the laminate box under compressive loads. Group m represents the response of matrix while Group/represents the response of fibres. 39 Figure 3-18 Force-Displacement response of Group m and Group / while in parallel under compressive loads 40 Figure 3-19 A schematic of the Rubble Box and the basic elements inside it 42 Figure 3-20 Force-displacement and stiffness-displacement responses for an infinite number of units in the rubble box 44 List of Figures Figure 3-21 A schematic of the analog model which shows arrangement of elements and boxes. 46 Figure 3-22 Force-displacement response of Spring/Lock elements in series 48 Figure 3-23 Force-displacement response of the analog model. First compressed, then unloaded and finally loaded in tension to complete failure 51 Figure 3-24 Force-displacement response of the analog model (Unloading after the yielding) 52 Figure 3-25 Force-displacement response of the analog model. First pulled in tension, then unloaded and finally loaded in compression to complete failure 54 Figure 4-1 Damage growth-potential function) diagrams for tensile and compressive loading 59 Figure 4-2 (Modulus reduction-damage growth) diagrams for compressive and tensile loading. Modulus reduction due to matrix saturation is more pronounced in compression comparing to the reduction in tension 60 Figure 4-3 A schematic of stress-strain response of composites under compressive loads which showing areas associated with formation of kink bands and damage band broadening 61 Figure 4-4 Calculation of the main fracture energy from the area under the stress-strain response of the laminate box 62 Figure 4-5 The unloading path after the saturation of each kink band. Only the stress-strain response of the rubble box is shown here 63 Figure 4-6 Stress contour in front of the notch and Stress-Strain response of one element 65 Figure 4-7 Stress contour in front of the notch while damage is growing. Damage height grows from hc to hi and finally to hui- The maximum damage height, JJUL, is the damage height at the ultimate strain 66 Figure 4-8 Damage height increment due to the damage length increment 67 Figure 4-9 Calculation of the main fracture energy for one element from the stress-strain response of that element 68 Figure 4-10 Comparison of experimental results (Jackson and Ratcliffe, 2004) and analytical solutions for critical stress intensity factor functions 72 List of Figures Figure 4-11 Crack band method. Both the master and the scaled cure are shown here... 73 Figure 4-12 Modified crack band method. Both the master and the scaled cure are shown here 74 Figure 4-13 Different unloading paths for compressive failure. The left diagram shows unloading from the peak stress point while the peak stress and damage initiation coincide. The right diagram shows the unloading path for an element in which the peak stress and damage initiation do not coincide. 75 Figure 5-1 . Stress-Strain response of one element obtained from the implemented constitutive model in LS-DYNA. Some of the input parameters are obtained from Vogler and Kyriakides (1999) while the damage parameters are estimated 81 Figure 5-2 Failure modes in sandwich columns (Fleck and Sridhar, 2002). From left to right, Euler macrobuckling, core shear macrobuckling, face sheet microbuckling and face sheet wrinkling 84 Figure 5-3 Notched sandwich panel under double eccentric compression (Bayldon, 2003a, 2003b) 85 Figure 5-4 Test geometry for notched sandwich panels under double eccentric compression 87 Figure 5-5 Initiation of kinking with matrix cracking. Photoelastic image for the notched sandwich panel under eccentric compression (Bayldon, 2003a). 89 Figure 5-6 Initial damage height is equal to 3.5 mm for kinking and delamination. Photoelastic image for the notched sandwich panel under eccentric compression (Bayldon, 2003a) 90 Figure 5-7 Propagation of the kink band and delamination in horizontal and vertical directions. Height of the damaged zone is almost equal to 6 mm. Photoelastic image for the notched sandwich panel under eccentric compression (Bayldon, 2003a) 90 Figure 5-8 Experimental size effect result for CDC sandwich panels under eccentric loading 97 List of Figures Figure 5-9 Gauss integration points for CDC8. Only half of the specimen is shown here 98 Figure 5-10 Stress contour at the notch tip for CDC2. Triangles are the Gauss integration points of the finite element model. The zero distance point is extrapolated based on the stress at other points 99 Figure 5-11 Stress contour at the notch tip for CDC4. Triangles are the Gauss integration points of the finite element model. The zero distance point is extrapolated based on the stress at other points. 99 Figure 5-12 Stress contour at the notch tip for CDC8. Triangles are the Gauss integration points of the finite element model. The zero distance point is extrapolated based on the stress at other points 100 Figure 5-13 Stress contour at the notch tip for CDC16. Triangles are the Gauss integration points of the finite element model. The zero distance point is extrapolated based on the stress at other points 100 Figure 5-14 Finite element model for simulating the CDC8 experiment. Only half of the specimen is modeled 103 Figure 5-15 Simulation of CDC experiments with different fracture energies or with different saturation strains 104 Figure 5-16 Simulation of CDC experiments with different damage initiation strains while fracture energy is equal to 20.9 mJ/mm2 105 Figure 5-17 Test geometry for Soutis et al. experiments (2002). "b" is the length of the tab section 110 Figure 5-18 Modified ICSTM compression test fixture 111 Figure 5-19 Open hole panel under tensile loads 116 Figure 5-20 Finite element model for the half of the 50x50 mm specimen with the hole diameter equal to the 10 mm 117 Figure 5-21 Stress contour at different load stages for the open hole specimen in Figure 5-20. As damage progresses, the notch tip stress drops after the remote stress reaches 236 MPa 118 Figure 5-zZ Modifying the Stress-Strain curve based on the element width 119 Xll l List of Figures Figure 5-23: Modifying the Stress-Strain curve based on the element height. The < scaling is shown for an element height less than hc 119 Figure 5-2f Analytical and numerical strength predictions of 50x50 mm open hole panels under uniaxial compressive loads 121 Figure 5-25 Notch size effect results for different fracture energy values 122 Figure 5-26 LS-DYNA strength predictions for different saturation strains in 50x50 mm panels. Fracture energy is equal to 25.4 mJ/mm2 123 Figure 5-2f. Specimen size effect for different fracture energy values. (Hole diameter/ Specimen width = 0.2). 124 Figure 5-2$ Specimen size effect for different damage saturation strains. (Hole diameter/ Specimen width = 0.2) 125 xiv NOMENCLATURE Superscript C Compression T Tension Subscript 0 0 Relating to the remote stress 1,2,..,/,..,« Relating to the 1ST, 2 n d, ith, nth, etc. element app Relating to the applied force c Relating to the characteristic material parameter core Relating to the sandwich panel core e Element F Final f In C O D A M , fibre f Relating to the fracture energy face Relating to the sandwich panel face fb Relating to the broadening fracture energy fm Relating to the main fracture energy g In C O D A M , gap 8 Relating to the remote stress / Initiation inc Relating to the interval strains in kink band formation j Relating to the force of j active elements Nomenclature L Laminate box m In CODAM, matrix max Relating to the maximum stress M Relating to the modified parameter new Related to the modified Young's modulus P Plastic peak Relating to the peak stress plateau Relating to the plateau stress R Rubble box S Saturation tot Relating to the total stiffness UL Ultimate UN Unloading V Relating to the lock element Relating to the lock element gap y Relating to the slider yielding force Latin Symbols a Modification factor for Young's modulus in element width effect b Width of the laminate Cfr Length of the failure process zone d Deformation E Young's modulus F Force F(E) Damage potential function (effective strain) Fj., FM Strength of fuse element q Number of failed elements G Fracture energy Nomenclature h Damage height j Number of active elements K,L,S,T,U In CODAM, strain interaction constants k While used without index, Scaling factor k While used with index, Stiffness k0 Inter-laminar shear strength L Length of the damage band n In CODAM, the ration of characteristic height to the element height n In compressive fracture energy, the power of the damage height function P Applied force RE In CODAM, Ratio of remaining modulus to the initial modulus T Time U Strain energy Greek Symbols a Fibre rotation during the kink band formation P Kink band inclination 7 Fracture energy density To Composite yield strain A Increment 5 Displacement s Strain a Stress Initial misalignment of the fibres CO Current damage parameter xvii ACKNOWLEDGEMENTS First, I would like to thank my family members for their unconditional love, encouragement and support. I would like to thank my dear parents, Til la and Mohammad for providing me a life full of joy and inspiration. I would like to thank my sister, Nazli, for loving me so much. I would like to express my gratitude to my compassionate brother, Nima, for his important suggestions, remarks and countless patient hours spent in reviewing this thesis which aided me to improve it significantly. I would also like to thank my Yalda, who supported me in all means during the last two years. I would like to sincerely and especially thank my supervisors, Dr. Reza Vaziri and Dr. Anoush Poursartip, for their guidance, encouragement and constructive criticisms. Their valuable suggestions and constant support have made this thesis a better product. I would like to express my deep appreciation to all my friends in Composite Group specially Carla McGregor who assist me to accomplish such a wonderful joint study. I would not have made it without your support. My biggest personal thanks goes to my friends in Vancouver as they have provided me such a wonderful surrounding to concentrate on my studies. I appreciate you all for listening to my constant thesis stress and strain. Let God bless you all. Finally, my best thanks of all goes to God, who has always been there for me. I thank you for all of your inspirations that guided me to learn a tiny part of the science. xviii Chapter 1: Introduction 1.1. Background Composite materials are widely being used in industrial applications, e.g. aerospace and automotive industries. The growing popularity of composite materials is mainly due to their cost-effective manufacturing of large scale structures, light weight, high specific stiffness and energy absorbing capabilities under high intensity impact loading. They have also been identified as a suitable class of materials for use in naval and offshore structures where they can replace more conventional metallic materials. Their use eliminates the maintenance costs associated with corrosion protection required for metal structures in deep-water applications. Owing to their relatively small compressive strength compared to their tensile strength, designers have been understandably cautious about using these materials in such applications where compression is the dominant mode of loading. Often used as face sheets in sandwich components, composite materials are steadily gaining popularity as spacecraft and marine construction materials. These composite components are used in aircraft primary and secondary structures, such as primary wing and fuselage structures of Boeing aircraft, floor panels for military aircraft, structure of racing yachts, superstructures for fast passenger ferries, control surfaces of submarines and internal ship hull stiffeners where they undergo axial compression as the main loading condition. Because of the multitude of damage mechanisms involved, composite materials have superior specific energy absorption properties when compared with common engineering metallic materials. Composites are therefore the preferred materials to be used in impact-related applications in which high specific energy absorption is a critical design criterion. 1 Chapter 1: Introduction There are several issues related to the behaviour of composite materials that need to be addressed. One of the important issues is the prediction of failure and damage propagation from an imperfection under compression. Composite components are likely to contain many voids (stress raiser sites) introduced either intentionally, e.g. cut-outs and fastener holes, or unintentionally, e.g. fabrication defects and imperfections, mismatch between different fibre orientations, misalignment of fibres and accidental damage. These defects can act as potential sites for development and propagation of cracks under compressive loading. Despite these problems, the reduction of the mechanical properties of composite components caused by defects and the subsequent reduction in their compressive load carrying capacity have received little attention in the literature. It is well known that the tensile strength of composite materials stem from the high tensile strength of the fibres. In compression, however, the matrix and fibre-matrix interface play important roles because they must provide lateral support for the fibres and prevent them from undergoing microbuckling. Furthermore, under compressive loads or impact loads, composite materials can deform and fail in combinations of several interacting failure modes such as delamination and fibre microbuckling that emanate from voids. While much research work has been conducted on the tensile failure and damage behaviour of composite panels containing holes, relatively little work has been done on the effects of holes and voids on the compressive behaviour of composite components. Therefore there is a need for a more in-depth study of the nonlinear behaviour of composite materials under compressive loading in order to gain a better understanding of their load carrying capability in such loading scenarios. This understanding can then guide the formulation of physically realistic constitutive models that when implemented in finite element codes can be used to simulate the response of composite structures under compressive loads. As stated before, one of the main advantages of using composite materials is their high specific energy absorption capability. For example, tubular structures made of composite materials are being considered as frame structures in automotive applications. A key requirement in such cases is to absorb the impact energy of accidental collisions through axial crushing of the tubes. Under such loading scenarios, a significant volume of the 2 Chapter I: Introduction composite component undergoes a full range of compressive stress-strain response and fails completely. Failure associated with impact usually appears in the form of one or several combined failure mechanisms such as fibre fracture, matrix cracking, debonding and delamination. No matter which mechanism prevails depending on the specific application, damage always results in a reduction in strength and stiffness of the structure. To assess impact energy absorption capability or damage tolerance of a structural component, the micromechanical behaviour of failure process under compression needs to be well understood and models need to be developed that capture this behaviour in numerical analyses of such structures. 3 Chapter 1: Introduction 1.2. Motivation During the past forty years, many studies have been focused to improve our understanding of compressive failure in composite materials. It is well recognized that the compressive failure of these materials is mainly caused by a combination of localized buckling of fibres, kinking, and delamination (Sivashanker, 1998). Recently, various micromechanical aspects of the kinking failure as the main compressive failure mechanism have appeared in the literature. Based on the studies of Sutcliffe and Fleck (1994) and Fleck et al (1997), a good understanding of the fracture mechanics aspects of kinking has been gained. These studies revealed the crack- like behaviour of the kink band which was confirmed by Moran et al (1995) who discovered the phenomenon of band broadening. A brief review of literature, reveals the lack of a consistent description for composite materials failure under compression and tension. Although the micromechanical behaviour of composite materials under tensile and compressive loading induced damage have been studied thoroughly, little has been done to relate the damage propagation behaviour of these materials in tension to damage propagation in compression. Also little work has been reported on models that take into account the micromechanical behaviour of composites under load reversals from tension to compression and vice versa. Recently, a plane-stress continuum damage mechanics based model for composite materials (CODAM) was developed at UBC ( Williams, 1998; Williams et al., 2003; Floyd, 2004). This model was implemented in the non-linear finite element code LS-DYNA and its predictive capability for size effect was evaluated for tensile loading applications (Floyd, 2004). It has been shown that CODAM predicts a size effect for the tension specimen geometries.that is in good agreement with the experimentally observed size effect. The goal of this thesis is to construct a physically meaningful model within the framework of CODAM, capable of describing the micromechanical damage propagation behaviour of composites under different loading conditions from initiation of damage all 4 Chapter I: Introduction the way to its fully saturated damage state. It is also desired to relate the damage effect in one loading condition, such as axial compression, to the response of material in other loading conditions. This research is mainly focused on studying the response of composites under tension, compression and reversal loadings. The practical importance of the model is that it will enable the designers to predict the performance of large structures by doing experiments on small composite parts rather than performing the costly full-scale experiments. Besides the cost effective method of predicting the performance, we can obtain the desired safety level on large structures by changing and improving the parameters that matter most, rather than improving everything, which could be a costly proposition. 5 Chapter 1: Introduction 1.3. An Outline of This Study An outline of this thesis is briefly presented here. In Chapter 2, a literature review of composite materials related to the compressive failure is presented. Both unidirectional and multidirectional composite failures are considered. This chapter is concluded with a detailed micromechanical description of the behaviour of composite materials during the compressive failure. In Chapter 3, a physically meaningful damage mechanics model, namely an analog model, to simulate the response of composite materials under compression, tension, and cyclic loads (load reversals) is constructed. Based on the response of various elements used to construct the model appropriate formulations are derived to relate the characteristic parameters of material to the response of composites. In Chapter 4, some issues, which need to be considered before implementing the analog model in LS-DYNA, are discussed. These issues are focused on damage growth, energy concept, and scaling law for the compressive response of composites. Chapter 5 is dedicated to verification and validation of the analog model. Verification consists of simulating the response of a laminate to check the model implemented in LS-DYNA. The successful verification is obtained by studying the stress-strain response of one element. Validation also consists of comparing the predictions the experimental results obtained from the literature consisting of sandwich panels under eccentric compression (Bayldon, 2003a, 2003b) and open hole panels under compression (Soutis et al., 2002). Finally, Chapter 6 presents conclusions and recommendations for future work. 6 Chapter 2: Compressive Failure of Fibre Composites 2.1. Introduction Compressive failure of composite materials has been an area of considerable research over the past forty years. This is mainly due to the fact that compressive failure is a design-limiting feature of fibre composite materials. For instance, the compressive strengths of unidirectional carbon-epoxy laminates are often less than 60% of their tensile strengths as a result of compressive failure mechanisms that are rather sensitive to fibre misalignment. It is acknowledged that the compressive failure of composites is usually caused by microbuckling of fibres, also known as kinking. This failure mode is particularly sensitive to the shear properties of the matrix and also the degree of misalignment of the fibres (Argon, 1972; Budiansky, 1983; Fleck, 1997). Fibre misalignment usually causes the kinking to localize in areas of imperfection and therefore reduces the compressive strength of composite materials compared to their tensile strength. In this chapter, the main compressive failure mechanisms of composites for both unidirectional and multidirectional laminates are summarized. Emphasis is placed on studying the kinking failure as the main failure mechanism of unidirectional fibre composites. Micromechanics of kinking propagation are also described in the following section. For multidirectional composites the main failure mechanism is usually a combination of kinking, delamination and off-axis matrix cracking. 7 Chapter 2: Compressive Failure of Fibre Composites 2.2. Failure Mechanisms in Unidirectional Composites Long fibre composites are usually designed taking advantage of their high axial stiffness and strength. For this reason, the fibres are usually made from strong and stiff materials, such as graphite or silica glass. At the same time, to provide the composite with adequate in-plane strength and ductility, the matrix needs to possess high toughness properties. Also as described before, the axial compressive strength of composites is relatively low due to weak matrix properties and misalignment of fibres. These two sources are responsible for most of the individual mechanisms of compressive failure. In 1993, Budiansky and Fleck presented a study on compressive failure of fibre composites. They focused on studying kinking as the main failure mechanism in unidirectional laminates. Subsequently, Fleck (1997) presented a comprehensive study on compressive failure mechanisms of unidirectional and notched multidirectional composite laminates. In his study, he categorized the main mechanisms of compressive failure for unidirectional composites as follows (see Figure 2-1). 1- Elastic microbuckling. This is a shear buckling instability, in which the matrix deforms in simple shear. 2- Fibre crushing. This failure is due to the waviness of the fibres embedded in a soft matrix and occurs at the fibre level. The compliant matrix provides inefficient lateral support for the fibre, which leads to fibre buckling. 3- Splitting. In this mechanism, for matrix materials with low toughness, matrix cracks form parallel to the main fibre direction. 4- Buckle delamination. This is actually debonding and buckling of a surface layer from the sub-surface. This failure mechanism often occurs due to low matrix toughness and a large surface flaw. Also, post-impact compressive loading can induce buckle-delamination growth due to large debonding after the impact. 8 Chapter 2: Compressive Failure of Fibre Composites 5- Shear band formation. In this failure, matrix cracking or yielding occurs in a band oriented at about 45° to the loading axis. 6- Plastic microbuckling or kinking. This mechanism is a shear instability, which occurs when matrix deforms nbnlinearly. After cracking or yielding of the matrix, instability of fibres leads to the fibre rotation and kink band formation. Each of these mechanisms is reviewed in the following sections. (a) (d) 0 single fiber fiber -(«) shear planes Figure 2-1 The main compressive failure mechanisms of unidirectional composites, (a) Elastic microbuckling, (b) Kinking, (c) Fibre crushing, (d) Splitting, (e) Buckle delamination, (f) Shear band formation (Fleck, 1997). 9 Chapter 2: Compressive Failure of Fibre Composites 2.2.1. Elastic Microbuckl ing Rosen (1965) pioneered an analysis to model elastic microbuckling by assuming elastic bending of the fibres and elastic shearing of the matrix. Rosen assumed that elastic microbuckling occurs in two possible modes: Either a transverse buckling mode, where matrix undergoes strains transverse to the axial direction or a shear buckling mode, where the matrix shears in the axial direction (see Figure 2-1). In practice, with fibre volume fraction greater than 0.3, the shear mode gives lower failure loads. Assuming there is no misalignment in fibres and no fibre bending stiffness, Rosen calculated the critical load at which the fibres deflect into a sinusoidal shape. This formulation was later confirmed by Jelf and Fleck (1992) by modeling composite materials made from spaghetti rods in a silicone matrix. Rosen's analysis (1965) has been used and modified in many studies, e.g. Hahn and Williams, 1986; Johnson and Ellen, 1974, 1975a, b, 1976. 2.2.2. Fibre Crushing Fibre crushing occurs when the matrix is sufficiently stiff and strong to inhibit fibre microbuckling. This mechanism occurs when the uniaxial strain in the composite equals the crushing strain, which depends on the material properties and the geometry. Typical value of crushing strain for Kevlar and carbon fibres is 0.5% (Fleck, 1997). Fibre crushing in the form of kinking can also occur within fibres of small width (Piggott and Harris, 1980, Gibson and Ashby, 1988). 2.2.3. Splitting Under uniaxial loading, this mechanism occurs when the stiffness of the matrix is greater than the stiffness of the fibres and when the composite has a low toughness and high porosity (Kaute et al., 1996). It is a tensile cracking mode developed from imperfections such as voids or inclined flaws. This can happen in compression as well, where under increasing compressive loads tensile stresses around imperfections increase and the specimen fails along a macroscopic shear band. 10 Chapter 2: Compressive Failure of Fibre Composites 2.2.4. Buckle Delamination Delamination in unidirectional laminates mainly occurs when a surface layer is debonded over a certain length. Debonding is due to one of the following reasons: 1. The manufacturing process 2. Surface impact 3. Out of plane loading by waviness of the fibres Unlike multidirectional laminates, where delamination can occur frequently due to the lay-up, in unidirectional laminates with no imperfections the occurrence of this failure mode is rare. In fact, studies on delamination have primarily focused on the effect of imperfections or the effect of impact on development of delamination in unidirectional laminates (see for example Kim and Sham, 2000). Many studies also have been focused on characterizing the post-buckling behaviour of delaminated laminates (Kardomateas and Schmueser, 1988; Kardomateas, 1993; Gu and Chattopadhyay, 1995). 2.2.5. Shear Band Formation This failure mechanism may occur for polymer-matrix fibre composites with very low fibre volume fraction. Fried (1963) showed that failure occurs in a band oriented at about 45° with respect to the loading direction (Figure 2-1). This failure mode essentially occurs for unreinforced matrix material and is not expected to happen for composites with practical fibre volume fractions. 11 Chapter 2: Compressive Failure of Fibre Composites 2.2.6. Plastic Microbuckling or Kinking This is the dominant failure mode for unidirectional composite laminates. The compressive strength is controlled by plastic shear deformation in the matrix and fibre misalignment. During the kinking process the supporting matrix material undergoes plastic deformation. As a result, this failure mechanism is also called plastic microbuckling. In this section, a review of the studies on kinking process done in the literature are presented. Also, micromechanics of various stages during kink band formation for both thermosets and thermoplastics are discussed. 2.2.6.1. History of Kink Band Analysis Kinking as one of the main compressive failure mechanisms has been extensively studied in the literature in the past forty years. As mentioned before, Rosen (1965) presented a simple formulation to calculate the compressive strength of elastic fibres embedded in an elastic matrix, while neglecting the bending stiffness of the fibres. Argon (1972) extended Rosen's formula by considering plastic yielding and misalignment of the fibres. In this study, he derived the following formula for the compressive strength of composites failing by kinking mode: a = k0/j0 (2-1) where, ko is the inter-laminar shear strength and ^ 0 is the initial misalignment of the fibres. Budiansky (1983) extended Argon's formula by assuming an elastic-perfectly plastic material and proposed the following formula for kinking critical stress: a = - — (2-2) in which, yo is the composite yield strain. Budiansky assumed that kink band inclination, (3, is equal to zero (Figure 2-2). Based on this work, several studies were carried out to measure the misalignment in the fibres. For example Yurgatis (1987) found that most of 12 Chapter 2: Compressive Failure of Fibre Composites the fibres in a carbon fibre-PEEK unidirectional composite were oriented within ±3° of the mean fibre direction. Figure 2-2 Kink band formation under compressive load for a unidirectional laminate, (a) Before failure, (b) After failure. Various improvements were made by subsequent studies to analyse the post-buckling behaviour of the fibres, inelastic behaviour of the matrix, and imperfections. These works provided an understanding of the factors governing the kink band initiation, inclination of the kink band, height of the kink band, fibre rotation, shape of the assumed yield surface, kink band propagation, and fracture toughness of this failure mechanism (Fleck and Budiansky, 1993; Soutis and Fleck, 1993; Fleck et al., 1995; Kyriakides et al., 1995; Moran and Shih, 1995, 1996, 1998; Sivashanker and Fleck, 1996; Sutcliffe and Fleck, 1994, 1997; Sutcliffe et al., 1996; Shu and Fleck, 1997; Vogler and Kyriakides, 1997, 1999, 2001; Budiansky et al., 1998; Sivashanker, 1998; Soutis et al., 1999; Bazant et al., 1999; Berbinau et al., 1999; Niu and Talerja, 2000). Several studies have been carried out to understand the micromechanics of this failure mechanism. Guynn and Bradley (1989) compared the damage zone to a crack with a plastic zone. They applied the Dugdale model (1960) to predict the size of the buckled 13 Chapter 2: Compressive Failure of Fibre Composites region as a function of the compressive load for both carbon-epoxy and carbon-PEEK laminates. They concluded that the Dugdale model, in which a constant stress was assumed over the damage zone, is unable to accurately predict the compressive failure stress. Following this study, Soutis et al. (1991) presented a cohesive zone model to analyse the damage propagation and failure of the orthotropic laminates due to kink band formation. In their study, the damage zone was represented by a line crack loaded on its faces with a traction, which varies linearly with the crack displacement. The experimental studies of Sutcliffe and Fleck (1994), Sivashanker et al. (1996) and Fleck et al. (1997), based on strain measurement of the kink band, showed that the axial stress across the kink band decreases with the distance from the band front and reaches a plateau equal to about 50 percent of the maximum stress. Such a crack-like behaviour of the band was then confirmed by Moran and Shih (1995). They also discovered band broadening; whereby the damage zone grows under a constant remote stress. Budiansky, Fleck and Amazigo (1997) presented a formula for the kink band critical stress based on the kink band fracture energy by analysing the propagation of an out-of-plane kink band (Figure 2-3). Bazant et al. (1999) showed that this formula does not account for the compressive strength size effect of the specimen. In this study, they calculated the fracture energy from the area under the stress-strain curve and above the plateau stress based on the analysis of Palmer and Rice (1973). In fact in the Bazant et al. study (1999), the formation of kink band was assumed to be the result of shear cracking of the matrix, while neglecting the effect of the bending stiffness of the fibres. Finally, they derived a formula to account for the specimen size effect on the compressive strength. Niu and Talreja (2000), reviewed the classical results of kink band compressive strength. They reached the previous results by using a generalized Timoshenko beam model that was developed for a two-dimensional periodic matrix-fibre-matrix laminate. They presented a matrix shear instability based mechanism for kink band formation. They also modified the Argon-Budiansky kinking formula to account for the combination of the axial compressive stress and transverse shear perturbation. 14 Chapter 2: Compressive Failure of Fibre Composites Chapter 2: Compressive Failure of Fibre Composites 2.2.6.2. Kinking Formation Several researchers, such as Moran et al. (1995,1998), Liu et al. (1996), Sivashanker et al. (1996), and Sivashanker (1998), have studied the micromechanical stages of kink band formation process in unidirectional thermoset and thermoplastic matrix composites. Various stages in the kink process were recorded by video camera, which is depicted in Figure 2-4 for the formation of an out-of-plane kink band in the Moran and Shih study (1995). Displacement (d) Figure 2-4 Overall behaviour of unidirectional fibre composites under compressive loads in the fibre direction when kinking occurs (Moran et al., 1995) The initial stress-strain response is linear elastic. However, at some point the matrix in the highly stressed zone in front of the notch begins to fail and loading curve begins to 16 Chapter 2: Compressive Failure of Fibre Composites show some non-linearity. The failure mechanism in the matrix depends on the type of material, i.e. thermoset or thermoplastic. For thermoset materials, such as epoxy, matrix cracking occurs under a combination of axial and shear loading. On the other hand, for thermoplastics such as PEEK, matrix begins to flow plastically. Moran et al. (1995) named this early inelastic deformation as "incipient kinking". The incipient kinking is followed by the early stage of kink band formation, which involves progressive fibre bending/rotation and plastic shearing or cracking of the matrix within a narrow band. Fleck et al. (1995) showed that this initial damage height (Figure 2-5) is of the order of 10-15 times a fibre diameter. This was also confirmed by Moran et al. Damage Height Figure 2-5 Initial damage height in kinking process (Fleck et al., 1995). Fibres rotate under compressive load. During these early stages, the peak or critical stress is reached. Experimental observation (Moran et al., 1995, 1998; Liu et al., 1996) showed that the peak stress is typically no more than 5 or 10% higher than the incipient kinking stress. This is because after the matrix cracks, the fibres cannot carry any more loads due to instability. This leads to the peak stress being reached just after the matrix cracks or yields. The progressive matrix failure in the form of cracking or yielding and fibre bending leads to a strain-softening response of the material within the band and the load subsequently begins to drop. During this stage, at some point, due to the stiffening of the composite 17 Chapter 2: Compressive Failure of Fibre Composites shearing response at large shear strains, the rotation of the fibres stops. This phenomenon is known as "fibre lock-up", shown in Figure 2-6. Time T< T 1 A T, T T i 2 i , Fibre rotation terminated by matrix strain hardening Rapid fibre i rotation caused1 by geometric ( softening i Gradual fibre bending and rotation Figure 2-6 Schematic showing the behaviour of a single fibre during the early stages of kink band formation (Moran et al., 1995). After the lock-up stage, the kink band starts to broaden. Band broadening is accompanied by the rotation of the fibres, and matrix cracking or yielding moving towards the undamaged interior material. During this stage, the kink band orientation remains fixed. The final height of the kink band is set when the fibres snap (Sivashanker et. al., 1996). Moran et al. reported a width of approximately 30-50 times the fibre diameter for the final band height, while Sivashanker et al. (1996) observed a factor of around 100. 18 Chapter 2: Compressive Failure of Fibre Composites 2.3. Failure Mechanisms in Angle-ply Composites Some studies have been done on the micromechanics of damage propagation in multidirectional composites in compression (Soutis et al., 1993, 2002; Sivashanker and Bag, 2001; Sivashanker 2001). Sivashanker (2001) presented a detailed explanation of micromechanical behaviour of three different multidirectional lay-ups of carbon-fibre epoxy composites. He identified the compressive failure process as out-of-plane microbuckling of the [0], off-axis layer damage and interface delamination between layers. He observed that microbuckling in multidirectional laminates is accompanied by delamination in the vicinity of the microbuckle zone. He reported the main feature of the fractured multidirectional laminates as a zigzag pattern in off-axis plies, which he classified as parallel splits along the fibre direction and out-of-plane kink band normal to the splits. For the band broadening stage, he reported a growth of delamination combined with the growth of kink band height. In his study, microbuckling and delamination are never presented independently, but always as parts of a damage unit. It is further proposed that delamination is not a cause for microbuckling, but in fact is the consequence of it. He also observed and measured the constant plateau stress in the band broadening stage. He mentioned that this constant bridging stress is mainly associated with steady-state band broadening in [0] layers and steady-state delamination crack growth under a constant remote stress. Microscopic images of the Sivashanker's study are presented in Figure 2-7. 19 Chapter 2: Compressive Failure of Fibre Composites Figure 2-7 (a) Overview of arrested damage zone, (b) Band broadening in progress, (c) A close-up view of the microbuckle band, (d) 45 degree off-axis microbuckle band linking the 0/45 degree delaminated interfaces, (e) Out-of-plane 0 degree microbuckle band extending into the 45 degree off axis layer. (Sivashanker, 2001) 20 Chapter 2: Compressive Failure of Fibre Composites 2.4. Summary A brief review of studies on compressive failure of composites was presented in this chapter. While extensive research has been done on the unidirectional laminates failure under compressive loads to understand their micromechanical behaviour, little has been done with respect to compressive failure of multidirectional laminates. The work by Sivashanker (2001) indicates that compressive failure mechanism is a combination of kinking, delamination and off-axis matrix cracking. Lack of consistent models for unidirectional and multidirectional composites failure under compressive loads is clear in the literature. Soutis et al. (1993, 2002) applied his crack bridging model to analyse the failure of notched multidirectional laminates under compressive loads. However, there is no detailed explanation on how the model relates to the micromechanical behaviour. Band broadening, which is an important stage in the compressive failure has not been studied thoroughly in the past and there is no physical model to explain and account for this stage. The need to construct a model to analyse the compressive failure of multidirectional laminates that could relate to the observed micromechanical behaviour such as incipient kinking and kink band broadening is the driving force to build an analog mechanical model of the response in the next chapter. 21 Chapter 3: Analog Model 3.1. Introduction This chapter is mainly focused on constructing a physically meaningful "analog model", that is able to simulate the behaviour of composite materials during the failure process in tension, compression and under reversal loads. The main advantage of constructing such a model is that it will enable us to explain and study the true behaviour of composite materials under various loading conditions. As mentioned in Chapter 2, a comprehensive model to simulate the micromechanical behaviour of composite materials during the compressive failure does not exist. The main reason for this is perhaps that various failure mechanisms exist for different loading conditions and geometries. Even for a specific geometry and loading condition, the lay up of the laminate plays an important role in the damage propagation process. All these effects make it difficult to construct a physically meaningful model capable of capturing the true behaviour of composite materials. The method that has been used in this chapter to construct the analog model involves incorporating the compressive behaviour of composite material during the failure process into the existing tensile failure model, CODAM (Williams, 1998; Williams et al. 2003; Floyd, 2004). Data from the literature was used to distinguish between compressive and tensile failure mechanisms. There are some primary differences between the failure mechanisms in tension and compression. Tensile failure is due to fibre breakage and matrix cracking or yielding for thermoset and thermoplastic matrix materials respectively (Floyd, 2004). Often however, depending on the lay-up of a multidirectional laminate, the failure is a combination of delamination and fibre breakage. 22 Chapter 3: Analog Model For the compressive failure on the other hand, the primary failure mechanism is kink band formation which is the formation of matrix cracking or yielding and fibre rotation (McGregor, 2005). Like tensile failure, sometimes a combination of different failure mechanisms such as kinking and delamination lead to the complete failure (Sivashanker, 2001; Sivashanker and Bag, 2001). But unlike tensile failure, as Moran and Shih (1998) observed, there is little evidence of fibre breakage during the formation of the kink band. Fibre breaking in compression was observed after the formation of the first kink band during the band broadening process by Sivashanker et al. (1996) and Vogler and Kyriakides (1997). After the formation of the kink band, the damage zone, under further application of displacement, propagates into the undamaged interior materials. This broadening of the kink band which combines with fibre breakage and splitting results in a plateau stress after the softening in the overall stress-strain response of the composite material. This is another difference between compressive and tensile failure. Upon unloading the damaged material, there will be a permanent deformation in the compressive specimen which is not the case in the tensile specimen. The presence of the constant deformation in the compressive damage process is due to the friction between the new cracked surfaces and rotation of the fibres in the kink band. After rotation of the fibres in the kink band, the new cracked surfaces will appear. Upon unloading the friction between these cracked surfaces leads to plastic deformation. The construction of the analog model has to be such that it will incorporate all these behaviours of composites material under compressive loads using basic elements, such as springs and sliders, which will be chosen based on the behaviour of composite materials in the damage process. Besides slider and spring elements, we need some other basic repeating elements in the model to simulate band broadening and plastic deformation as described in the next section. The combination of all elements will result in construction of two primary collections or boxes of elements which can then simulate the overall behaviour of a composite material under tension, compression and reversal loads. Each of these boxes represents a physical 23 Chapter 3: Analog Model behaviour of the compressive failure such as matrix cracking as described in the following sections. After building the boxes, the model will be presented and its response will be examined under compression, tension and reversal loads. The final section will be dedicated to the model behaviour explanation. 24 Chapter 3: Analog Model 3.2. Elements Each of the elements used in constructing the analog model will be described in this section. It is noteworthy that in addition to the normal elements used in the literature, such as springs and sliders, introduction of new elements has been necessary in order to describe the behaviour of composite materials. The response of each element will be described in detail. 3.2.1. Rigid Link This element doesn't deform, hence called rigid, and transfers the applied load to the next adjacent element. The governing equation for this element is given below. F = F„, (3-1) where, F will be used to denote the applied load. A schematic of the element behaviour is also shown in Figure 3-1 below. This element will be shown with a bold solid line in the analog model. ' + 0 0 Figure 3-1 The behaviour of a rigid link element under compressive and tensile loads. 3.2.2. Spring and T/C Spring Spring element is the classic linear spring, which can be represented with the following equation: 25 Chapter 3: Analog Model F = kS (3-2) where, k represents the spring stiffness. The behaviour of a spring is shown in Figure 3-2. F A Figure 3-2 The symbol and behaviour of a spring element under compressive and tensile loads. Another form of spring used here is the so-called Tension/Compression (T/C) spring, which unlike a normal spring has different stiffnesses under tension and compression. Schematics of element symbols and element behaviours are shown in Figure 3-3. F Figure 3-3 The symbol and behaviour of a T/C spring element under compressive and tensile loads. 26 Chapter 3: Analog Model 3.2.3. Slider This element is used to show the state of band broadening in kink band failure or plastic yielding. In fact in the broadening state, under a constant load, damage propagates into the interior undamaged area (Moran and Shih, 1996). This element has a rigid-perfectly plastic behaviour. In other words, it acts as a rigid link until reaching a limit load and then it yields and exhibits a zero tangential stiffness. This element is formulated as follows: FapP KP|<N) -F.. 5 = 0 S>0 S<0 (3-3) where, Fy is the yield load. Schematics of the element symbol and behaviour are shown side-by-side as follows. F Figure 3-4 The symbol and behaviour of a slider element under compressive and tensile loads. 3.2.4. Fuse and T/C Fuse The fuse element is similar to the slider, but after reaching the limit load it fails and cannot carry any loads. The definition of this element is necessary to model the matrix and fibre cracking and breakage. 27 Chapter 3: Analog Model T/C Fuse has two different values for compressive and tensile failures. This element will be used to simulate fibre failure. The difference in tension and compression is because of the fact that in compression a fibre can fail due to a variety of phenomena such as instability, while in tension the failure is due to the tensile breakage. These two elements are defined as follows: Fapp {-Fc<FapP <FJ) 0 {Fapp=FT) 0 ( F a p p = - F c ) 5 = 0 5<0 (3-4) where, indices T and C denote limit states in tension and compression, respectively. Schematics of element symbols and element behaviours are shown in Figure 3-5 and Figure 3-6. F t pT_ p c Figure 3-5 The symbol and behaviour of a fuse element under compressive and tensile loads. F F T M FT4= FC -Fc Figure 3-6 The symbol and behaviour of a T/C fuse element under compressive and tensile loads. , 28 Chapter 3: Analog Model 3.2.5. Gap This element represents a distance, shown as 5 , between two elements. A gap element cannot carry any tensile loads but if that distance is reduced to zero as a result of compressive deformation, the gap can close and become a rigid link and thereafter can carry only compressive loads. The definition of this element is necessary to simulate the behaviour of the damaged material in compression. In compression, after the formation of a damaged zone the material can carry a specific amount of load due to friction or yielding in the damaged matrix. The governing equation of this element is as follows: f 0 8 > Se F = \F 8 = 8 (3"5) A schematic of element symbol and element behaviour is shown in Figure 3-7. Figure 3-7 The symbol and behaviour of a gap element under compressive and tensile loads. 3.2.6. Lock As mentioned before, one difference between compressive failure and tensile failure is the presence of plastic deformation in compression in unloading. This requires us to define the lock element. This element acts as a rigid link when the applied load is less than a defined tensile load capacity, FT. Once this tensile limit has been reached the element splits apart and does not transfer any more load unless it is forced back to zero 29 Chapter 3: Analog Model deformation by other elements. For this element we also define a gap, Svg, before the load carrying capacity is reached. This element is defined as follows: \FapP(Fapp<Fv) S=5y \0(Fapp=Fv) 5>S, (3-6) A schematic of the both element symbol and behaviour is shown below. I o FV4I Figure 3-8 The symbol and behaviour of a lock element under compressive and tensile loads. 30 Chapter 3: Analog Model 3.3. Collection of Elements or Boxes Element boxes are simply a collection or combination of previously defined elements in parallel or series and are able to model more complex behaviour. Two boxes are currently defined in the analog model, called the 'laminate box' and the 'rubble box'. Each of these boxes is used to represent one aspect of the physical behaviour of the composite during damage; the laminate box models the behaviour of the intact laminate and the rubble box represents the behaviour of the damaged material in compression. The analog model can then be constructed as a combination of these two boxes and some other simple elements. In this section a mathematical representation of the boxes is presented. It is noteworthy that the laminate box is an analog model which reacts differently under compressive and tensile loads. The rubble box on the other hand, only reacts under the compressive loads. 3.3.1. Laminate Box To model the response of the intact laminate as damage progresses the laminate box is introduced. As discussed in Chapter 2, as damage progresses in the laminate the overall stress-strain response of the laminate shows a softening behaviour, which is different under tension and compression. Under a tensile load, the softening is due to the reduction in effective area due to fibre and matrix cracking. In compression, on the other hand, this softening happens due to the matrix cracking or yielding and fibre instability. In addition, the instability and rotation of the fibres combined with fibre cracking and splitting contribute to the overall softening behaviour under compression. In fact, in compression, matrix cracking leads to the instability of fibres which is not the case in tension. The overall reduction in the laminate tangential stiffness which forms the softening behaviour has been modeled using T/C fuses and T/C springs. The choice of T/C spring elements is based on the idea that the composite laminate acts differently under tension and compression. The laminate box consists of a collection of infinite number of basic units in parallel as shown in Figure 3-9. Each basic unit consists of two T/C spring elements and two T/C 31 Chapter 3: Analog Model fuse elements as shown in Figure 3-10 below. One of the T/C springs and T/C fuses in series, denoted as "m", simulates the response of the matrix, while the other series, "f\ represents the behaviour of the fibres. The element notations are similar to what was presented before. m f m f m f 777777777777777777777777777777777777777777777777-1 2 n Figure 3-9 Laminate box and the arrangement of basic units. A l l the elements are T /C springs and T /C fuses. 777777 Figure 3-10 Basic unit as used in the laminate box. "w" represents the matrix response and " / ' represents the fibre response in a representative volume element. 32 Chapter 3: Analog Model The overall response of the basic unit can be easily obtained and is shown as follows. F f fails Figure 3-11 Force-displacement response of a basic element in the laminate box. 3.3.1.1. Laminate Box Equations The equations describing the behaviour of the matrix response under tensile loads will now be derived as an example. Other responses, such as fibre under compressive loads can also be derived following the same procedure. First, the springs and fuses for the matrix and the fibre are grouped together and labelled as "w" and "f \ respectively, in the following figure. 33 Chapter 3: Analog Model m f FT \//777777777777777777777777777777777777777777777T l \ 2 3 A 1 2 3 n Figure 3-12 Grouping of matrix and fibre elements in the laminate box under tensile loads. To derive the appropriate equations for group m, the following simplified notation is used: n F, AF 8, = number of springs = kTm = spring stiffness (m) = Fl = tensile strength of the first fuse (m) = AFm = increment of the tensile fuse strength (m) = 5]Lm = deformation at first (initiated) spring failure (ni) = 5TSL = deformation at last (saturated) spring failure (m) The response of the finite number of elements in group m is therefore given as follows: n(ke)S 5<5, {n-q\kt)S S,<S<SS 0 8 > S.. (3-7) where, q is the number of failed springs. 34 Chapter 3: Analog Model As an example, the general response for four elements is shown in Figure 3-13. F = nke5, + q= l q=2 q=3 Figure 3-13 The response of group " A M " under tensile loads in which the group consists of four elements. We can also replace q by ^'J^g- After substituting this value and further simplification, we arrive at the following equation for the response of a finite number of elements in group m. F f r p 5 A AF/ k.8 Sr < S < 8S (3-8) The next step is to determine the response of group m as the number of elements approaches infinity. This can be easily done by taking the limit of the Equation 3-8 when n approaches infinity. The result is: F = k„ 5, <S <5S (3-9) where, ktot represents the total stiffness of all elements. By defining the following constants, the total response shown in Figure 3-14 can be attained. 3 5 Chapter 3: Analog Model a = nke = km c = Kts, b 5 c / \ a \ 8, 5S Figure 3-14 The response of group under tensile loads in which group consists of an infinite number of elements. The response of group / is identical and we only need to replace the corresponding parameters. The overall response of the laminate box in tension can then be found by taking into account that m and /are in parallel. The constitutive equation of the laminate box under tension is shown below. F = k„ 8 + k„ 5 "- m , L f -si 8 + kl S •m J f 8lL -5 ^ sLn, -5L 8 + k] ( ST -si •f J 0 + kjL , 8V, V S L f • S'lLf si <s<si, SL„, < S < S S L 5 > Si (3-11) where, k\ = nke = Total initial tensile stiffness (m) 36 Chapter 3: Analog Model 8TU = Tensile deformation of first spring failure (initiation) (m) 8'SI = Tensile deformation of last spring failure (saturation) (m) k\L f = nke^ = Total initial tensile stiffness if) SjL = Tensile deformation of first spring failure (initiation) if) S'sl = Tensile deformation of last spring failure (saturation) if) In the overall response, when the loads are applied the material initially shows a linear response, which indicates that there is no damage. Then, by increasing the load damage starts at some point. Usually the matrix cracking in tension occurs before the fibre cracking. This means that the damage initiation strain for matrix in tension is smaller than that of the fibre's. Therefore, after the load reaches the matrix damage initiation strain the overall response becomes a nonlinear function. This nonlinear function is simulated by using a parabolic function in the analog model as a consequence of our previous assumptions. After damage initiation, the damage propagation results in decreasing tangential stiffness and the overall softening behaviour. For the above equations, considering the fact that matrix damage initiation strain is less than the corresponding value for fibre and assuming that the matrix damage saturates before the fibre is damaged, the total response of the laminate box and each element group under tensile loads (displacement controlled) can be shown schematically in Figure 3-15. F • 8 Figure 3-15 Force-displacement response of the laminate box under displacement controlled tension, m and /represent the response of the matrix and fibre respectively. 37 Chapter 3: Analog Model To obtain the response of the laminate box under compression the same procedure can be followed. Therefore, assuming the box under compression shown in Figure 3-16, the constitutive equations are derived and presented as Equations 3-12. F c m f 1 2 3 n 1 2 3 n Figure 3-16 Grouping of elements in the laminate box under displacement controlled compression. F = kf, S + kn .8 USL , ( 8C USL, •si -8 8 + kfL f8 m J \ 8 + k' f 8% -8 m J •sfLfJ 0 + k, ( Sc -5- ^ •f ) 8< < 8 < <8< 'SL <S< <8 'SL (3-12) The following parameters have been used in the above equations: kfL = Total initial compressive stiffness (m) 38 Chapter 3: Analog Model Sf, = Compressive deformation for first spring failure (initiation) (m) Sg, = Compressive deformation for last spring failure (saturation) (ni) kf, = Total initial compressive stiffness (J) 5fL = Compressive deformation for first spring failure (initiation) (f) Sg, f= Compressive deformation for last spring failure (saturation) (J) Schematics of the response of each element group and the laminate box under compressive displacements are shown in Figure 3-17 and Figure 3-18, respectively. Similar to the response under tension, the overall response is simply the sum of the matrix and fibre responses. It is also noteworthy that in compression, unlike tension, the effect of matrix cracking or yielding on the overall response is much more pronounced than the effect of fibre cracking. In other words, the loss of modulus due to matrix cracking is more than the loss of modulus due to fibre cracking. The reason for the greater role of matrix cracking in compressive failure is the increased fibre instability after the matrix is cracked, i.e. the loss of transverse matrix support leads to the rotation and failure of fibres. This is shown in the following figures. Figure 3-17 Individual responses of Group m and Group/of elements in the laminate box under compression. Group m represents the response of the matrix while Group / represents the response of the fibres. 39 Chapter 3: Analog Model Figure 3-18 Force-Displacement response of combination of Group m and Group / i n parallel, and subjected to displacement-controlled compression. It is also desirable to study the response of the laminate box under reverse loading. In such cases, the strain-stress response of a damaged specimen would be different in many aspects from the undamaged specimen. First difference is the loss of modulus in reverse loading. After one element in the box fails, the total stiffness of the laminate box decreases and it results in a strain softening behaviour of the laminate box. Upon unloading and reloading in the reverse direction, the failed element cannot carry any load. This means that after damage propagation, the modulus of the laminate box decreases permanently. Another difference is the change of the damage initiation strain in the damaged specimen. The laminate behaves linearly up to the previous strain in the cycle at which unloading took place. After reaching that strain, a new damage area propagates in the laminate and the response of the laminate follows a strain softening curve. In the laminate box, this behaviour has been simulated by using T/C fuses. Because in the previous loading cycle, all the fuses between the damage initiation and the strain at unloading failed; the new damage initiation strain would coincide with the strain value at the previously unloading point. 40 Chapter 3: Analog Model Lastly, the peak stresses are different. Upon re-loading a previously damaged specimen the peak stress drops. This drop is due to the fact that the stiffness of the damaged material is less than that of an undamaged material. In fact, the drop of the peak stress along with the change in the damage initiation and laminate modulus all show that the new damaged material is a softer material compared to the undamaged material (see McGregor, 2005 for more details). 41 Chapter 3: Analog Model 3.3.2. Rubble Box The rubble box represents the behaviour of the damaged material in compression. Upon reaching the damage initiation strain under compressive loads, the matrix starts to crack or yield resulting in the formation of rubble and cracked surfaces. After the damage propagates under further compressive loads, the friction between the newly cracked surfaces causes an increase in load carrying capacity of the specimen. This damaged material can carry compressive loads. This phenomenon, however, only occurs for damage propagation under compression. The behaviour of the rubble box can be represented with an analog model consisting of gap and spring elements to model the compressive load carrying response with an increasing stiffness. The gap is active before damage initiation. After damage initiates new surfaces form resulting in an increased stiffness of the damaged material. The rubble box cannot carry any load in tension as described above. This model is equivalent to an infinite number of parallel gaps and identical spring elements to represent the stiffening response of the compressed rubbles as illustrated in Figure 3-19. R /////////// Figure 3-19 A schematic of the Rubble Box and the basic elements inside it. . For this box, it is assumed that the gaps are different in size, but the difference between two neighbouring elements, i.e. the interval, is constant. Also, all spring elements are 42 Chapter 3: Analog Model assumed to have an identical stiffness, ke. Therefore, the interval and the total stiffness of the rubble box, equal to the sum of the stiffnesses of all the springs, are given by: A5 = (Se -5e )ln h„ b j K, = nh (3 -13 ) Thus, we can write the following equations for the rubble box, in which j is the number of springs for which the corresponding gaps have been closed: 7 = 0 J = l j = 2 j = n F , = 0 Fj=keAS F, = k AS + 2k AS J L F, = k„AS + 2k AS +... + nkAS (3 -14) where, the corresponding forces, Fj, are effective when each gap is closed. Therefore, in a generic step we can write down the following equations for the load and the total displacement: Fj=J0 + \)/2xktAS S=S +JA5 (3 -15 ) Using Equation 3-13 and Equation 3-15, we can write: j = (S-Sg])/AS = n(S-5g])/(5gn-Sg]) (3 -16) Now by combing the above equations, we can write the response of the rubble box for a finite number of gap and spring elements under compressive loads, as follows: (3 -17 ) 43 Chapter 3: Analog Model The constitutive equation of the rubble box for an infinite number of springs can be easily obtained by taking the limit of the above equation when n approaches infinity. The result is given below. 0 5 < \8n 2 (8SR-8m) SIR < \5\ < 5SR (3-18) kKUS -(SSR + SIR) 5 > SSI where: SIR: Initial Rubble Deformation = 5V * i SSR: Saturation of Rubble box = 5 ksR.- Final Rubble Stiffness = k A schematic of the rubble box behaviour and the manner in which its stiffness increases as the gaps close is shown in Figure 3-20. F ->SR HR Figure 3-20 Force-displacement and stiffness-displacement curves for an infinite number of units in the rubble box An issue that needs clarification is the role of the rubble box after saturation. After the saturation, which represents the maximum damaged area in the specimen being loaded, upon further loading the damage propagates into the interior undamaged material. In other words, unlike tension, the damage height starts to grow under compressive loads. 44 Chapter 3: Analog Model During unloading, the damaged material becomes uncompressed. This leads to a decrease in the friction force between the cracked surfaces, i.e. some jagged surfaces become disengaged. This behaviour has been simulated by opening of gaps after unloading. 45 Chapter 3: Analog Model 3.4. Analog Model In this section, the analog model capable of modeling the behaviour of composite material under compression, tension and load reversal is represented. This model is assembled using the elements and boxes that have already been defined. Each element simulates an aspect of composite micromechanical behaviour during the failure process, which will be described. We will also introduce the model formulations for various loading conditions. 3.4.1. Model Construction The proposed analog model of the composite material is shown below. In this model, a rubble box and a lock-spring element are placed in parallel, and in turn both are placed in series with a slider element to form the left part of the model shown in Figure 3-21. This part of the model is also in parallel with a laminate box. A summary of the analog model elements is given here. The laminate box simulates the behaviour of the intact laminate in the process of damage propagation and shows the 5 Figure 3-21 A schematic of the analog model which shows arrangement of elements and boxes. 46 Chapter 3: Analog Model strain-softening behaviour of the composite material under tensile and compressive loads. Group m and Group / of elements inside the laminate box represent the response of matrix and fibres, respectively. As discussed before, the response of the matrix and fibres in tension and compression are different and this difference is modeled by T/C spring elements. Matrix failure ( matrix cracking or yielding) is also simulated by fuse elements. The rubble box simulates the response of the damaged material under compression. After damage initiates, further loading causes more new cracked and jagged surfaces, which results in an increased friction force between the surfaces. This increased force is modeled by using a combination of gap-spring elements in parallel. The lock-spring element is used to simulate the plastic deformations under compressive loads. This deformation occurs due to the rotation of the fibres in the kink band. After the fibres rotate, the friction between the cracked surfaces doesn't allow the fibres to rotate back to their original direction upon unloading. This phenomenon only exists under compressive loading because there is no fibre rotation in tensile failure. The response of the slider is related to the response of the rubble box because we assumed that the saturation of the rubble box leads to yielding of the slider. After the saturation of the first kink band, the kink band propagates into the interior undamaged material to form additional kink bands. As mentioned in the previous chapter, many experiments (Moran et al., 1995; Liu et al., 1996; Moran and Shih, 1998) have confirmed the band broadening phenomenon. The slider simulates the damage band propagation. After application of compressive load, an increase of force in the rubble box and spring leads to activation of the slider, since the yielding strain is assumed to coincide with the saturation strain of the rubble box. This assumption also means that the saturation of the damaged zone for a band with constant height, results in the damage propagation into the undamaged interior material and an increasing damage height for compressive failure. It should also be noted that the left part of the model cannot carry any tensile loads. In fact upon loading in tension, only the laminate box is load bearing. 47 Chapter 3: Analog Model As described before, upon unloading after the propagation of the damage and loading in the reverse direction, the modulus of the laminate decreases and the damage initiation strain moves further (see McGregor, 2005 for more details). This behaviour is modelled by using the T/C fuses. 3.4.2. Model Formulation Formulations of the rubble box and the laminate box were presented in the previous section. The formulation for the lock-spring element can be easily obtained and is presented below. A schematic of the lock-spring response is also shown in Figure 3-22. (3-19) 3 1 Figure 3-22 Force-displacement response of Spring/Lock elements in series. 48 Chapter 3: Analog Model For the slider element, the following formulation has been derived, based on the aforementioned assumption that the saturation of the rubble box and yielding of the slider coincide. Fy=(^f + k)(SSR-Sm) (3-20) Having the formulation for all the elements and taking advantage of their arrangement in parallel or series we can derive the constitutive equations for the analog model. This is done based on the load applied to the analog model. To simplify the formulation, the following simplified notation is assumed: 5<f =  5lm =  5lf  =5*g  = 5 I R 5S =5SL„, =  SSR S'f  =5><'f S U L = S S L f kf, = very small kf = ku k, = k'n kT - kT 3.4.2.1. Load Reversal: Compression-Unloading Before Band Broadening-Tension The load can be divided into the following stages, and the corresponding formulations can be derived accordingly: 1. Loading in compression: c kfS \S\<\5) kfS^5 + k{S-Sf)+k^^l \8f\<\8\<\SUN\ ( 3 " 2 1 ) 49 Chapter 3: Analog Model 2. Unloading: F = kcM5 + k(5-5f) + ^-x^^- \sfU\sU\S, 2 (S'-Sf) {kcMS + k(S-Sf) Sj<S<SP where, kcM=kf(5f-5UN)/(5^-5f) 5 =kS< C M kit H~ k (3-22) 3. Loading in tension: Based on the modifications which account for failed elements, the final equations for loading in tension can be written as follows: F = kcM5 + k(5-5f) k ' M f 5 + k l i f 5 f 51-5-51-5] 5 + k'L 5 m J \ 5's-8 KSl-5] \ m J 6 + k] f ... \ S's -5 s 'f J 0 0<|£|<|cv| 0 < 5 < 51 Si, <5<8lUi 5'Mr <5<5's 5>5l (3-23) After the damage propagation in compression, the total modulus decreases. Therefore upon further loading in tension, the modified modulus and damage initiation strain of the model in the above equations are given by: kTM =kj {5cs-5UN)l(5cs-5f) m m kL f =  kJf ( sui - 5UN) 1(8UL - 5f) kT kJ <8TS-Sj ) (3-24) S ' M = S S - J J k 50 Chapter 3: Analog Model From the above formulations, the force-displacement .diagram for the analog model can be schematically shown in Figure 3-23. For this diagram, it is assumed that the fibre and the matrix damage initiation strains in compression are identical. I F Figure 3-23 Force-displacement response of the analog model. First compressed, then unloaded and finally loaded in tension to complete failure. 3.4.2.2. Load Reversal: Compression-Unloading After Band Broadening-Tension After band broadening, the matrix fails completely. Therefore the corresponding equations can be derived accordingly: /. Loading in compression: F = kfS 8\<8f ( (^ + k)(8^-8f) \sc\<\S\<\5UN kf v7 7 8 + k(8-5f) + ^ . x y y r ' \8f\<\8\<\8^\ (3-25) (8f-8f) 2 (5cs-8f) I 'I I I I M 2. Unloading: F = k(8-8P)+^,-^4- \5„\<W<\5U o rxc xc\ r f r r n r w i < 3" 2 6) 2 (°S ~ ° l ) 51 Chapter 3: Analog Model where, SP=SUN - ( 5 S ~5f) 3. Loading in tension: Equations for the analog model response under compressive loads upon unloading from tension are derived and presented below. A schematic of the analog model response is shown in Figure 3-24. F k(S-5P) 0 5\ -8 v 's ' / J 0 5P -5, < 5 < 5P Q<\6\<\8P -sy 0<8<8' 8'Mf <8<S'S (3-27) 5 > Si Using the same procedure as in the previous loading case, the modified modulus and the new damage initiation strain in compression are derived as follows. k' =0 kL, = , (SUL - 8UN )!(8UL - 8f) (3-28) kl 8'L =tf--¥L(8l-8f ) J k, J 'f Figure 3-24 Force-displacement response of the analog model (Unloading after the yielding). 52 Chapter 3: Analog Model 3.4.2.3. Load Reversal form Tension to Compression The load can be divided into the following stages, and the corresponding formulations can be derived accordingly: 1. Loading in tension: F = k] 8 + ki 8 8 + kj 5 8 + k], 51-5 STS-Sj \ s f J 8<5] s; <s<s; 5] <8 <8,j (3-29) 2. Unloading: F = k'M8 0<8<8, UN where, k'M =k] (5l-5UN)l(STs-5] ) •k](5Ts-SUN)l{STs-S]r) f kT k1 +kT M  + K M , (3-30) 3. Loading in compression: Equations for the analog model response under compressive loads upon unloading from tension are derived and presented below. A schematic of the analog model response is also shown in Figure 3-25. F = kM5 k^5 + k(5-5f) + C\2 (5-5)') (5Cs-5) (5Cs-5f) 2 (5cs-5f) _ - , ,^5-5f) + ^-x 2 (5<-Sf) C\2 kSR(5-5y) (^L + k)(8C-Sf) 0 H<KC| kc|<|<5|<|^  5LM<5<5-\5s \<5 < 8UN 5UL < 5 (3-31) 53 Chapter 3: Analog Model Using the same procedure as in the previous loading case, the modified modulus and the new damage initiation strain in compression are given by: Figure 3-25 Force-displacement response of the analog model. First pulled in tension, then unloaded and finally loaded in compression to complete failure. 54 Chapter 3: Analog Model 3.5. Further Remarks on the Analog Model 1- Band broadening occurs under a constant stress. This so-called plateau stress has been measured in previous studies in the literature. It is noteworthy that as Gupta et al. (1998) and Sivashanker et al. (1996) have shown, at a certain strain, ultimate strain, the steady-state band broadening is superseded by the sliding action of the two halves of the specimen along one of the kink band fracture planes. 2- For a multidirectional laminate, the failure mechanism is different. As Soutis and Spearings (2002) showed for a plain weave [45/0/90] laminate the main failure mechanism is delamination, while for a [0/90] laminate kink band is the main failure mechanism. The failure of a multidirectional laminate has been studied in recent years (Soutis et al., 1993, 2002; Sivashanker and Bag, 2001; Sivashanker 2001). These studies revealed that a combination of kink band and delamination lead to the laminate failure. The band broadening process is simply the propagation of delamination and kink band into the interior undamaged material. 3- Differences in the failure mechanisms of thermosets and thermoplastics were considered in construction of the analog model. For thermoplastic materials, the formation of kink band is due to the yielding of the matrix and rotation of the fibres. In thermosets, on the other hand, the matrix cracking leads to the softening and rotation of the fibres. The response of the thermosets and thermoplastics are also different under tensile loads. In thermoplastics, unlike thermosets, the saturation of the matrix damage in tension occurs after the saturation of the fibre damage. This phenomenon is due to the yielding of the matrix. 4- Finally, it is noteworthy to mention that the analog model presented here has been developed in collaboration with Carla McGregor. For a complete description of the analog model and its response, including description of multidirectional laminate failure, failure in thermosets and thermoplastics, and the formulation of analog model in stress-strain domain the reader is referred to McGregor (2005). 55 Chapter 4: Damage Growth, Energy Concept and Scaling L a w To complete the construction of a comprehensive constitutive model, there are some issues that need to be considered before implementing the analog model in a finite element code. These issues, which are discussed in this chapter, are as follows: 1 - Compressive damage growth 2- Compressive fracture energy 3- Compressive scaling law First a brief review of CODAM damage growth theory and energy concept (Williams, 1998; Floyd, 2004, McClennan, 2004) as it pertains to tensile loading is presented. Then, a compressive constitutive model based on the analog model is developed that predicts damage propagation under compressive loads. At the end of this chapter, scaling laws for finite element modeling of compressively loaded specimen are presented. This topic is explored in an effort to overcome the mesh sensitivity problem, which can occur due to different element sizes used to model strain-softening behaviour (Floyd, 2004; McClennan, 2004). 56 Chapter 4: Damage Growth, Energy Concept and Scaling Law 4.1. Damage Growth One of the main objectives in damage modeling is to relate the damage growth to the state of strain, using a single mathematical expression. In the analog model, following the procedure used in CODAM (Williams, 1998; Williams et al. 2003; Floyd, 2004), a linear relationship is assumed between the potential function, F, which represents the state of the strain as a potential force for damage growth, and the damage parameters, associated with matrix cracking or yielding and fibre breakage. The state of damage, which is a linear function of fibre and matrix damage will then be used to calculate the modulus reduction in each stage of loading. It is also important to note that damage is an irreversible process. In other words, the effect of damage on matrix and fibre cracking and breakage is permanent and on load reversal, damage will continue to grow from the previous state. As a result, damage parameters increase monotonically. Therefore, the key to relate the stage of loading in compression or tension to the reversal loading is to properly transfer damage parameters. Now, to define damage growth in the analog model, following the CODAM definitions (Williams, 1998, Floyd, 2004) the potential function as an equivalent strain is given by: F(e) = K f c \ \ ^ J + V ^ J 7X y T u (4-1) Therefore in each stage, the damage parameter for the matrix and fibre under compressive and tensile loading are as follows: F(s)-F(s)cI F(sf, Matrix damage in compression (4-2) F(s)-F(s)Tl m_ F(s)Ts-F(s)T, Matrix damage in tension (4-3) 57 Chapter 4: Damage Growth, Energy Concept and Scaling Law c F(s)-F(ef, _.. , co: = —r Fibre damage in compression (4-4) F(e)-F(e)Tl co1,• = — Fibre damage in tension (4-5) 1 F(s)'s-F(s)T, ' f where in all the equations, the damage parameter is related to the initiation and saturation potential functions. In the next step, damage due to fibre and matrix saturation (see Floyd, 2004) is defined as follows: m f (4-6) as =\-cos m f To obtain these parameters, we have to consider the failure mechanism of the laminate. For example, assume a unidirectional lamina under compressive load in a displacement-control test. The threshold of damage is either shear cracking (thermosets) or plasticity in the matrix (thermoplastics), followed by formation of the kink band. At the onset of saturation of the kink band, the lock-up stage, almost the entire confined matrix in the kink band is damaged and therefore damage due to the saturation of the matrix will be significant. On the other hand, considering the same specimen under the tensile load, the matrix damage in the 0 degree layer is negligible (see Floyd, 2004). For other lay-ups, the comparison of the role of the matrix and the fibres in compressive and tensile failure modes specifies the compressive damage saturation parameters. For example, in a 45 degree layer, under tensile load, the damage due to matrix saturation would be around 50%. For compression, however, this parameter would depend on the failure mechanism. For an oblique kink band, the saturation would show a significant damage in the matrix, while for delamination the'matrix damage is not pronounced. The damage parameter in the lamina or sub-laminate is a combination of damage in the fibre and matrix. The percentage of matrix .damage or fibre damage in the final 58 Chapter 4: Damage Growth, Energy Concept and Scaling Law combination depends on the damage due to fibre and matrix saturation. This can be shown by a linear equation as follows: oh = ohs ahm + ah]' cocf For compression (4-7) of = col o)l + 0)1 o)] For tension These two parameters as a function of the potential function are depicted in Figure 4-1. It is also noteworthy that damage due to matrix saturation, oh , in compression is much m higher than in tension, col . As explained before, this is due to the fact that in the m kinking process in compression after the matrix damage saturates, almost all of the volume of the laminate is damaged. In other words, the saturation of the matrix damage coincides with the saturation of the kink band formation while this is not the case in tension. Figure 4-1 Damage growth-potential function) diagrams for tensile and compressive loading. Now, the relation between damage and modulus reduction in compression and tension, or the effect of damage on elastic constants can be easily obtained. The difference between tension and compression is that, in compression modulus reduction due to matrix damage is more pronounced than fibre damage. This is because after matrix damage in compression (not considering the steady-state loading) there will be no support for fibres and they will lose their load-carrying capacity. On the other hand, this is not the case in tension because the tensile strength of the fibre is larger than that of the matrix, therefore after the saturation of the fibre damage the specimen does not carry any further load. This 59 Chapter 4: Damage Growth, Energy Concept and Scaling Law is also shown in the following figure where RE is the ratio of remaining modulus to the initial modulus. T T C C Tension Compression Figure 4-2 (Modulus reduction-damage growth) diagrams for compressive and tensile loading. Modulus reduction due to matrix saturation is more pronounced in compression compared to the corresponding reduction in tension. In summary, the first step in calculating the stresses in a damaged laminate is to find the potential function. Damage parameters can then be calculated from the potential function. Finally, modulus reduction due to damage can be obtained from the damage parameters. The following equation is used to calculate the stress (Williams, 1998; Williams et al., 2003; Floyd, 2004): cr = REEs (4-8) in which, the modulus is modified by the modulus reduction factor, RE. 60 Chapter 4: Damage Growth, Energy Concept and Scaling Law 4.2. Compressive Fracture Energy Defining the fracture energy under compressive loads becomes an important issue in studying the size effect. Due to the complexity of the damage mechanisms in compression it is difficult to determine the fracture energy involved in the progression of damage. From the previous sections it is clear that damage grows in a bidirectional way. One direction is associated with the transient damage growth and the other one with the damage band broadening. In the analog model, this behaviour has been captured by using the laminate box and the slider. While the former represents the transient damage growth and formation of the first kink band, the latter represents the damage broadening normal to the transient damage which will lead to the formation of parallel kink bands. In other words, there are two forms of fracture energy involved in compression. One is the main fracture energy corresponding to the formation of the first kink band, which governs the peak stress and the size effects. The other one is band broadening fracture energy, which is required to form additional kink bands or to expand the damage band. This is depicted in Figure 4-3. o Area associated with the formation of the first kink band Formation of second kink band Formation of third kink band e Aplastic Figure 4-3 A schematic of stress-strain response of composites under compressive loads showing areas associated with formation of kink bands and damage band broadening. 61 Chapter 4: Damage Growth, Energy Concept and Scaling Law 4.2.1. Main Fracture Energy As Bazant & Planas (1998) showed, in the strain softening response of a quasi-brittle material, the strain energy release rate is the area under the stress-strain curve limited to the unloading path from the peak stress, as illustrated in Figure 4-4. Then, the fracture energy would be the area multiplied by the damage band height. This height is assumed to be a constant value before the matrix damage saturation and then starts to grow with the formation of additional kink bands. o A Gpeak Aplastic i^nitiation s^aturation Figure 4-4 Calculation of the main fracture energy from the area under the stress-strain response of the laminate box. This area is the fracture energy required to saturate the matrix cracking (or plasticity for thermoplastics) and fibre rotation in compression.. As stated before, this is the required energy to form the first kink band. Note that the small amount of energy associated with the fibre splitting and cracking is negligible compared to the matrix cracking energy. 62 Chapter 4: Damage Growth, Energy Concept and Scaling Law 4.2.2. Band Broadening Fracture Energy In this section, the fracture energy associated with the band broadening state is considered. As described in the previous sections, no energy is associated with the rubble box. In fact, the rubble box can be interpreted as the elastic response of the cracked rubbles. Slider is, however, different and its activation is followed by some plastic deformation, which is an indication of energy dissipation. One difference with the main fracture process is the damage band height; the formation of additional kink bands leads to the growth of the damage height. The calculation of the fracture energy will be discussed here. Note that the effect of the fibre splitting is assumed to be negligible again. At each strain after the matrix damage saturation strain, the unloading path is exactly similar and parallel to the loading path. The unloading path at the saturation of each kink band is shown in Figure 4-5. Saturation of the second kink band "plateau Saturation of the first kink band Saturation of the i t h kink band Unloading Figure 4-5 The unloading path after the saturation of each kink band. Only the stress-strain response of the rubble box is shown here. By assuming that the intervals between kink band saturations are the same and that the heights of all kink bands are equal, the energy per unit area required to form the i t h kink band can be calculated as follows: (s -s ) G, = cr lateau x (s - s )xhc= cr „ x U L s x hc (4-9) n-l 63 Chapter 4: Damage Growth, Energy Concept and Scaling Law And the total fracture energy is the sum of all kink band energies which is calculated as follows: GJh=G,x(n-\) = G p l m m u x (sUL - ss )xhc (4-10) where, the saturation strain, ss, and ultimate strain, eUL, are the same as what we had in the analog model. This energy is required to form additional kink bands, i.e. to broaden the damaged zone into the interior undamaged material. It is also noteworthy that at each strain, by using Equation 4-10, we have the following equation to calculate the broadening fracture energy: GJb=0-platemx(£-£s)xhc (4-11) and we also have: = — (4-12) where h and s, are the parameters related to a specific point in the stress-strain curve response. Therefore: r - (h-hc)x(eUL-es) _ (h-hc)x(n-Y)sinc ^ lb ~ °~ plateau* , ,  X "c ~ °plateau X , 1 X I X K Ki-K (n-\)hc (4-13) Gfl =° plateau* ^nc  X ( h ~ h c ) which is another way to define broadening fracture energy at each strain. In this equation, s,„c is the interval strain between complete kink band formations, depicted in Figure 4-5. 64 Chapter 4: Damage Growth, Energy Concept and Scaling Law 4.3. Compressive Fracture Energy in the Damage Propagation Process In this section, an analytical calculation of the compressive fracture energy as a function of damage band length is presented. Data from the literature (Jackson and Ratcliffe, 2004) is used to validate the analytical solution. As a result, the fracture energy concept of the analog model, presented in the previous section, will be validated. 4.3.1. Compressive Damage Propagation Jackson and Ratcliffe (2004) presented a study to experimentally measure fracture energy for kink band growth in sandwich specimens with sharp notches. They used woven carbon-epoxy composite in a [(0-90)(±45)] 2 lay-up. A schematic of their test configuration and the stress-strain response of composite material under compression is shown below. Figure 4-6 Stress contour in front of the notch and Stress-Strain response of one element. For this specimen, under increasing remote stress, the stress intensity factor in front of the notch increases. Therefore, as a result of an increase in stress intensity factor, the notch tip stress reaches the critical stress (peak stress). At this moment, compressive damage 65 Chapter 4: Damage Growth, Energy Concept and Scaling Law initiates. The propagation of the damage under increasing far field displacement, is divided into three different stages as shown in Figure 4-7. _ "peak °plateau Figure 4-7 Stress contour in front of the notch while damage is growing. Damage height grows from hc to hL and finally to /);;. The maximum damage height, ltvi, is the damage height at the ultimate strain. These stages are: 1- In a failure process zone (FPZ) in front of the notch, the matrix cracks and the fibres rotate to form the kink band. Stress in this zone decreases, until it reaches the plateau stress. Damage height also increases form zero in front of the FPZ to a characteristic damage height hc, at the notch tip. 2- Damage zone propagates horizontally under the remote displacement. Meanwhile, damage height also increases as a result of damage band broadening. The failure 66 Chapter 4: Damage Growth, Energy Concept and Scaling Law band is divided into two zones: FPZ and band boarding zone, in which stress is equal to the plateau stress aplaleau. 3- After damage height reaches the ultimate damage height, stress drops to zero. Damage height does not increase in this section. Therefore, three different zones can be distinguished: FPZ, band broadening zone and zero stress zone, where damage height is constant. 4.3.2. Compressive Fracture Energy Here, appropriate equations are derived to calculate the fracture energy required to propagate the damage zone for the damage propagation mechanism explained above. Assume f(L) is the damage height function in front of the notch, where L is the length of the damage zone. The shape of this function depends on loading, geometry and material properties. For an increment of length dL in the damage zone, according to Figure 4-8, the strain energy required to propagate damage into new undamaged areas is (shaded in Figure 4-8): N L rt H-dL Figure 4-8 Damage height increment due to the damage length increment. AU AUjam tj + AUd broadening (4-14) -> AU = AU, +AU2 where Ui is the strain energy required to grow the damage over an incremental length dL in front of the damage zone and U2 is the energy needed to expand the damage zone to a new height. 67 Chapter 4: Damage Growth, Energy Concept and Scaling Law To calculate Ui, the fracture energy of an infinitesimal element is used as follows: •AU. = GfxbxdL = ycxhcxbxdL (4-15) In which b is the laminate width and Gf is the main fracture energy, which can be calculated from Figure 4-9: "peak "plateau y c = G f / h c Figure 4-9 Calculation of the main fracture energy for one element from the stress-strain response of that element. To calculate the strain energy required to expand the damage zone, U2, first we have to calculate the new damaged volume, shaded area, in Figure 4-8. This is calculated for linear damage height function and also for generalized damage height function. After finding the strain energy, the fracture energy can then be derived. 4.3.2.1. Linear Damage Height Function The linear damage height function is derived below: h(L) = axL h(ch) = axch =hc -^h(L) = ^xL ch (4-16) where a, the constant of the linear function, is calculated from the boundary condition. Therefore, the damage height increment, dh, due to the damage length increment, dL, is: 68 Chapter 4: Damage Growth, Energy Concept and Scaling Law 2dh = h(L + dL) - h(L) = ^ x(L + dL)-^xL = ^ xdL (4-17) And the new damaged volume is: h Avolume = 2dh xLxb = — xdLxLxb (4-18) Cu Therefore, the strain energy for the damage height increment, based on Equation 4-13, can be calculated as follows: h A E / 2 = o-plaleau xsm c x^xbxLxdL (4-19) c Now we can calculate the total strain energy as: h AU = AUy +AU2 = ycxhcxbxdL + a hlcmxsmcx^-xbxLxdL U(L) =\AU=\(ycxhcxb + o-plalemi xs i m x^-xbxL)dL (4-20) h L2 U(L) = ycxhexbxL + a lateau x sinc x^xbx — ch 2 Therefore the total fracture energy can be derived as: i i j K L2 f „ n ycxhcxbxL + a laleau x sinc x — xbx — G(L) = "W= C-± 2-bxL bxL (4-21) h L L -> G(L) = Yc x hc + a laleau x^x- = hcx(yc + a x smc x —) ch 2 2ch 4.3.2.2. Generalized Damage Height Function Assuming that n is the order of the damage height function, a generalized damage height function can be written as: 69 Chapter 4: Damage Growth, Energy Concept and Scaling Law h(L) = \xL" ch (4-22) Using Taylor series (Appendix A) expansion, the damage height, dh, can be written as: 2dh = h(L + dL) - h(L) = \x(L + dL)" -L-x(L)" A - x [(!)" + n(L)->df,]- A - x A x (Z)- ' dL nh,. c, (4-23) Therefore, the new damaged volume is A,o,ume = \2dhxbxdL=\\ (^xiLy-'ch xbxdL =^£.L" xdL (4-24) As a result, the strain energy increment is: b x h,. AU = AUi + AU2 =ycxhcxbxdL + a „,„ u m x sinc x -S-L" x dL (4-25) And the total fracture energy is U(L) _ \AU G(L) = bx L bxL = hcx (jc + aplaleau x einc L" {n + \)xch" ) (4-26) 70 Chapter 4: Damage Growth, Energy Concept and Scaling Law 4.3.3. Summary of Fracture Energy Functions The damage height function can be derived for different values of n. The different damage height functions and the corresponding fracture energies are summarized as follows: Table 4-1 Damage height and fracture energy functions' for different values of n in Equation 4-26. Damage Height Function Fracture Energy Function ^xL cb K X £ inc \ x l } cb K X (Yc + ^plalea,, X Einc *?>•> fcb K X (Xc + U plal.au X ^inc 2JL h c x(L)n „ n cb K x (y + <j , , V / c plateau L" x ) (n + \)xch" 4.3.4. Comparison with Experimental Results The following parameters are used for comparison of the model with experimental results. These parameter are taken from the literature for the material and lay-up that are used in Jackson and Ratcliffe (2004): Table 4-2 Material parameters used to verify fracture energy functions. Parameter Symbol Amount Source Young's Modulus E 57.2 GPa Soutis et al., 2002 Critical Stress Intensity Factor KIC 35 MPaVm Jackson and Ratcliffe, 2004 Damage Initiation Strain £i -0.01 Soutis etai., 1993 Damage Saturation Strain Es -0.03 Soutis etai., 1993 Length of failure process zone Cb 5 mm Soutis et al., 2002 Plateau Stress °" plateau 20 MPa The difference between the material used in all other experiments and the one by Jackson and Ratcliffe (2004) is the lay-up, which is woven in the latter experiment. Because of 71 Chapter 4: Damage Growth, Energy Concept and Scaling Law this difference and based on the Sivashanker experiment (1998) who measured the plateau stress to be 100 MPa, here the plateau stress is assumed to be 20 MPa. This difference is due to misalignment of the fibres in the woven laminate. Also, the thickness of the laminate in Sivashanker was twice the laminate thickness in Jackson and Wade study (2004), which is another reason for reducing the plateau stress. By using the above parameters and assuming the e,„ c = -0.05, we can plot the following diagram for critical stress intensity factor in which a parabolic function, better correlates with the experimental results . 150 100 V x OH 50 Experiments Linear Strain Field Parabolic Strain Field - 4 1 1 (- -4 1_ 10 20 30 40 Damage Length (mm) 50 60 Figure 4-10 Comparison of experimental results (Jackson and Ratcliffe, 2004) and analytical solutions for critical stress intensity factor as a function of damage length. 72 Chapter 4: Damage Growth, Energy Concept and Scaling Law 4.4. Scaling Law Following the element height scaling in tension (Floyd, 2004) the scaling law for compression is constructed here. Two methods are considered for scaling the element stress-strain curve for different element heights: Crack band method and modified crack band method, both of which are briefly explained here. It is noteworthy that, only the response for the first kink band formation is scaled here. In fact, the change in the height of the element doesn't require scaling of the plateau stress. Therefore, for different element sizes, only the ultimate strain is scaled to preserve the area below the plateau stress. 4.4.1. Crack Band Method This method is based on the approach presented by Bazant (Bazant and Planas, 1998), in which it is assumed that the work done by both the master and the scaled curves are equal. This method also leads to an invariant fracture energy for both cases (depicted in the following diagram) as well as work done at any intermediate value of strain corresponding to a partially damaged state. The detail for this method is not provided here as this method will not be used in this thesis. "peak Master Curve Scaled Curve Figure 4-11 Crack band method. Both the master and the scaled curves are shown here. 73 Chapter 4: Damage Growth, Energy Concept and Scaling Law 4.4.2. Modified C r a c k Band Method This method is based on assuming the same fracture energy for the master and scaled stress-strain curves and by scaling the strains after the peak point strain, epeak. This method is shown schematically in Figure 4-12. Aplastic Speak Aplastic Speak Figure 4-12 Modified crack band method. Both the master and the scaled cure are shown here. The height scaling in tension using this method has been presented by Floyd (2004) who proposed this method because of its efficient numerical implementation. The details of the procedure can be found in this reference and will not be presented here. 4.4.3. Scaling Factor in Compression In this section, we will present the scaling law for compression, using the procedure presented by Floyd (2004), in tandem with the analog model. In the analog model, the peak point strain is either the same as the initiation strain or half of the saturation strain, whichever is larger. The difference is that in the former case there is no permanent deformation upon unloading from the peak stress point as shown in Figure 4-13. The scaling factor for these two cases will be obtained separately. 74 Chapter 4: Damage Growth, Energy Concept and Scaling Law Figure 4-13 Different unloading paths for compressive failure. The left diagram shows unloading from the peak stress point while the peak stress and damage initiation coincide. The right diagram shows the unloading path for an element in which the peak stress and damage initiation do not coincide. 4.4.3.1. Scaling Factor for an Element with Peak Stress at the Damage Initiation Strain Using hr/he = n as the ratio of the characteristic height to the element size, the shaded area below the stress-strain curve from the above figure can be calculated as: s, x Es, \ Es- 2 E -ds = Es, 12 A ( _ 3 2 3 (4-27) s E E-J- + — (ss- e, )(2e, + ss) I o The area below the scaled curve is: e 2 E ye=E^- + - k(ss -e,){2s,+ ES ) (4-28) 2 o Now, by equating the fracture energies for the two curves, we obtain: Ye x K = Yc x K -^Ye =nxyc -> s E E~Y + -^ k(es - £i )(2s,' + ss ) = m £ E 1 o (4-29) 75 Chapter 4: Damage Growth, Energy Concept and Scaling Law Rearranging the above equation, the following equation for calculating the scaling factor, L is obtained: «x ( f i . 2 + £,£s +£ " / 2 ) -3 f / 2 (4-30) 4.4.3.2. Scaling Factor for an Element with Peak Stress not at the Damage Initiation Strain Using the same notation, the area under the stress-strain curve shown as the shaded area in Figure 4-13 is: y,. = — — x ^ + EE — d£ = 2 4(£S-£,) £SI2 £v - £, (4-31) EES (£S/2-EF) | E ? ; g s . 8(e s-ff/) es-e, 12 And the area under the scaled curve is: EE^ts^ 12-ep) , E E.3 y, = — + kx x-±-S(£S-£,) £S-£, 12 (4-32) Again, if the fracture energies are equated, the following equations are derived: ye xhe =ycxhc -+ye =nxyc -» EES (ES/2-£P) [ / t _ E ,£S = S(Es-£,) es-e, 12 EEJ(E^ 12-Ep) E EJ* nx \ — ^ — ^ — + x-^-8(e s -£,) ss -£, 12 The scaling factor for this case is as follows: In - 3 3 £ p k = - + -(1 - » ) — 4 2 e. (4-33) (4-34) 76 Chapter 4: Damage Growth, Energy Concept and Scaling Law 4.4.4. Summary of Scaling Factor Equations As a summary, in this section the scaling factor equations which will be used to modify the element stress-strain curves due to the element height effect are given below: 1- While peak stress is at the damage initiation strain: « X ( £ \ , 2 + £,£., +£,2)-3S, 2 k = 's 2 ' 2—- (4"3S) 2- While peak stress is not at the damage initiation strain: In-3 3 eP k = —r- + - 0 - " ) — (4-36) 4 2 £s These equations will be used to effectively model the damage propagation process with a finite element code (see Section 5.2. and Section 5.3.). 77 Chapter 5: Mode l Verification and Validation The focus of this chapter will be on verifying and validating the presented composite compressive damage model. This model has been implemented in the commercial finite element code, LS-DYNA, the predictions of which will be discussed here. To verify the model, data from the literature will be used to simulate the response of a unidirectional laminate. The successful simulation of the laminate response by using one element, will verify the implementation of the model in LS-DYNA. Subsequently , LS-DYNA predictions for the behaviour of composite components under compression will be compared with experimental results to validate the model. The following two experimental studies will be used to validate the model: 1. Notched sandwich panels under eccentric compression (Bayldon, 2003a, 2003b). 2. Open hole plates under compression (Soutis et al., 2002). The first step to validate the implemented model is to derive appropriate material damage parameters by studying the damage mechanism in each composite panel. These parameters will then be used as input to the model. At the end of each simulation, the experimental results will be compared with the analog model predictions and conclusions will be drawn. The successful prediction of composite component responses would validate the analog model and allows effective use of the model to predict responses of other composite components under tension, compression and load reversals (see McGregor, 2005, for other simulations). 78 Chapter 5: Model Verification and Validation 5.1. Model Verification Vogler and Kyriakides (1999) presented a study on the axial propagation of kink band in fibre composites under compression. From their study on AS4/PEEK unidirectional composite panels, material parameters for compression are derived and explained below. These parameters are listed in Table 5-1. The modulus loss due to matrix damage in compression is assumed to be 97% in this case. This assumption is the result of kink band formation in the unidirectional laminates, which leads to the complete failure of the panel after the saturation of matrix cracking (see Section 4.1. for more details). Table 5-1 Compressive damage parameters for A S 4/PEEK unidirectional panels. Parameter Symbol Amount Matrix damage initiation m -0.0082 Fibre damage initiation r -0.0082 Matrix damage saturation m -0.013 Compression Fibre damage saturation ' h -0.05 Damage due to the matrix saturation ' m 0.4 Damage due to the fibre saturation c 0.6 Modulus loss due to the matrix damage \-R-F 0.97 Modulus loss due to the fibre damage 1 - RCR 0.03 Plateau stress ^ plateau 430 MPa Other assumptions have also been made for tensile damage parameters as shown in Table 5-2 (see Floyd, 2004, for details on how to estimate tensile parameters). It is also noteworthy that for both tensile and compressive parameters, the assumed damage due to the fibre and matrix saturation are derived from fibre volume fraction of 60% in all panels (see Floyd, 2004, and McGregor, 2005, for more details on similar estimations). 79 Chapter 5: Model Verification and Validation Table 5-2 Tensile damage parameters for AS4 /PEEK unidirectional panels. Parameter Symbol Amount Matrix damage initiation m 0.015 Fibre damage initiation 1 f 0.02 Matrix damage saturation m 0.035 Compression Fibre damage saturation 0.035 Damage due to the matrix saturation co's m 0.4 Damage due to the fibre saturation 0.6 Modulus loss due to the matrix damage m 0.15 Modulus loss due to the fibre damage \-RTF LL 0.85 The undamaged elastic material parameters were also reported for AS4/PEEK panels in the Vogler study, as follows: Table 5-3 Mater ia l parameters for AS4 /PEEK unidirectional panels. Unidirectional AS4/PEEK Amount Longitudinal Young's modulus 128 GPa Transverse Young's modulus 10.57 GPa Shear modulus 5.79 GPa Poisson's ratio 0.3 Having all the material and damage parameters, we can model the behaviour of the laminate by using the constitutive model implemented in LS-DYNA. The response of a shell element under uniform uniaxial and in-plane displacement in tension and compression is shown in Figure 5-1. From this figure, it is obvious that the compressive strength is smaller than the tensile strength. This is due to misalignment of the fibres, which leads to formation of kink band and failure of the laminate (Fleck, 1997). As the shell element response in this figure follows the expected pattern for the material response, it verifies the implementation of the model in LS-DYNA. This means that by feeding appropriate parameters into LS-DYNA, one can reproduce the desired response. 80 Chapter 5: Model Verification and Validation The so-called plateau stress, damage initiation strains and damage saturation strains are also clearly shown in this figure. I _ 1 i _ 1 5 0 0 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.01 0.02 0.03 0.04 Strain Figure 5-1 Stress-Strain response of one element obtained from the implemented constitutive model in LS-DYNA. Some of the input parameters are obtained from Vogler and Kyriakides (1999) while the damage parameters are estimated. 81 Chapter 5: Model Verification and Validation 5.2. Simulations of Notched Sandwich Panels under Eccentric Compression Experiments on sandwich panels under eccentric compression with various notch and specimen sizes were carried out at the North-Western University (Bayldon, 2003a, 2003b). The choice of sandwich panel structures and application of double-eccentric loads were to prevent elastic buckling of specimens while concentrating the stress at the notch tip. During all tests, strain gauges were used to measure the strains at the edges of the specimens, except the notched edge. Photoelastic images were also obtained from the front face of each sandwich panel to demonstrate the strain field around the notch. Final measured loads were reported as the specified strength for each specimen. It was observed that a combination of kinking and delamination, starting from the notch tip, were the main failure mechanisms in all the experiments. The main goal of this section is to simulate these experiments using the compressive damage model developed in this study. For this purpose, after reviewing the compressive failure in the sandwich panels, damage parameters describing the failure process in these panels will be derived. Then, the nominal strength of these panels, predicted by the model implemented in LS-DYNA will be compared with the experimental results. 82 Chapter 5: Model Verification and Validation 5.2.1. Sandwich Panel Failure The use of sandwich panels in the aeronautical and marine structures has gained popularity during recent years. Light weight and high bending stiffness are the two main reasons for the increasing popularity of such panels. Sandwich panels are usually made of metallic or composite face sheets and a polymeric, metallic, or wood foam core. Despite the fact that several studies have been performed to evaluate the load carrying capacity of sandwich structures (Mouritz and Thomson, 1999; Mirazo and Spearing; 2001; Soutis and Spearing, 2002; Fleck and Sridhar, 2002), few predictions are offered by current theories. One major problem in predicting the load carrying capacity is the interaction of different failure modes. Sandwich panel failure due to delamination, face wrinkling, buckling and fracture of skin and core are often combined. As a result the failure response becomes a mixture of buckling, fracture and damage. Another unresolved problem of sandwich panels is the role of shear in highly deformable cores. This problem magnifies considering the high value of the ratio of the elastic modulus of the face to core (Bazant, 2003). When subjected to compressive loads, sandwich panel failure can occur by variety of mechanisms, depending on the material parameters and panel geometry. Fleck and Sridhar (2002) constructed collapse mechanism maps to study the dependence of failure mechanism on these specifications. In their study, the following four distinct failure modes for sandwich panel were specified. They are schematically shown in Figure 5-2: 1. Face sheet microbuckling. Face sheet microbuckling, or kink band, is perhaps the most common mode of failure for defective or notched sandwich panels. For this.failure mechanism, the critical load required to initiate the failure decreases when the size of the defect or crack length increases (Mouritz and Thomson, 1999). 83 Chapter 5: Model Verification and Validation I I I I t t t t Figure 5-2 Failure modes in sandwich columns (Fleck and Sridhar, 2002). From left to right, Euler macrobuckling, core shear macrobuckling, face sheet microbuckling and face • sheet wrinkling. 2. Euler macrobuckling. The Euler buckling of a sandwich panel is simply a function of bending stiffness and the length of the panel. For a long panel with constant cross section, the Euler.buckling becomes a common failure mode. On the other hand, for a sandwich panel with significant shear deformation the role of core shear modulus will also become important in defining the buckling load (Engesser, 1889; Haringx, 1942). Such a situation arises for the continuum built-up columns or for highly orthotropic fibre composite columns (Bazant, 2003). 3. Shear macrobuckling. The core shear macrobuckling is a regular failure mode for crack-free sandwich panels that are unstable under compression. This failure mode is a function of the shear modulus of the core (Fleck and Sridhar, 2002). 4. Face wrinkling. Skin wrinkling or short wavelength elastic buckling of the face sheets is also a common failure mechanism for sandwich panels when there is no defect on the face sheet. 84 Chapter 5: Model Verification and Validation 5.2.2. Test Geometry In all the experiments, double eccentric compressive loads are applied on notched sandwich panels. The load is applied by loading plates through the ball bearing on the top and bottom capture plates. As a result of this configuration, load is applied through two eccentric pin jointed ends. In order to prevent slipping of these ball bearings, small detents are arranged. There are also two side plates at both end of the panels to prevent transverse movement. The test is a displacement controlled experiment in order to study the strain softening behaviour of the composite material. Eccentricity is introduced to concentrate the compressive stress at the notch tip and ensure progressive failure of the material. At the same time, this loading condition will prevent the panels from buckling or back-face failure. A schematic of the test arrangement is shown in Figure 5-3 below. Capture Plate Sandwich Panel Loading Plate Loading Ball Side Plate Figure 5-3 Notched sandwich panel under double eccentric compression (Bayldon, 2003a, 2003b). 85 Chapter 5: Model Verification and Validation There are two sets of experiments on notched sandwich panels denoted as ECC and CDC. ECC or notch size effect test is a set of experiments on a specific sandwich panel geometry with three different notch sizes. The face material is a woven glass-epoxy while the core is a Divinycell foam. The specification of the ECC specimens are shown in Table 5-4. Table 5-4 Test geometries of notch size effect experiments (ECC). All numbers are in mm. Core Face Notch Notch Specimen Height Width . . . „, . . Thickness Thicknes Diameter Depth ECC1 80 40 25.4 4.23 6.35 3.175 ECC2 80 40 25.4 4.23 12.7 3.175 ECC3 80 40 25.4 4.23 25.4 3.175 CDC or specimen size effect test is a set of experiments on four different specimen sizes. Each successive specimen is twice as large in all dimensions as the next smaller specimen. The smallest specimen face sheets are two-ply laminates of woven glass-epoxy, while the other three sizes are made of 4, 8 and 16 ply laminates. The specifications, for CDC tests are shown in Table 5-5. Table 5-5 Test geometries of specimen size effect experiments (CDC). All numbers are in mm. c . T I • ,„•,., Core Face Notch Notch Specimen Height Width Thickness Thicknes Diameter Depth CDC2 20 10 6.25 0.55 3.175 1 CDC4 40 20 12.5 1.1 6.35 2 CDC8 80 40 25 2.2 12.7 4 CDC 16 160 80 50 4.4 25.4 8 Eccentricity, as shown in Figure 5-4, for ECC tests is 0.1 in both directions, while for CDC tests is 0.077 in z direction and 0.139 in y direction. The point of load application can be easily calculated by multiplying the eccentricity by the specimen size in that direction. A schematic of the eccentricity and other sandwich panel sizes is shown in Figure 5-4. 86 Chapter 5: Model Verification and Validation Figure 5-4 Test geometry for notched sandwich panels under double eccentric compression. 87 Chapter 5: Model Verification and Validation 5.2.3. Failure Mechanisms of Woven Glass Laminates During the past years, there has been increasing demands to use textile reinforcement. One of the reason for this is perhaps to take advantage of their through thickness arrangement of fibres which enhances interlaminar strength and toughness. The other advantage of using woven material is gaining more compression after-impact strength; the improved interlaminar shear property is considered the main reason for this improved strength. However, as Yang et al. (2000) showed, a woven laminate may have a lower total compressive strength compared to a [0, 90] non-crimp laminate as a result of fibre waviness. In fact, the compressive strength decreases with the increasing fibre misalignment or waviness (Argon, 1972; Budiansky, 1983). Despite the fact that substantial research has been done to understand the behaviour of woven graphite fibre composites, the amount of research on the woven glass composites is relatively less. While more brittle, the response of woven glass composites to compressive loads is almost linear up to failure, compared to nonlinear response of woven carbon (Zhou and Davies, 1995; Fleck et al., 1995). A woven laminate under compression may fail by various failure mechanisms. In fact, geometry and material properties, laminate orientation, and laminate imperfections such as fibre misalignment or fibre yarn waviness contribute to type of failure mechanism. However, a notched specimen prevented from buckling, would most likely fail by a combination of kinking and delamination (Khan et al., 2002 ) The difference between the failure of a woven laminate and failure of a [0, 90] non-crimp laminate is that in the woven one, formation of out of plane kink band is most likely as opposed to in-plane kinking in the other one. This difference is because of the 90 degree fibres in woven laminates, which provide the 0 degree fibres with extra in-plane resistance. The formation of in plane kink band is also a function of thickness and lay-up of the laminate (see Section 2.2.6.1.). Out of plane kinking is the main failure mechanism in CDC and ECC experiments. Although delamination and fibre splitting can also be observed in some cases. 88 Chapter 5: Model Verification and Validation The failure process starts with matrix cracking from the notch tip. In the case of an un-notched specimen, the point where waviness and misalignment in the fibre is greatest, is the starting point of matrix cracking. In tandem with matrix cracks, rotation of fibres leads to the formation of out of plane kink band. Meanwhile, delamination propagates. Kink band and delamination propagate in horizontal direction and in the mean time the failure band expands, moving toward undamaged interior materials. The failure mechanism is unstable and sudden due to the brittle response of woven glass. The damage initiation and broadening have been confirmed by photoelastic images, which show the starting damage height to be around 3.5 mm and then broaden up to almost 10 mm (Bayldon, 2003a). The photoelastic images are presented in Figure 5-5, Figure 5-6 and Figure 5-7. Figure 5-5 Initiation of kinking with matrix cracking. Photoelastic image for the notched sandwich panel under eccentric compression (Bayldon, 2003a). 89 Chapter 5: Model Verification and Validation Figure 5-6 Initial damage height is equal to 3.5 mm for kinking and delamination. Photoelastic image for the notched sandwich panel under eccentric compression (Bayldon, 2003a). Figure 5-7 Propagation of the kink band and delamination in horizontal and vertical directions. Height of the damaged zone is almost equal to 6 mm. Photoelastic image for the notched sandwich panel under eccentric compression (Bayldon, 2003a). From the photoelastic images it is also clear that out of plane kink band is the main failure mechanism in notched sandwich panel experiments, because in-plane kink band does not propagate horizontally (see Figure 2-3). The width of the specimen is considerably larger than the thickness of the face sheets which provides less resistance in the direction of out of plane kink band. oo Chapter 5: Model Verification and Validation 5.2.4. Material The sandwich panel face sheets are made of woven glass-epoxy pre-preg (7781 style glass and Bryte 250 resin). The pre-preg was manufactured by hot melt film coating and then oven cured under vacuum consolidation (Bayldon, 2003a, 2003b). There was noticeable porosity reported in some cases, caused by dry plies. The porosity showed no effect on the elastic properties of the laminate but reduced its compressive strength. Face sheets were laid up as woven [0/90] layers. Core material was also H250 Divinycell commercial foam. For the CDC specimens, material properties for face sheets and foam core are shown in Table 5-6 and Table 5-7. Table 5-6 Properties of the foam core. A l l properties are in the loading direction. Core Young's Shear Modulus Poisson's Density Modulus (MPa) (MPa) Ratio (Kg/m3) H250 Divinycell 400 108 0.32 250 Table 5-7 Properties of the face sheets. Properties in both directions are the same. Face Young's Shear Modulus Poisson's Modulus (MPa) (MPa) Ratio Woven Glass-Epoxy 24300 12000 0.07 There are several damage parameters required to properly model the failure process using the implemented model. In the following, these parameters will be discussed in detail. From the photoelastic pictures, we can estimate the damage height. Photoelastic coating is a thin, transparent plastic of uniform thickness, which when attached to the surface of a test part and viewed with polarized light shows the strain field in that surface. Therefore, we can estimate the damage height from Figure 5-5 and Figure 5-6 to be approximately 3.5 mm at the initiation of damage. The damage height at this stage is the characteristic height parameter, which will be used in the calculation of the main fracture energy. To calculate the broadening energy, we need to measure the height of the damage zone after 91 Chapter 5: Model Verification and Validation complete failure. The damage height is much larger at ultimate strain due to damage band broadening. For this material, we assume that the peak stress is at the damage initiation strain. In other words, we postulate a very brittle behaviour for this material according to the experimental results (there is no nonlinearity before the peak stress). This assumption leads us to the calculation of the damage initiation strain based on an elastic model. LS-DYNA is then used to model different CDC specimens with a linear elastic material model. By comparing the numerical results with the experimental results, we can estimate the damage initiation, , to be -0.022. The estimation of this parameter will be described in the next section. According to this assumption and also based on the formulation of the analog model, matrix damage saturation should not be greater than twice the initiation or -0.044 (see Section A A3.2. ). On the other hand, the fact that woven-glass is a brittle material requires the damage saturation strain to be around the damage initiation strain. So, we initially choose the damage saturation strain, , to be -0.030 and we will revise this number later in the following section. The ultimate strain, in which all the fibres would fail, is chosen to be -0.09 based on other experiments on the woven lay-up under compression, such as Gupta et al. (1998). The formation of the first kink band corresponds to the matrix cracking saturation. This allows us to calculate the damage due to matrix saturation based on the matrix volume fraction (Floyd, 2004). Because the fibre volume fraction in CDC experiments is approximately 40%, the matrix volume fraction is 60%. As a result, at the matrix damage saturation state, the loss of 60% of the surface area is the total damage due to matrix saturation. In other words, the damage due to matrix saturation is cocs = 0.6. In the same m manner, damage due to fibre saturation, which is the state of ultimate failure, is coc: =0.4. v 92 Chapter 5: Model Verification and Validation Two other parameters, required to properly describe the failure process, are modulus loss due to matrix damage saturation and fibre damage saturation (Floyd, 2004). As described before, after the matrix damage saturation in compression, there is no lateral support for fibres and therefore they buckle. This means that after the formation of the kink band, the laminate loses its entire load carrying capacity. For this reason, the modulus loss due to matrix damage saturation is 100%. However, this parameter is chosen to be 97% due to the presence of the bending stiffness of the fibres. Therefore, the modulus loss due to fibre damage saturation at the ultimate strain is only 3%. The plateau stress has the least importance among all damage parameters in this analysis. Because of the brittle response of woven glass sandwich panels and the thin face sheets, the value of the plateau stress is considered to be negligible. This stress was not reported in the CDC experiments but from other experiments on the same material and lay-up, for example Yang et al. (2000), the amount of plateau stress is derived to be 20 MPa. This small plateau stress compared to the peak stress of approximately 500 MPa, confirms the negligible role of the plateau stress in compressive failure of thin woven glass-epoxy laminates. In other words, a small stress is adequate to propagate the damage into the interior undamaged material after the saturation of matrix cracking. On the other hand, due to the relation between the fracture energy and plateau stress (Section 4.2. ), a small amount of energy is required for kink band broadening in the woven glass-epoxy. This statement on the plateau stress would be altered in the case of multidirectional laminates. For example, Gupta et al. (1998) showed that for a [+45,-45] woven laminate under compression, the amount of plateau stress is almost equal to the peak stress. As for the case of thick laminates, the plateau stress and the required energy to propagate the damage band will increase. The latter assumption was also confirmed by Khan et al. (2000), in which the plateau stress was reported to be almost one third of the peak stress. Now that we have an estimation of all the damage parameters, we can calculate the main fracture energy due to kinking in sandwich panels. Based on the calculation of fracture energy for the case that damage initiation and peak stress coincide (Section 4.4.3.1. ) the fracture energy is derived as follows: 93 Chapter 5: Model Verification and Validation = ^IEL + £f E s El-^de = Es,2 / 2 + • E ( 3 2 3 A v 6 2 3 y £ , 2 £ J E4- + -(^.-£;)(2ff /+eA.) z o In this equation the Young's modulus of the laminate is the modulus loss due to matrix damage saturation (Section 4.4.3.1. ). As explained before, this is equal to 97% of the laminate modulus. Therefore:-24.3 x 0.97 x + " X - (0.03 - 0.022)(2 x 0.022 + 0.03) = 8.03 x 103 KPa So, knowing the damage height and the area under the stress-strain curve, the critical strain energy release rate can be obtained as follows: Gf =ycy-hc =8.03x103 x3.5 = 28.1 mJ/mm2 This value for fracture energy changes by altering the assumption for damage saturation strain. As discussed above, damage saturation strain should be just after damage initiation strain. On the other hand, lowering the damage saturation strain would decrease the critical strain energy release rate. The minimum fracture energy, equal to 20 mJ/mm2, is obtained by lowering the saturation strain to -0.022, a value equal to initiation strain. The summary of all parameters is shown in Table 5-8. It should be noted that the damage initiation in fibres and matrix are the same (see Section 3.4.2.1. for more details). 94 Chapter 5: Model Verification and Validation Table 5-8 Damage parameters for woven glass-epoxy laminate. Parameter Symbol Amount Matrix damage initiation m -0.022 Fibre damage initiation -0.022 Matrix damage saturation -0.030 Fibre damage saturation -0.09 Damage due to the matrix saturation m 0.6 Damage due to the fibre saturation cocs 0.4 Modulus loss due to the matrix damage m 0.97 Modulus loss due to the fibre damage 1 - RCF 0.03 Compressive toughness Gf 28.1 mJ/mm2 Damage height K 3.5 mm Plateau stress ^ plateau 20 MPa 95 Chapter 5: Model Verification and Validation 5.2.5. Finite Element Model 5.2.5.1. Experimental Results Before modeling CDC specimens with LS-DYNA, we should obtain a representation of strength versus size diagram for experimental results. This diagram will be used to compare the numerical CDC size effect results against. By applying eccentric compression on a sandwich panel, stress on one edge of the panel will be larger than the stress on the other edges. Therefore, we can choose the maximum applied stress under the top rigid plate as a measure of the sandwich panel strength. Due to the linear response of the sandwich panel before complete failure, the maximum strain instead of the maximum stress can be used. Now, based on Figure 5-4, the maximum strain can be calculated as follows. First, we have to calculate the effective modulus of the specimen: IEr x t, + E xt ~ face face core core ^ E = — J- (5-1) h Bending stiffness in both the y and z directions can be obtained as a sum of the core and the face bending stiffnesses, as followings. (EI)y = 2{EfaJfave) + (EC0Jcnra) = 2Eface_ x + Ecorc x t^L (5.2) {EI), = b x [Eface (h3 - tl,)+ Emjlc ]/12 (5-3) Assuming that the final load on the sandwich panel is P, the maximum strain is: P bxte.xP) hx(e7x P) £ = 1 L J H — (5-4) max bhE 2{EI)y 2(EI)Z 96 Chapter 5: Model Verification and Validation Therefore, by choosing the width of the panels as the size parameter and by calculating maximum strains in all panels, for non-porous CDC experiments we can plot the maximum strain as a function of characteristic size of the panel (width) on a logarithmic scale as shown in Figure 5-8. -4.2 -4.3 -4.4 -4.6 1 J -4.7 -4.8 1 -4.9 H -5.0 J 2.5 3.0 3.5 4.0 4.5 CDC Experimental data • Linear fit Ln (Width mm) Figure 5-8 Experimental size effect results for CDC sandwich panels under eccentric loading. 5.2.5.2. Damage Initiation Strain As mentioned in Section 5.2.4, damage initiation strain can be derived from the elastic response of the CDC specimens. This assumption is valid because of the linear response of woven glass laminate. In fact, because we assumed that the damage initiation and the peak stress coincide we can use the elastic results of CDC experiments to derive the damage initiation strain from the notch tip strain. In this section, all of the CDC experiments will be modeled with elastic materials and the load-displacement response will be compared with the experimental results. 97 Chapter 5: Model Verification and Validation First, we will obtain the stress contour at the notch tip of the CDC experiments at failure. By extrapolating from the stresses at the element integration points, we can derive the notch tip stress. This approximate method has to be used, because there is no closed form analytical solution available for calculating the stress concentration factor of notched sandwich panels under eccentric compression. The integration points are shown for CDC8 in Figure 5-9 below. 3 4 Y "HBHE3 Figure 5-9 Gauss integration points for CDC8. Only half of the specimen is shown here. After finding the notch tip stress, we can simply derive the failure strain from an elastic analysis. Stress contours of CDC specimens are shown in Figure 5-10, Figure 5-11, Figure 5-12 and Figure 5-13. 98 Chapter 5: Model Verification and Validation Figure 5-10 Stress contour at the notch tip for CDC2. Triangles are the Gauss integration points of the finite element model. The zero distance point is extrapolated based on the stress at other points. Figure 5-11 Stress contour at the notch tip for CDC4. Triangles are the Gauss integration points of the finite element model. The zero distance point is extrapolated based on the stress at other points. 99 Chapter 5: Model Verification and Validation 0.0 2.0 4.0 6.0 Distance (mm) 8.0 10.0 Figure 5-12 Stress contour at the notch tip for CDC8. Triangles are the Gauss integration points of the finite element model. The zero distance point is extrapolated based on the stress at other points. Figure 5-13 Stress contour at the notch tip for CDC16. Triangles are the Gauss integration points of the finite element model. The zero distance point is extrapolated based on the stress at other points. 100 Chapter 5: Model Verification and Validation As a summary, the notch tip strain of the CDC experiments under the failure load is shown in Table 5-9 below. Table 5-9 Damage initiation strains for different CDC experiments based on their elastic analysis. Specimen Damage Initiation Strain CDC2 -0.0240 CDC4 -0.0227 CDC8 -0.0216 CDC 16 -0.0251 According to this table, we choose the damage initiation strain equal to -0.022 which is an average value. This strain can be altered later, if needed. However, the maximum initiation strain should not exceed -0.025. 5.2.5.3. Scaling Factor Considering both the element width effect and the height effect based on the works of Floyd (2004) and McClennan (2004), we have to decrease the Young's Modulus and increase the saturation strain to preserve the same fracture energy for different element sizes without altering the peak stress at the notch tip (see Section 4.4. for complete details). Here, we assume the modified modulus due to element width effect as follows: E„ew=E/a (5-5) where a is the modification factor for the element width effect obtained from the numerical stress concentration factors in Section 5.2.5.2. Fracture energy can also be evaluated based on the damage propagation formulation, when the peak stress and damage initiation coincide (Section 4.4.3.1. ). In this case, we assume that there is no plastic deformation after unloading from the peak stress. This assumption is valid because there is no damage before the peak stress. Therefore, the strain energy release density (specific energy) is: 101 Chapter 5: Model Verification and Validation YC=EC-^- + ^ {sc-eci)(2sc+e<:) (5-6) Using this equation, we can derive the area under the stress-strain curve of an element as follows: Y, = ^ ^ - + ^ k ( £ c s - s o i l s ' ; +s cs) (5-7) a 2 6a And because the fracture energy for different element sizes should be kept constant (Bazant and Planas, 1998), we have: Yc x K =Yex  2K (5-8) The factor of two is applied on the element size, because we are modeling half of the CDC specimens. Therefore, the scaling factor is: h EL sf •ycx-S. a—J— 2h„ a 2 k = fpZ . ' (5-9) - ^ - « - < ) ( 2 < + < ) Using this equation, the results for the CDC specimens are derived as shown below. Table 5-10 Scaling factors for notched sandwich panel under eccentric compression. Specimen Modified Young's Modulus (MPa) Scaling Factor CDC2 20369 4.756 CDC4 19597 1.252 CDC8 22091 0.871 CDC 16 17173 1.822 102 Chapter 5: Model Verification and Validation 5.2.6. Simulations As described in Section 5.2.4., the least known parameter among the damage parameters is the damage saturation strain. It should be noted that modifying the saturation strain while keeping all the other parameters constant changes the fracture energy. It is quite obvious that by increasing the saturation strain the material becomes tougher and less size dependent. The strength versus size curve for this material is less steep compared to a more brittle material. Decreasing the saturation strain, on the other hand, has the effect of decreasing the fracture energy, which then results in having a more brittle material. As a result, size dependency would be more pronounced. In other words, we can control the size dependency of the simulations by changing the saturation strain. It is also noteworthy that based on the previous discussions we cannot change the saturation strain to a value higher than twice the damage initiation strain. Also the lower bound for the damage saturation strain is the damage initiation strain. As mentioned before, for CDC experiments we will only model half of the specimens. The finite element mesh for CDC8 is shown in Figure 5-14 as an example. Figure 5-14 Finite element model for simulating the CDC8 experiment. Only half of the specimen is modeled. 103 Chapter 5: Model Verification and Validation CDC experiments are simulated with different fracture energies. The change of fracture energy results from changing the damage saturation strain, as explained above. The result of changing the fracture energy is shown in Figure 5-15. Ln (Width mm) F i g u r e 5-15 S i m u l a t i o n o f C D C e x p e r i m e n t s w i t h d i f fe ren t f r ac tu re energies o r w i t h d i f fe ren t s a t u r a t i o n s t r a ins . From this figure, it is evident that by reducing the fracture energy we get results that are closer to experiment. In fact, the best result was obtained when saturation strain was equal to -0.023 corresponding to a fracture energy of 20.9 mJ/mm2. This number indicates a very brittle response for the woven glass laminate. This finding also matches our understanding of the behaviour of this material. It is also noteworthy that for the circular blunt notch in these panels, the change of fracture energy does not change the size dependency as much as it does for the sharp notches. McClennan (2004) confirmed this for sharp and blunt notches. 104 Chapter 5: Model Verification and Validation On the other hand, the outcome of changing damage initiation strain while keeping the fracture energy constant will be the same size dependency results for CDC experiments, but with a lower value of strength. In other words, changing the damage initiation moves the line of length-displacement parallel to its original position. This is confirmed in Figure 5-16. -4.3 -4.4 -4.5 -4.6 £ -4.7 & J -4.; -4.9 -5 1 0 2.5 i i 3.0 3.5 4.0 l 4.5 5 • e, = -0.022 HI ^ v s O N s ^ H Experiment • a / / ^ V V ^ s . e, = -0.020 / £ / H e, = -0.021 • e, = -0.020 Xe, = -0.021 •Experiment Ae, = -0.022 Ln (Width mm) Figure 5-16 Simulation of CDC experiments with different damage initiation strains while fracture energy is equal to 20.9 mJ/mm2. In these simulations, three different damage initiation strains are used with the fracture energy constant and equal to 20.9 mJ/mm2. This choice is based on the previous set of simulations for different fracture energies. It is noteworthy that from the above figure the best fitted result for simulation can be obtained by using the same brittle material parameters in the previous section, while changing the fracture energy. So, for both simulations the best results is obtained with the saturation strain equal to -0.023 and the 105 Chapter 5: Model Verification and Validation fracture energy equal to 20.9 mJ/mm . As a summary, the material parameters that give the best results are shown in Table 5-11. Table 5-11 Damage parameters for woven glass-epoxy laminate which give the best result for C D C simulations. Parameter Symbol Amount Matrix damage initiation m -0.022 Fibre damage initiation '•, -0.022 Matrix damage saturation m -0.023 Fibre damage saturation V -0.09 Damage due to the matrix saturation ' m 0.6 Damage due to the fibre saturation 0.4 Modulus loss due to the matrix damage \-RcF 0.97 Modulus loss due to the fibre damage 0.03 Compressive toughness G f 20.9 mJ/mm2 Damage height K 3.5 mm Plateau stress ^plateau 20 MPa 106 Chapter 5: Model Verification and Validation 5.2.7. Summary Main findings in simulating the notched sandwich panels under eccentric compression are summarized here: 1- Woven glass-epoxy face sheets fail by a combination of out of plane kinking and delamination, due to laminate lay-up and thickness. , 2- Formation of kink band the resulting failure are sudden. 3- Plateau stress is negligible for a woven [0, 90] laminate. 4- Energy required to expand the damage zone is small, compared to the energy required to form the first kink band. 5- Glass-epoxy used in this experiment is a brittle material. 6- For circular blunt notches, changing the fracture energy in the analog model does not affect the size dependency of specimens. 7- The analog model successfully showed a size effect for CDC experiments, which matches the experimental results. 8- Analog model is validated as a result of successful prediction of size dependency in these specimens. Chapter 5: Model Verification and Validation 5.3. Simulation of Open Hole Plates under Compression 5.3.1. Introduction Soutis et al. (1993) investigated the compressive fracture properties of carbon fibre/epoxy laminates. Compressive strengths of both unnotched and notched carbon-epoxy panels for a wide rang of lay-ups were reported in their experiments. They reported that the failure mechanisms in all laminates were due to the microbuckling in the [0] plies, delamination between off-axis and [0] plies, and plastic deformation of the off-axis plies. All laminates were symmetrically laid up and consisted of 24 plies with equal size for both the unnotched and notched experiments. Unnotched experiments showed that the failure strain (strain at the peak stress) was independent of lay-up configuration. They concluded that based on this observation a critical strain to failure or maximum strain criterion can predict the failure point with adequate precision. For notched specimens, the failure strength, damage zone size at failure, and notch size effect supported predictions of the cohesive crack model of Soutis et al. (1991). To use the cohesive crack model, they measured the compressive fracture toughness of centre-cracked specimens from which they calculated the compressive toughness. They assumed that the response of each specimen is linear elastic before the failure load. Then, by using the failure load they measured the compressive fracture toughness. Following these works, Bazant et. al. (1999) presented a size effect study on compression strength of fibre composites failing by kink band propagation. In this work, size effect law, proposed by Bazant (1983) for quasi-brittle materials, was verified. Bazant generalized this law to notch-free specimens attaining the maximum load after a stable growth of kink band and verified this law using Soutis et al. experiments (1993). In his verification, he used the fracture energy and the length of the fracture process zone measured solely from the maximum load. 108 Chapter 5: Model Verification and Validation Soutis et. al. (2002) presented another investigation on the compressive strength of carbon fibre-epoxy laminates. They used the same material as their previous work (in 1993) but with a [(±45/0/90)3]S lay-up. All multidirectional laminates were approximately 3 mm thick with different specimen and notch sizes. Failure was sudden and occurred within the gauge section. Failure mechanism was also a combination of kinking and delamination in all specimens. In these experiments, they studied the effect of both notch size and specimen size on the compressive strength of panels. Notched panel strength was predicted successfully as a function of hole size and width, using Soutis et al. (1991) linear cohesive zone model. The model predicted the strength by using independently measured laminate parameters from compressive study of unnotched specimens. In this section, the more recent experiments by Soutis et al. (2002) will be simulated using the analog model. To estimate the damage material parameters a procedure similar to what was used before (Section 5.2.) will be utilized. These estimations then will be verified by using the implementation of the analog model in LS-DYNA. Fracture energy, as the main damage parameter, will be derived from the measurement of compressive fracture toughness in Soutis et al. experiments (1993). Using these parameters, notch size effect and specimen size effect diagrams will also be derived and compared with experimental results. It is expected that the results for a blunt notch will be less dependent on the fracture energy compared to those for a sharp notch (see McClennan, 2004). 109 Chapter 5: Model Verification and Validation 5.3.2. Test Geometry Several specimens were cut from carbon epoxy panels, 3 mm thick, to carry out both the un-notched and notched experiments. There were 4 different specimen sizes, each with different circular notch sizes drilled at the center of the specimens. At least four specimens were tested for each configuration. The circular holes were drilled using a tungsten carbide bit to minimize fibre damage and delamination around the notch boundary. There were also glass-fibre epoxy reinforcement tabs bonded at both ends of the specimens for load application. A schematic of the test specimen geometry is shown in Figure 5-17. Specimen sizes and notch diameters are also shown in Table 5-12. 4 b i i*— Is » . — 9 + h > 1 a c r - • Figure 5-17 Test geometry for Soutis et al. experiments (2002). "b" is the length of the tab section. All experiments were quasi-static compressive tests with a displacement controlled rate of 1 mm per minute. Several experiments were stopped during the damage growth, before complete failure, to study the damage around the notch boundary. There were also anti-buckling devices employed to prevent the specimens from buckling. Anti-buckling devices contained a window around the notch tip in order to study the damage propagation, while restraining the specimen from general bending. Frictional effects between the anti-buckling devices and specimens were minimized by using Teflon tapes to line the inner faces of the fixture. 110 Chapter 5: Model Verification and Validation Table 5-12 Specimen sizes with different notch sizes for Soutis et al. experiment (2002). Specimen Sizes (mm) Hole Diameter (mm) a/W 30x30 1.5 0.05 3 0.1 6 0.2 9 0.3 12 0.4 15 0.5 50x50 2.5 0.05 5 0.1 10 0.2 15 0.3 20 0.4 25 0.5 70x70 7 0.1 14 0.2 21 0.3 28 0.4 35 0.5 90x90 9 0.1 18 0.2 27 0.3 36 0.4 It is also noteworthy that the load introduction to the specimens was mainly done by end loading using a modified ICSTM fixture as shown below. Figure 5-18 Modified ICSTM compression test fixture. 111 Chapter 5: Model Verification and Validation 5.3.3. Material The material used in this experiment was a multidirectional laminate autoclaved from Toray T300 carbon fibres and embedded in Ciba-Geigy BSL 924C epoxy resin. The pre-preg tapes were laid up by hand into a [(±45/0/90)3]S quasi-isotropic lay-up with an approximately 3 mm thickness for all specimens. From the material parameters of the unidirectional lay-up, we can calculate the laminate parameters in the load direction, as given in the following table. Table 5-13 Material parameters for the carbon-epoxy laminate in the compressive load direction. [(±45/0/90) 3] s Amount Longitudinal Young's modulus 63 GPa Transverse Young's modulus 63 GPa Shear modulus 24 GPa Poisson's ratio 0.315 We now need to estimate the damage parameters for the failure process. The key to derive these parameters is first to study the damage mechanism in these laminates. As mentioned, for all laminates, failure was sudden and occurred within the gauge section and was a combination of kinking and delamination. The damage initiation strain is the strain, in which matrix cracking, fibre rotation and delamination start to propagate. In the experiments by Soutis et al. (2002) on notched specimens, this strain was measured to be around 80% of the strain at the peak stress point. The average strain at the peak stress point was reported to be almost 1.0 %. Therefore, the damage initiation strain can be estimated to be -0.008. This number is the damage initiation strain for both fibre and matrix. As described before, this is due to the fact that matrix cracking and fibre rotation initiate at the same strain in the kinking process. The saturation of the kink band (before broadening) is related to the strain at the peak stress. Because damage propagation is simulated as a parabolic function in the analog model (see Section 3.4.2.) the damage saturation strain is always equal to either twice the 112 Chapter 5: Model Verification and Validation strain at the peak stress point or the damage initiation strain, whichever is larger. From this explanation, the damage saturation strain is estimated to be around 2.0%, which is twice the strain at the peak stress point. The compressive fracture toughness for a variety of lay-ups for carbon fibres T800 embedded in Ciba-Geigy BSL 924C epoxy resin was measured by Soutis et al. (1993). The only difference between this and their other experiment (2002) is the T300 carbon material used in the latter. Now the question arises that whether the measured compressive fracture toughness can be used for the same lay-up and matrix material, but with different carbon fibres. It is noted that because the Young's modulus in both cases are almost the same, in kinking and delamination failure mechanisms where the fracture energy mainly depends on the matrix cracking, we can use the same amount for compressive toughness. This amount for [(±45/0/90)3]S lay-up was reported to be 40 MPa ml/2 (Soutis et al., 1993). Based on this number and by knowing the Young's modulus for this quasi-isotropic lay-up the fracture energy can be calculated as below: Gf = A: 2 /£ = 402/62.00 = 25.4 kN/m = 25A mJI mm2 (5-10) As explained in the previous section, the two important parameters in describing damage propagation in notched sandwich panels are damages saturations in the matrix and. the fibre. As an estimate of damage at the saturation stage, we assume that the entire matrix in the [0], 50 % in each of the +45 and -45, and almost nothing in the [90] layers will be damaged and therefore the damage parameter due to the matrix damage saturation would be 0.5. This estimation is based on the idea that the damage in the [0] layer is due to kink band formation. In the +45 and -45 layers, a combination of delamination and kinking is the main damage mechanism. In the [90] layer, we assume delamination is the main failure mechanism. Accordingly, an equal amount should be assigned to the damage due to the fibre damage saturation (Floyd, 2004) and therefore this parameter is equal to 0.5 following the same explanation. To model the behaviour of laminates in the damage propagation process we also need to derive the modulus loss due to damage saturation in matrix and fibres. At the matrix damage saturation state, there is 100% damage in [0] layers due to saturation of the kink 113 Chapter 5: Model Verification and Validation band. As described before, saturation of kink band and saturation of matrix cracking coincide. This means that after the saturation of damage in matrix, 100% of the modulus in [0] layer will be lost. In the +45 and -45 degree layers, a combination of delamination and kinking decreases the modulus of layers to 50% and therefore only 50% of fibres carry compressive loads at this stage. This is due to the fact that only 50% of the matrix in these layers cracks and therefore 50% of fibres still have side resistance from buckling. In [90] layers, there is almost no matrix cracking. This means that the modulus loss due to the matrix cracking in [90] layers is zero. Now, by using ply discount method (Floyd, 2004) we can derive the remaining modulus at the saturation of matrix damage equal to 13.71 GPa. This modulus is the sum of 50% modulus in +45 and -45 degree layers and 100% modulus in 90 degree layers. The calculation of the modulus loss due to the fibre and matrix damage saturation is given below: c = 1 _ E ~ n g = ] _ _ 3 _ 1 _ 0 7 8 undamaged U Z • y y (5-11) l-Rf =1-0.78 = 0.22 Based on these two parameters and by knowing the amount of fracture energy, we can simply calculate the area under the stress-strain curve to calculate the height of damage zone. This characteristic height is the height of damage zone at the saturation of kinking and delamination, before broadening. Assuming that there is no plastic deformation upon unloading from the peak stress point and by using Equation 4-32, we have the following equation for strain energy release density: es „ „ ™ 0.020 y - = 7 / 4 8 x £ L — ^ — = 7/48 x (62.99x0.78) m scs-ect (0.020-0.008) (5-12) - > X c =4.78 MPa = 4.78 xlO 3 KPa Therefore, the characteristic height of damage is: hc = Gf lyc = 5.32 mm (5-13) 114 Chapter 5: Model Verification and Validation There are two more parameters required to capture the behaviour of the damage propagation in these laminates: plateau stress and ultimate strain. Since there is no data reported in Soutis et al. (2002) on these two parameters we have to use similar experiments. Sivashanker (1998) has done an experimental study on the compressive response of the same lay-up and material as in Soutis et al. (2002), except that they used T300 carbon. We can safely ignore this difference as the damage mechanism in these panels mainly depends on the matrix properties. In this work, the plateau stress was reported to be around 100-133 MPa. Therefore, a plateau stress of 100 MPa will be used here. Sivashanker (1996) also presented a study on the response of unidirectional composites with the same material as in Soutis et al. (2002). In their study, the ultimate strain was reported to be around 3-4%. This should be used as a lower bound in the current study, due to the effect of off-axis plies. For this reason, the ultimate stain is chosen to be 5.0%. As a summary, the following parameters will be used to simulate the damage propagation in the laminates. Table 5-14 Damage parameters for |(±45/0/90)3] s carbon-epoxy laminate. Parameter Symbol Amount Matrix damage initiation 'f. -0.008 Fibre damage initiation '•/• -0.008 Matrix damage saturation m -0.020 Fibre damage saturation -0.05 Damage due to the matrix saturation < 0.5 Damage due to the fibre saturation 0.5 Modulus loss due to the matrix damage \-Rf J m 0.78 Modulus loss due to the fibre damage I - Rf: 0.22 Compressive toughness Gr 25.4 mJ/mm2 Damage height K 5.32 mm Plateau stress ®~ plateau 100 MPa 115 Chapter 5: Model Verification and Validation 5.3.4. Stress-Strain Curve Scaling As described in Section 5.2. , we need to properly scale the stress-strain curve in the finite element model based on the element height and width. For each element, the stress-strain response is the response of Gauss point in that element. Based on the notch tip response from the analytical solution (following section), we can modify the stress-strain response at the Gauss point to accurately capture the behaviour at the notch tip (see McClennan, 2004 for more details). In this section, first we introduce the analytical solution for stress concentration factor at the notch tip and then by using the stress contours resulting from the finite element model, we can modify the response of the element. This is in addition to scaling the stress-strain curve of an element according to the element height (Floyd, 2004). 5.3.4.1. Analytical Solution To calculate the stress concentration factor at the notch tip, we can use the formulation for the panel under tension. A schematic of the panel is presented in the Figure 5-19 below. * = Point of maximum stress Figure 5-19 Open hole panel under tensile loads. Based on the Heywood formula (Peterson, 1974), the stress concentration factor, Kg, is: 116 Chapter 5: Model Verification and Validation 2+ 1 K„ = • W (5-14) If w 5.3.4.2. Stress Contour Using LS-DYNA, we can obtain the stress contour around the notch tip. This stress contour will be used to calculate the ratio between the notch tip stress and stress at the nearest Gauss point. This ratio can then be used to modify the stress-strain response of each element in the analysis (McClennan, 2004). A schematic of the finite element mesh, for half of the specimen, and the stress contour at different load stages are shown in Figure 5-20 and Figure 5-21. I i i I I I Symmetry line I Figure 5-20 Finite element model for the half of the 50*50 mm specimen with the hole diameter equal to the 10 mm. 117 Chapter 5: Model Verification and Validation In Figure 5-21, the specified points are at the Gauss points in the finite element model. It is seen that when the far field stress is equal to 195 MPa, there.is no damage in the laminate. The notch tip stress at this loading stage can be calculated from the above analytical formulation. At other loading stages, damage starts to propagate and therefore the stress level at the notch tip drops. In such cases, the notch tip stress is extrapolated from the stress at other Gauss points. —•— Remote Stress= 195 MPa A Remote Stress= =307 MPa - H — Remote Stress= =236 MPa 3400 5 10 15 20 Distance from the notch tip (mm) 25 Figure 5-21 Stress contour at different load stages for the open hole specimen in Figure 5-20. As damage progresses, the notch tip stress drops after the remote stress reaches 236 MPa. 5.3.4.3. Scaling In order to consider the width effect, we need to lower the Young's modulus and increase the saturation strain forward to get the exact stress-strain curve at the notch tip, and preserve the fracture energy in the meantime (Floyd, 2004; McClennan, 2004). This has been demonstrated in Figure 5-22. 118 Chapter 5: Model Verification and Validation For the height effect, on the other hand, we have to modify the stress-strain response of the element after the peak stress according to the scaling law (Section 4.4.). This is also shown in Figure 5-23 below. Master Curve Modified Curve Figure 5-22 Modifying the Stress-Strain curve based on the element width. Master Curve Modified Curve Figure 5-23 Modifying the Stress-Strain curve based on the element height. The scaling is shown for an element height less than hc. Based on these two figures, the modified Young's modulus and scaling factor for each model are derived. These parameters are shown for the simulation of 50x50 mm specimen in Table 5-15 below. 119 Chapter 5: Model Verification and Validation Table 5-15 Modified parameters for the simulation of 50x50 mm specimens. Notch Diameter (mm) Kg Modified Young's Modulus (GPa) k-scaling Factor 2.5 3.0 49.94 4.4 5 3.0 50.0 4.4 10 3.1 56.75 3.8 15 3.3 57.53 3.7 20 ' 3.7 56.95 3.8 25 4.3 65.57 3.8 120 Chapter 5: Model Verification and Validation 5.3.5. Simulation Results The results of simulations for the specimen strengths are shown in Figure 5-24. In this figure, there are also two LEFM bounds. One is the "hole insensitive" bound, which is obtained by multiplying the peak stress by the minimum net area for the notched specimen. The other one is the hole sensitive bound, which is obtained from the calculation of stress concentration factor. By calculating the stress concentration factor from analytical solution we can estimate the far field stress, while at the notch tip stress reaches the critical stress. In this figure, the LS-DYNA prediction lies between the minimum and maximum of the experimental results. 0_7 ~ ( Figure 5-24 Analytical and numerical strength predictions of 50*50 mm open hole panels under uniaxial compressive loads. 121 Chapter 5: Model Verification and Validation 5.3.5.1. Notch Size Effect For the 50*50 mm specimen, the notch size effect is presented in this section. The simulation results for different fracture energy values are shown in Figure 5-25. 0.2 -I 1 1 1 1 1 1 0 0.1 0.2 0.3 0.4 0.5 0.6 Hole Diameter/Specimen Width Figure 5-25 Notch size effect results for different fracture energy values. It is clear that for different fracture energy values, the resulting curves are almost parallel to each other. Also as expected, by increasing the fracture energy, the predicted specimen strength increases. For Gf = 25.4 mJ/mm2, the predictions tie between experimental bounds. We can also obtain the results for different saturation strains, as shown in Figure 5-26. The results are presented for only two saturation strains. It is also obvious that the best result is obtained when saturation strain is equal to -0.020. This confirms our first estimation of the saturation strain. It can also be observed that by increasing the 122 Chapter 5: Model Verification and Validation saturation strain, the resulting curve moves up but remains almost parallel to the original curve. 150 H 1 1 1 1 r— 1 0 5 10 15 20 25 30 Hole Diameter (mm) Figure 5-26 LS-DYNA strength predictions for different saturation strains in 50*50 mm panels. Fracture energy is equal to 25.4 mJ/mm2. 123 Chapter 5: Model Verification and Validation 5.3.5.2. Specimen Size Effect In this section, the effect of specimen size on its compressive strength is studied. Different specimen sizes are examined for a constant hole diameter to specimen width ratio of 0.2. Results for two different fracture energy levels are shown in Figure 5-27. For fracture energy equal to 25.4 mJ/mm2, the predictions fit better with the experimental results. This number is the estimate that is obtained from the study on the failure mechanism and measurement of compressive toughness (Section 5.3.3.). 450 400 350 CS & 300 C a 250 200 150 -tt-Gj-25.4 mJ/mm2 Experimental Results —•— Min. Experiment -X— Max. Experiment X Gj-= 30 mJ/mm2 Gf = 25.4 mJ/mm2 Gj- = 30 mJ/mm2 i 20 30 40 50 60 70 Specimen Width (mm) 80 90 100 Figure 5-27 Specimen size effect for different fracture energy values. (Hole diameter/ Specimen width = 0.2). We can obtain the results for different damage saturation strains. This has been depicted in Figure 5-28 for two different saturation stains, but with a constant fracture energy of 25.4 mJ/mm2. Also it is shown that for saturation strain equal to -0.020 the predictions are closer to the experimental results. 124 Chapter 5: Model Verification and Validation It is noteworthy that for both the specimen size effect and the notch size effect the best results are obtained by using damage parameters derived in Section 5.3.3.. 450 400 350 rfi 300 c 250 200 150 • Min. Experiment Experimental Results -Hr- Max. Experiment e, = -0.020 - X - ES = -0.022 es = -0.022 Es = -0.020 20 30 40 50 60 70 Specimen Width (mm) 80 90 100 Figure 5-28 Specimen size effect for different damage saturation strains. (Hole diameter/ Specimen width = 0.2). 125 Chapter 5: Model Verification and Validation 5.3.6. Summary of Observations In summary, the main findings in simulation of the open hole panels under compression are presented in point form here: 1- The experimentally observed specimen size effect and notch size effect in [(±45/0/90)3] carbon-epoxy panels were simulated effectively using the constitutive model. 2- The results obtained from the LS-DYNA simulations showed that the estimate of damage parameters obtained from the study of failure mechanism can adequately predict the experimental results. 3- The model was validated by successfully simulating the damage process in angle-ply laminates that fail by kinking, delamination and off-axis matrix cracking. 4- It was confirmed that the damage-based constitutive model is capable of predicting both specimen size effect and notch size effect in all similar panels. 126 Chapter 6: Conclusions and Future Work 6.1. Conclusions Through reviewing the literature, the main differences between the tensile and compressive failure mechanisms were distinguished. One important difference between the two modes of loading is the wide range of failure mechanisms in compression which depends on geometry and material parameters. Even for the same geometry and material, the laminate lay up plays an important role in the compressive damage propagation process. Another main difference between the tensile and compressive failure mechanisms is the band broadening state in compression. After saturation of the damage zone in compression, under a constant remote stress, or plateau stress, the damage band propagates into the interior of the undamaged material. This phenomenon suggests a 2D damage propagation in compression, consisting of transient and steady-state damage growth. By taking all these differences into account, an analog model was constructed based on the experimental studies by Moran et al. (1995). This mechanical analog model is made of basic elements, such as springs and fuses, to simulate the behaviour of a composite material during the damage process caused by uniaxial compression, tension and load reversals. By grouping the basic elements, two distinct systems (boxes of elements) were constructed. Each box represents an aspect of the physical behaviour in the failure process. The laminate box represents the behaviour of the intact laminate in the damage propagation process, while the rubble box represents the elastic response of the damaged material. 127 Chapter 6: Conclusions and Future Work The more challenging issue in constructing the model was to develop an energy concept in the compressive failure process. The fracture energy is divided into two parts: the main fracture energy and the broadening fracture energy. The main fracture energy was used to modify the stress-strain response of the element following the crack band theory (Bazant and Planas, 1998; Floyd, 2004). The successful implementation of the model in the finite element code, LS-DYNA, allowed using the model to predict the response of composite components undergoing compressive failure process. Comparison of the numerical results and experimental results from two different studies (Soutis et al., 1993, 2002; Bayldon, 2003a, 2003b), was used to validate the model. These comparisons served to validate not only the predictions of the compressive strength but also the influence of the specimen size on these strength values. As a result of this work, the implemented constitutive model is proposed as an effective tool to predict the response of composite structures under compression, tension and load reversals. The successful prediction of the size effect in the validation of the model show its usefulness for predicting the response of large composite components. The conclusions can be summarized as follows: 1- Extensive data on the failure of unidirectional composite laminates under compressive loads is available in the literature. 2- Few studies have been done with respect to the compressive failure of multidirectional laminates. 3- In thermoset matrix composites, the formation of kink band under compressive loads is due to matrix cracking, which leads to softening and rotation of the fibres. 4- In thermoplastic matrix composites, the formation of kink band under compressive loads is due to the yielding of the matrix, which leads to rotation of the fibres. 128 Chapter 6: Conclusions and Future Work 5- There are two fracture energies involved in compression failure: main fracture energy and broadening fracture energy. The main fracture energy is associated with the formation of the first kink band. The band broadening fracture energy is the energy required to form additional kink bands or to expand damage into the interior undamaged material. 6- The model is capable of simulating the damage process in multidirectional laminates that fail by kinking, delamination and off-axis matrix cracking. 7- The model is capable of predicting both the specimen size effect and notch size effect in multidirectional composite panels. In other words, the model can be used to predict the response of large composite structures undergoing damage propagation in compression or tension. 129 Chapter 6: Conclusions and Future Work 6.2. Future Work The nature of the load application on composite components requires the study of the damage and failure process under a 3D state of stress. We cannot apply the ID analog model to predict the response of a specimen under tri-axial loading condition due to the effects such as Poisson's ratio or interaction of tension and compression. This reveals the need to construct a mutiaxial stress-strain model to simulate the 3D response of composite components under different loading conditions. A real structure undergoes a variety of stresses. Upon impact, fatigue, off-axis loading and other loading applications the role of shear and bending stresses become important in determining the response of composite components. At present, there are no constitutive models capable of accounting for all different loading conditions, including a combination of compression, tension, bending and shear. To achieve this goal, a comprehensive literature review is required to identify failure mechanisms under shear and bending loads. By studying the differences and similarities between these failure mechanisms and compressive and tensile failure mechanisms a 3D damage model could be constructed. It is also desirable to implement this enhanced model in a finite element code in order to provide the ability of predicting the response of composite structures in the failure process. The future work can be summarized as: 1- Extending the analog model to 3D. 2- Constructing a model to account for compressive, tensile, shear and bending failure of multidirectional composites. 3- Implementing the model in a finite element code and validating the model. 130 References Argon, A., (1972). "Fracture of composites", Treatise on Materials Science and Technology, Vol. 1, pp. 79-114.. Bayldon, J., (2003a). "Effect of defects in GA composite: size effect testing", North Western University, Presentation. Bayldon, J., (2003b). "Effect of defects on GA aircrafts", Center for Intelligent Processing of Composite Materials, Presentation. Bazant, Z.P., (1983). "Fracture in concrete and reinforced concrete", IUTAM Prager Symposium on Mechanics of Geomaterilas: Rocks, Concrete, Soil., pp. 281-316. Bazant, Z.P., (2003). "Shear buckling of sandwich, fibre composite and lattice columns, bearings, and helical springs: Paradox resolved", Journal of Applied Mechanics, Transactions ASME, Vol. 70, No. 1, pp. 75-83. Bazant, Z.P., Kim, JJ.H., Daniel, I.M., Becq-Giraudon, E. and Zi, G., (1999). "Size effect on compression strength of fibre composites failing by kink band propagation", International Journal of Fracture, Vol. 95, pp. 103-141. Bazant, Z.P. and Planas, J., (1998)."Fracture and Size Effect". CRC Press LLC. Berbinau, P., Soutis, C , Guz, I.A., (1999). "Compressive failure of 0 degree unidirectional carbon-fibre-reinforced plastic(CFRP) laminates by fibre microbuckling", Composites Science and Technology, Vol. 59, pp. 1451-1455. Budiansky, B., (1983). "Micromechanics", Computers and Structures, Vol. 16, pp. 3-12. Budiansky, B., Fleck, N.A., (1993). "Compressive failure of fibre composites", Journal of Mechanics and Physics of Solids, Vol. 41, No. 1, pp. 183-211. 131 References Budiansky, B., Fleck, N.A., Amazigo, J.C.O, (1998). "On kink-band propagation in fibre composites", Journal of the Mechanics and Physics of Solids, Vol.46, No.9, pp. 1637-1653. Dow, J.F., Gruntfest, I.J, (1960). "Determination of most needed, potentially possible improvements in material for ballistic and space vehicles", General Electric Co. Report, No. TIS R60SD389. Dugdale, D.S., (1960). "Yielding of steel sheets containing slits", Journal of Mechanics and Physics of Solids, pp. 100-104. Engesser, F., (1889). "Die Knickfestigkeit gerader Sf'abe.", Zentralblatt des Bauverwaltung, Vol. 11, pp. 483.-486. Fleck, N.A., (1997). "Compressive failure of fibre composites", Advances in Applied Mechanics, Vol. 33, pp. 43-113. Fleck, N.A., Deng, L., Budiansky, B., (1995). "Prediction of kink width in compressed fibre composites", Journal of Applied Mechanics, Transactions ASME, Vol. 62, No. 2, pp. 329-337. Fleck, N.A., Jelf, P.M., Curtis, P.T., (1995)."Compressive failure of laminated and woven composites", Journal of Composites Technology & Research, Vol. 17, No. 3, pp. 212-220. Fleck, N.A., Sridhar, I., (2002). "End compression of sandwich columns", Composites -Part A: Applied Science and Manufacturing, Vol. 33, No. 3, pp. 353-359. Floyd, A.M., (2004). "An engineering approach to the simulation of gross damage development in composites laminates", Ph.D. Thesis, Department of Civil Engineering, The University of British Columbia. Fried, N., (1963), "The compressive strength of parallel filament reinforced plastics- the role of the resin", Proceedings of 18th Annual Meeting of the Reinforced Plastics Division, Society of Plastic Industry, Section 9-A, pp. 1-10. 132 References Gibson, L.J., Ashby, M.F., (1988). "Cellular solids: structure and properties", Pergamon, New York. Gu, H., Chattopadhyay, A., (1995). "Delamination buckling and postbuckling of composite cylindrical shells", Collection of Technical Papers AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, No. 5, pp. 3095-3105. Gupta, V. , Anand, K., Grape, J., (1998)." Failure of woven carbon-polyimide laminates under off-axis compression loading", Acta Materialia, Vol. 46, No. 2, pp. 711-718. Guynn, E.G., Bradley, W.L., (1989). "Detailed investigation of the micromechanisms of compressive failure in open hole composite laminates", Journal of Composite Materials, Vol. 23, No. 5, pp. 479-504. Hahn, H.T., Williams, J.G., (1986). "Compression failure mechanisms in unidirectional composites", ASTM Special Technical Publication, pp. 115-139. Haringx, J.A., (1942). "On the buckling and lateral rigidity of helical spring", Proc, Konink. Ned. Akad.Wetenschap., Vol. 45, pp. 533. Jackson, W.C., Ratcliffe, J.G., (2004). "Measurement of fracture energy for kink-band growth in sandwich specimens", Proceedings of Composite Testing and Model Identification, Paper No. 24. Jelf, P.M., Fleck, N.A., (1992). "Compression failure mechanisms in unidirectional composites", Journal of Composite Materials, Vol. 26, No. 18, pp. 2706-2726. Johnson, A.M., Ellen, S.D. (1974). "A theory of concentric kink. And sinusoidal folding and of monoclinal flexuring of compressible, elastic multilayers. I introduction", Tectonophysics, Vol. 21, pp. 301-339. 133 References Johnson, A.M., Ellen, S.D. (1975a). "A theory of concentric kink. And sinusoidal folding and of monoclinal flexuring of compressible, elastic multilayers. II initial stress and nonlinear equations of equilibrium", Tectonophysics, Vol. 25, pp.261-280. Johnson, A.M., Ellen, S.D. (1975a). "A theory of concentric kink. And sinusoidal folding and of monoclinal flexuring of compressible, elastic multilayers. Ill transition from sinusoidal to concentric-like chevron folds", Tectonophysics, Vol. 27, pp.l-38. Johnson, A.M., Ellen, S.D. (1976). "A theory of concentric kink. And sinusoidal folding and of monoclinal flexuring of compressible, elastic multilayers. IV development of sinusoidal and kink folds in multilayers confined by rigid boundaries", Tectonophysics, Vol. 30, pp. 197-239. Kardomateas, G.A., (1993). "Initial post-buckling and growth behaviour of internal delaminations in composite plates", Journal of Applied Mechanics, Transactions ASME, Vol. 60, No. 4, Dec, pp. 903-910. Kardomateas, G.A., Schmueser, D.W., (1988). "Buckling and postbuckling of delaminated composites under compressive loads including transverse shear effects", AIAA Journal, Vol. 26, No. 3, pp. 337-343. Khan, A.S., Colak, O.U., Centala, P., (2002). "Compressive failure strengths and modes of woven S2-glass reinforced polyester due to quasi-static and dynamic loading", International Journal of Plasticity, Vol. 18, No. 10, pp. 1337-1357. Kaute, D., Ashby, M.F., Fleck N.A., (1996). "Compressive failure in ceramic matrix composites". Advances in Applied Mechanics, Vol. .33, pp. 43-117. Kim, J.K., Sham, M.L., (2000)." Impact and delamination failure of woven-fabric composites", Composites Science and Technology, Vol. 60, No. 5, pp. 745-761. Liu, X.H., Moran, P.M., Shih, C.F., (1996). "Mechanics of compressive kinking in unidirectional fibre reinforced ductile matrix composites", Composites Part B: Engineering, Vol. 27, No. 6, pp. 553-560. 134 References McClennan, S.A., (2004). "Crack growth and damage modeling of fibre reinforced polymer composites", MASc. Thesis, Department of Materials Engineering, The University of British Columbia. McGregor, C.J., (2005). MASc. Thesis, Department of Civil Engineering, The University of British Columbia, in preparation. Mirazo, J.M., Spearing, S.M., (2001). "Damage modeling of notched graphite/epoxy sandwich panels in compression", Applied Composite Materials, Vol. 8, pp. 191-216. Moran, P.M., Liu, X.H., Shih, C.F., (1995). "Kink band formation and band broadening in fibre composites under compressive loading", Acta Metallurgica et Materialia, Vol. 43, No. 8, pp. 2943-2958. Moran, P.M., Shih, C.F., (1998). "Kink band propagation and broadening in ductile matrix fibre composites: Experiments and analysis", International Journal of Solids and Structures, Vol. 35, No. 15, pp. 1709-1722. Mouritz, A.P., Thomson, R.S., (1999). "Compression, flexure and shear properties of a sandwich composite containing defects", Composite Structures, Vol. 44, No. 4, pp.263-278. Niu, K., Talreja, R., (2000). "Modeling of compressive failure in fibre reinforced composites", International Journal of Solids and Structures, Vol. 37, No. 17, pp. 2405-2428. Palmer, A., Rice, J.R. (1973). "The growth of slip surfaces in the progressive failure of over-consolidated clay", Proceedings of the Royal Society of London, A332, pp. 527-548. Peterson, R.E., (1974)."Stress Concentration Factors", John Wiley & Sons. 135 References Piggott, MR., Harris, B., (1980). "Compression strength of carbon, glass and kevlar-49 fibre reinforced polyester resins", Journal of Materials Science, Vol. 15, No. 10, pp. 2523-2538. Rosen, V.W., (1965). "Mechanics of composite strengthening", Fibre Composite Materials. American Society of Metals, Metals Park, Ohio, pp. 37-75. Shu, J.Y., Fleck, N.A., (1997). "Microbuckle initiation in fibre composites under multiaxial loading", Proceedings of the Royal Society of London, Series A: Mathematical, Physical and Engineering Sciences, Vol. 453, No. 1965, pp. 2063-2083. Sivashanker, S., (1998). "Damage growth in carbon fibre-PEEK unidirectional composites under compression", Proceedings of the 1997 Symposium on Integrated Experimental-Computational Modeling of Advanced Materials, McNu'97, Vol. A249, No. 1 -2, pp. 259-276. Sivashanker, S., (2001). "Damage propagation in multidirectional composites subjected to compressive loading", Metallurgical and Materials Transactions A: Physical Metallurgy and Materials Science Physical Metallurgy and Materials Science, Vol. 32, No. l,pp. 171-182. Sivashanker, S., Bag, A., (2001). "Kink-band propagation in a multidirectional carbon fibre-polymer composite", Metallurgical and Materials Transactions A: Physical Metallurgy and Materials Science Physical, Vol. 32, No. 12, pp. 3157-3160. Sivashanker, S., Fleck, N.A., Sutcliffe, M.P.F., (1996). "Microbuckle propagation in a unidirectional carbon fibre-epoxy matrix composite", Acta Materialia, Vol. 44, No. 7, pp. 2581-2590. Soutis, C , Curtis, P.T., Fleck, N.A., (1993). "Compressive failure of notched carbon-fibre composites", Proceedings of the Royal Society of London, Series A: Mathematical, Physical and Engineering Sciences, Vol. 440, No. 1909, pp. 241-256. 136 References Soutis, C, Fleck, N.A., Smith, P.A., (1991). "Failure prediction technique for compression loaded carbon fibre-epoxy laminate with open holes", Journal of Composite Materials, Vol. 25, No. 11, pp. 1476-1498. Soutis, C, Lee, J., Kong, C, (2002). "Size effect on compressive strength of T300/924C carbon fibre-epoxy laminates", Plastics, Rubber and Composites, Vol. 31, No. 8, pp. 364-370. Soutis, C, Spearing, S.M., (2002). "Compressive response of notched, woven fabric, face sheet honeycomb sandwich panels", Plastics, Rubber and Composites, Vol. 31, No. 9, pp. 392-397. Sutcliffe, M.P.F., Fleck, N.A., (1994). "Microbuckle propagation in carbon fibre-epoxy composites", Acta Metallurgica et Materialia, Vol. 42, No. 7, pp. 2219-2231 Sutcliffe, M.P.F., Fleck, N.A.,' (1997). "Microbuckle propagation in fibre composites", Acta Materialia, Vol. 45, No. 3, Mar, 1997, pp. 921-932. Sutcliffe, M.P.F., Fleck, N.A., Xin, X.J., (1996). "Prediction of compressive toughness for fibre composites", Proceedings of the Royal Society of London, Series A: Mathematical, Physical and Engineering Sciences, Vol. 452, No. 1954, pp. 2443-2465. Vogler, T.J., Kyriakides, S., (1997). "Initiation and axial propagation of kink bands in fibre composites", Acta Materialia, Vol. 45, No. 6, pp. 2443-2454. Vogler, T.J., Kyriakides, S., (1999). "On the axial propagation of kink bands in fibre composites: Part I experiments", International Journal of Solids and Structures, Vol. 36, No. 4, pp. 557-574. Vogler, T.J., Kyriakides, S., (2001). "On the initiation and growth of kink bands in fibre composites: Part I. Experiments", International Journal of Solids and Structures, Vol. 38, No. 15, pp. 2639-2651. 137 References Williams, K.V., (1998). "A physically-based continuum damage mechanics model for numerical prediction of damage growth", Ph.D. Thesis, Department of Materials Engineering, The University of British Columbia. Williams, K.V., Vaziri, R., Poursartip, A., (2003). "A physically-based continuum damage mechanics model for thin laminated composite structures", International Journal of Solids and Structures, Vol. 40, pp. 2267-2300. Yang, B., Kozey, V., Adanur, S., Kumar, S., (2000). "Bending, compression, and shear behaviour of woven glass fibre-epoxy composites", Composites Part B: Engineering, Vol. 31, No. 8, pp. 715-721. Yurgatis, S.W., (1987). "Measurement of small angle misalignments in continuous fibre composites", Composites Science and Technology, Vol. 30, pp. 279-293. Zhou, G., Davies, G.A.O., (1995). "Characterization of thick glass woven roving/polyester laminates: 1. Tension, compression and shear", Composites, Vol. 26, No. 8, pp. 579-586. 138 Appendix A : Taylor Series for Damage Height Function Using Taylor series, we will simplify the following function to a polynomial: f(x) = (x + dy)" (A-l) where, dy is a very small number and n is a constant value. Taylor series is given below: •fn( \ f(x) = f(a) + f'(a)x(x-a) + ^ -^-(x-a)2 + ... (A-2) Now, by expanding the function about a = x - dy, we have: / (x) = (x - dy + dy)" + n(x -dy + dy)"~] x(x-x + dy) + nx(n-\)(x-dy + dy)n~2 x { x ~ X + d y ^ +... ^ Now considering the fact that dy is very small, we can derive: fix) = (x + dy)n « (x)n + n(x)"~] x dy (A-4) 139 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0063388/manifest

Comment

Related Items