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MODE I FRACTURE CRITERION AND THE FINITEWIDTH CORRECTION FACTOR FOR NOTCHED LAMINATED COMPOSITES By BOKWON LEE A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2003 Copyright 2003 by Bokwon Lee To my parents, Jonghan Lee and Jongok Woo ACKNOWLEDGMENTS I am grateful to Dr. Bhavani V. Sankar for giving me the opportunity of pursuing this research and for the insightful guidance he provided me, not only in the research work, but throughout the course of my graduate studies at the University of Florida. I would also like to express my gratitude to Dr. Peter G. Ifju and Dr. Ashok V. Kumar for serving as my supervisory committee members and for giving me advice. I must also acknowledge the Korea Airforce for giving me an opportunity to study abroad and meet wonderful people in the research world. The experience I have gained at the Center for Advanced Composites will be greatly helpful to me in my duties as an Airforce Logistics Officer. I wish to thank all my fellow students for the fun and support. I greatly look forward to having all of them as colleagues in the years ahead. The final acknowledgement goes to all of my family, Jonghan Lee, Jongok Woo, and Hongwon Lee, for their unconditional support and love. TABLE OF CONTENTS page A CK N O W LED G M EN T S ......................... ...................................................................iv LIST OF TABLES ........... .............................................. ................ vii LIST OF FIGURES ................... .............................. viii AB STRA CT ............................................................... ... ......... ............... xi CHAPTERS 1 IN TR OD U CTION ............................................... .. .... .................... .. 1 1.1 B background and O objectives ................................................ ............................ 1 1.2 Literature Review ........ ....................................... ........ .... .... .. .......... .. 3 2 LAYUP INDEPENDENT FRACTURE CRITERION.............. ........ .............. 7 2 .1 Intro du action ................... ................... ...................7......... 2.2 Stress Intensity Factor M easurements................................ ......................... ...... 8 2.2.1 Experim mental Background........................................................................... 8 2.2.2 Stress Intensity Factor Calculations............................................................ 10 2.3 Derivation of Stress Intensity Factor in the LoadCarrying Ply................................... 12 2 .4 R results and D iscu ssion ......... .. ............................................................. .............. 15 3 FINITE ELEM ENT ANALYSIS .............. .......................................................... 18 3.1 2D Finite Element Global M odel.......................... ................... .................. 18 3.2 2D Finite Element SubM odel ....................................................... ............ 26 3.3 Comparison and FE Results and Analytical Model................................. ............... 31 3.4 FiniteW idth Correction Factor ............................................. ............................ 34 3.4.1 Isotropic FiniteWidth Correction Factor.......... .................................... 34 3.4.2 Orthotropic FiniteWidth Correction Factor............... ............ .............. 35 3.4.2.1 Developing procedure for FWC solution ................. ................. ... 35 3.4.2.2 Anisotropy param eter, .............. ................................. .............. 42 3.5 Effect of Blunt Crack Tip ....................... .................................... .......................... 43 3.5.1 2D FE M odeling Procedure ........................................ ......... .............. 43 3.5.2 Results and Discussion .................... ............................. 44 3.6 Effect of L local D am age ......... .. .................... ....................................................... 46 3.6.1 3D FE Modeling Procedure for Delamination................ ................................ 49 3.6.2 3D FE Modeling Procedure for Axial Splitting .......................................... 51 3.6.3 R results and D discussion .............. ................................................................ 52 3.7 Summary of FE Results and Discussion................................... .............. 53 4 CONCLUSIONS AND FUTURE WORK ..... .................... .............. 55 4.1 C conclusions ........................................ 55 4.2 Future W ork ............................. .............. ...... 59 APPENDIX A LAMINATION THEORY ............ .... ..................................... 60 B MATHEMATICAL THEORY OF BRITTLE FRACTURE................. .................. 63 L IST O F R E F E R E N C E S ................................................................................................... 67 BIOGRAPHICAL SKETCH ............................................................ .......................... 69 LIST OF TABLES Table page 21 Material properties of unidirectional AS4/35016 graphite/epoxy ..............................8 22 Fracture toughness of AS4/35016 graphite/epoxy laminates.................. ............. 31 Jintegral value calculated from FE model of the [0/+45]s laminate ..........................38 32 Values of the anisotropy parameter, p, for the test specimens .................................40 33 The coefficients of semiempirical solution of the orthotropic finitewidth correction factor ................ ........ .............................. ...........................40 34 The finitewidth correction factors obtained from semiempirical solution...................41 35 Ranges of anisotropy parameter, p, of various composite materials........................43 41 Comparison of the fracture toughness of loadcarrying ply obtained from different m methods ............ ................ ....... ............... ........ .... 56 42 Comparison of experimental results with failure model predictions...........................58 LIST OF FIGURES Figure pge 21 Specimen specification and fiber directions............................................. ............ 8 22 Local coordinate system of crack tip area stress components ..............................12 23 Comparison of the fracture toughness of the loadcarrying ply .............................. 15 31 Scheme of twodimensional FE model of [0/45]s laminate.............. ............ 20 32 Finite element global model mesh and boundary condition ..................................21 33 Scheme of 2D FE global model analysis procedure................................ ...........22 34 Normal stress distribution in the global model in the [0/45]s laminate.....................24 35 Stress intensity factor in the global model in the [0/145]s laminate............. ..............24 36 qs distribution in the [0/45]s laminate under a load of 351.14 MPa .........................25 37 D different sizes of subm odel........................................................................ 27 38 Comparison of SIF obtained from different sizes of submodel..............................27 39 Finite element submodel mesh and linking with global model..............................28 310 Normal stress distribution in the submodel in case of [0/45]s laminate.................29 311 Stress intensity factor in the submodel in case of [0/45]s laminate...........................29 312 g, distribution in the [0/45], laminate under a load of 351.14 MPa..........................30 313 Comparison of normal stress distribution in the submodel and global model in case of [0/ 45], lam inate........... .................................... ........ ........ .............. 32 314 Stress intensity factor in the global model in case of [0/45], laminate.....................32 315 Stress intensity factor in the submodel in case of [0/45]s laminate.........................33 316 Comparison of the fracture toughness of principal loadcarrying ply...........................33 317 Comparison of the finitewidth correction factors in an isotropic plate computed by the finite element methods with the closed form solution...............................................37 318 Jintegral vs. contour line calculated from FE model of the [0/45]s laminate..............38 319 The finitewidth correction factors vs. ratio of crack size to panel width ...................39 320 The finitewidth correction factors vs. anisotropy parameter ................................39 321 The finitewidth correction factors as a function of anisotropy parameter, fp,and ratio of crack size to panel width, a/w .............. .......... ................. .... ........... 41 322 Anisotropy parameter, /, as a function of lamination angle for graphite/epoxy [ 0]s and [0/ ]s lam inates........... ....................................................... .. .... .... ..... 42 323 C rack tip shape profiles ................................................................. .....................44 324 Effect of crack tip shape on predicted fracture toughness of [0/45]s laminate ; 2D FE global model results ................................ ... .. ........ ............ 45 325 Effect of crack tip shape on predicted fracture toughness of [0/45]s laminate ; 2D FE subm odel results ..... .... .................................................... ...... .. .... ...... 45 326 VonMises stress distribution in [0/45]s laminate........ ........ ..................48 327 Comparison of stress intensity factor of the loadcarrying ply between 2D and 3D global m odel ..................................................................... .........49 328 Scheme of ply interface mesh in the [0/45]s laminate ......... ......... ............... 50 329 Estimated delamination area on interface between OE and +45E ply in the [0/145]s lam in ate .......................................................................... 5 0 330 Axial splitting failure mode in the loadcarrying ply................. ................ .............51 331 Estimated axial splitting area on interface between OE and +45E ply........................52 332 Effect of the local damage on normal stress distribution in the loadcarrying ply in the [0/45]s laminate ....... ............ .. .... ....... ................................... 53 333 Effect of the local damage on stress intensity factor in the loadcarrying ply ..............54 41 Comparison of the fracture toughness of the loadcarrying ply .............. ...............56 42 Comparison of experimental results with failure model predictions ...........................57 43 Comparison of experimental results with failure model predictions (log scale)............58 A Lam inated plate geom etry ......................................................... .............. 61 B1 Stress components in the vicinity of crack tip ................................................. 65 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science MODE I FRACTURE CRITERION AND THE FINITEWIDTH CORRECTION FACTOR FOR NOTCHED LAMINATED COMPOSITES By Bokwon Lee August 2003 Chair: Bhavani V. Sankar Major Department: Mechanical and Aerospace Engineering The purpose of this study was to investigate the applicability of the existing layup independent fracture criterion for notched composite laminates. A detailed finite element analysis of notched graphite/epoxy laminates was performed to understand the nature of stresses and crack tip parameters in finitewidth composite panels. A new laminate parameter 3 has been identified which plays a crucial role in the fracture of laminated composites. An empirical formula has been developed for finitewidth correction factors for composite laminates in terms of crack length to panel width ratio and / The effects of blunt crack tip and local damage on fracture behavior of notched composite laminates are investigated using the FE analysis. The results of experiments performed elsewhere are analyzed in the light of new understanding of crack tip stresses, and the applicability of the layup independent fracture criterion for notched composite laminates is discussed. It is determined that the analytical layup independent fracture model that considers local damage effect provides good correlation with experimental data, and the use of orthotropic finitewidth correction factor improves the accuracy of notched strength prediction of composite laminates. CHAPTER 1 INTRODUCTION 1.1 Background and Objectives Composite laminates are widely used in aerospace, automobile and marine industries. Because of their high strengthtoweight and stiffnesstoweight ratio, use of fiberreinforced composite materials in advanced engineering structures such as high performance aircraft has been increasing. Recently, laminated composites have been increasingly used in military fighter airframes and external surfaces because those designs are required to handle high aerodynamic forces, be lightweight for maximizing airtoair performance, minimize radar cross section, and withstand foreign object and battle damage. In addition, by properly sequencing the stacking layup, a wide range of design requirements can be met. Laminated composites posses distinctive advantages as mentioned above; however, fiberreinforced composites exhibit notch sensitivity, and this can be an important factor in determining safe design. The problem of predicting the notched strength of laminated composites has been the subject of extensive research in the past few years. Owing to the inherent complexity and the number of factors involved in their fracture behavior, several semiempirical failure criteria have been proposed and have gained popularity. Some failure criteria are based on concepts of linearelastic fracture mechanics (LEFM), while others are based on the characteristic length and stress distribution in the vicinity of the notch. In general, these models rely on a curvefitting procedure for the determination of a number of material and laminate parameters, involving testing of notched and unnotched specimens for each selected layup. One limitation commonly reported is that the fracture toughness in many cases is dependent upon laminate configuration, the specific fiber/matrix system, etc. The use of these failure criteria without properly accounting for this dependence has led to some mixed results. Awerbuch and Madhukar [1] reviewed some of the most commonly used fracture criteria for notched laminated composites. They collected data from several sources and different materials and evaluated the performance of each criterion. They concluded that the parameters are strongly dependent on laminate configuration and material system and must be determined experimentally for each new material system and laminate configuration. To overcome this limitation of those popular failure criteria, recently a layup independent fracture criterion was developed by employing linear elastic fracture mechanics concepts. The fracture toughness of the loadcarrying ply in the presence of fiber breakage was considered as the principal fracture parameter. The main objective of this research was to further verify and also improve the existing layup independent fracture criterion for notched laminated composites. Detailed finite element analyses, both 2D and 3D, of the laminate were performed to understand the average laminate stresses as well as stresses in individual layers. An analytical model for determining the principal fracture parameter, fracture toughness of the loadcarrying ply, was also developed. It has been found that factors such as crack bluntness, local damage such as fiber splitting, delamination, etc. in the vicinity of crack tip play a significant role in reducing the stresses in angle plies and thus increasing the stresses and stress intensity factor in the loadcarrying ply. A new laminate parameter, referred to as p/, has been identified. This parameter represents the ratio of the axial and normal stresses at points straight ahead of crack tip, and plays a crucial role in the failure of the notched laminate. A secondary objective was to develop a semiempirical solution for finitewidth correction factor (FWC) for laminated composites to improve the accuracy of prediction of notched strength. Due to the lack of a closed form solution for orthotropic finitewidth correction factor, most existing failure criterion used the isotropic finitewidth correction factor for prediction of notched strength of laminates. In some cases, the application of the isotropic finitewidth correction factors to estimate the anisotropic or orthotropic finitewidth correction factors can cause significant error. An attempt was made to reduce the error in predicting the notched strength by developing an orthotropic finite width correction factor for various laminates configurations. 1.2 Literature Review Numerous failure models have been developed for the prediction of the notched strengths of composite laminates. Due to the complication of analyzing the fracture behavior of notched composite laminates, a number of assumptions and approximations were contained in most commonly used failure models to predict the tensile strengths of notched composite laminates. In recent years several simplified fracture models have been proposed. The scope of this research was limited to laminated composites containing a straight center crack when subjected to uniaxial tensile loading. Waddoups, Eisenmann, and Kaminkski (WEKLEFM models) [2] applied Linear Elastic Fracture Mechanics to composites. Their approach was to treat the local damage zone as a crack, and apply fracture mechanics. Whitney and Nuismer [3] proposed the stressfailure models. These models named the AverageStress Criterion (ASC) and the PointStress Criterion (PSC) assumed that fracture occurs when the point stress or the average stress over some characteristic distance away from the discontinuity is equal to the ultimate strength of the unnotched laminate. The characteristic distances in point stress criterion and averagestress criterion, are considered to be material constants and the evaluation of the notched tensile strength is based on the closed form expressions of the stress distribution adjacent to the circular hole. Tan [4] extended this concept and developed more general failure models, the Point Strength Model (PSM) and Minimum Strength Model (MSM). These models successfully used to predict the notched strength of composite laminates subjected to various loading conditions. Such models using a characteristic length concept have been widely used to predict ultimate strength in the presence of notches. The main disadvantage is that the characteristic length is not a material constant and depends on factors such as the layup configuration, the geometry of the notch, etc. Therefore, the characteristic length obtained from tests on one laminate configuration may not be extrapolated to predict the failure of other laminates of the same material system. Mar and Lin [5] proposed LEFM fracture model called MarLin criterion, the damage zone model, and the damage zone criterion. They assumed that the laminate fracture must occur through the propagation of a crack lying in matrix material at the matrix/filament interface. It was able to provide good correlation with experimental data and at the same time is very simple to apply. However, the fracture parameter, which was used in MarLin criterion, depends on the laminate layup configuration. Therefore, the application of this fracture criterion requires experimental determination of the fracture parameter for each laminate. Chang and Chang [6] and Tan [7] proposed progressive damage models which were developed to predict the extent of damage and damage progression, respectively, in notched composite laminates. The models accounted for the reduced stress concentration associated with mechanisms of damage growth at a notch tip by reducing local laminate stiffness. Past experimental investigations, which were carried out by Poe [8], have revealed that fiber failure in the principal loadcarrying plies governs the failure of notched laminated composites. Poe and Sova [8, 9] proposed a general fracture toughness parameter, critical strain intensity factor, which is independent of laminate orientation. This parameter was derived on the basis of fiber failure of the principal load carrying ply and it is proportional to the critical stress intensity factor. Such an idea was adopted by Kageyama [10] who estimated the fracture toughness of the loadcarrying ply by three dimensional finite element analysis. Such analyses led to layup independent fracture criteria based on the failure mechanism of the loadcarrying ply, which governs the failure of entire laminate. Recently, Sun et al. [11, 12] proposed a new layup independent fracture criterion for composites containing center cracks. In their analysis, the fracture toughness of the loadcarrying ply was introduced as the material parameter. Such an analysis is approximate, since it does not take into account any stress redistribution caused by local damage and it used the isotropic form factor instead of an orthotropic finitewidth correction factor to account for finite width of the laminated specimens. There are several analytical and numerical methods [1315] to determine the orthotropic finitewidth correction factor including the boundary integral equation, finite element analysis, modification of isotropic finitewidth correction factor, etc. However, no closed form solutions are available. Therefore, the closed form solution for isotropic materials is frequently used for orthotropic material as well. Current study aims to further verify the layup independent model including new fracture parameter and improve the accuracy for the notched strength prediction of composite laminates by using orthotropic finitewidth correction factor. Chapter 1 provides an introduction and a literature review of fracture models for notched laminated composites. Chapter 2 describes a general concept of the layup independent fracture model and the analytical derivation of the stress intensity factor in the loadcarrying ply. Chapter 3 describes a FE modeling procedure including damage modeling and the development of the orthotropic finitewidth correction factor based on FE analysis. Conclusions and some recommendations for future work are presented in Chapter 4. CHAPTER 2 LAYUP INDEPENDENT FRACTURE CRITERION 2.1 Introduction Fibrous composite materials have high strength and stiffness as mentioned before. Under tension loading, however, most advanced laminated composites are severely weakened by notches or by fiber damage. Thus, a designer needs to know the fracture toughness of composite laminates in order to design damage tolerant structures. The fracture toughness of composite laminates depends on material property and laminates configuration. Consequently, testing to determine fracture toughness for each possible laminate configuration would be expensive and timeconsuming work. Thus, a single fracture parameter can be used to predict fracture toughness for all laminates configurations of the same material system. Poe and Sova [8, 9] proposed a general fracture toughness parameter, Qc, which was derived using a strain failure criterion for fibers in the loadcarrying ply. Sun and Vaiday [11] also proposed a single fracture parameter, stress intensity factor in the loadcarrying ply which was derived using classical lamination theory and LEFM theory. In this chapter, exact LEFM analytical expression for layup independent failure model is presented and it is compared with the similar methodology proposed by Sun and Vaiday [11] 2.2 Stress Intensity Factor Measurements 2.2.1 Experimental Background As mentioned earlier we have analyzed the results of fracture tests performed by Sun et al. [11, 12]. The material properties, laminate configurations used and failure load are presented here for the sake of completion. Nine different laminate configurations made from AS4/35016 (Hercules) graphite/epoxy were tested. The panels were made of unidirectional prepreg tape with a nominal thickness of 0.127 mm. The material properties for this unidirectional prepreg tape are shown in Table 21. The panel specifications, geometry, and coordinate system used for the present analysis is shown in Figure 21. All the test specimens reported here were fabricated using the hand layup technique and cured in an autoclave. Different laminate configurations were selected such that each one had at least one principal loadcarrying ply (OE), i.e., plies with fibers aligned along the loading direction. Table 21 Material properties of unidirectional AS4/35016 graphite/epoxy Young's M i ( ) Poisson's Shear Prepreg Young's Moduli (GPa) . Material Ratio modulus thickness EL ET VLT GLT, (GPa) (mm) graphite/epoxy 138 9.65 0.3 5.24 0.127 laver thickness 0125 mm LI P 4 tensile oad 254mm Figure 21 Specimen specification and fiber directions I ?\O fiber direction To make the crack, a starter hole was first drilled in the laminates to minimize any delamination caused by the waterjet. The crack was then made by a waterjet cut and further extended with a 0.2 mm thick jeweler's saw blade. The failure stress (load over nominal cross section of the laminate) for each laminate configuration tested are shown in Table 22. The fracture toughness estimated using the nominal stress intensity factor for the laminate is shown under the column "Laminate fracture toughness". This is computed using the formula KQ = aY(a/w)J (1) where a is the remote stress and Yis the finitewidth correction factor. Sun et al. [11, 12] used the Y for isotropic material which is equal to 1.0414 for the present case (a/w=0.2627). One can note that the fracture toughness estimated using this method is not the same for different laminates and hence cannot be considered as a material property. The last column of Table 22 is the fracture toughness of the loadcarrying ply calculated by Sun and Vaiday [11], which will be discussed in subsequent sections. Table 22 Fracture toughness of AS4/35016 graphite epoxy laminates [11] Failure Laminate fracture Fracture toughness of Notation Laminate stress toughness the loadcarrying ply configuration (MPa) K (Pa ) K (Pa ) S1 [0/90/45]s 343.00 44.83 3.05 115.66 S2 [45/90/0]s 323.27 42.19 1.87 108.82 S3 [90/0/45]s 316.05 41.25 + 1.46 106.43 S4 [0/+15]s 695.84 90.82 4.01 101.81 S5 [0/130]s 466.14 60.84 3.14 101.00 S6 [0/45]s 351.14 45.83 2.17 106.32 S7 [0/90]2s 446.76 58.31 5.30 109.04 S8 [45/0/+45]s 287.16 37.48 0.43 119.81 S9 [452/0/+45]s 248.40 32.42 0.60 123.64 2.2.2 Stress Intensity Factor Calculations A parameter commonly used to represent the notch sensitivity of materials is the critical stress intensity factor (S.I.F.) or the fracture toughness, KQ. Numerous investigations have attempted to determine the critical stress intensity factor for a variety of composite laminates, and those results indicate that it depends primarily on the material, laminate configuration, stacking sequence, specimen geometry and dimensions, notch length, etc. In addition, stress intensity factor is strongly affected by the extent of damage and failure modes at the notch tip. Therefore, it is a critical parameter to design composite laminates structure. The stress intensity factor of notched laminates can be calculated in three different ways. 1. From Eq. (1) using failure stress directly from the experiment 2. Finite Element Analysis I : From the normal stress distribution obtained from a series of finite element models using the following equation. KI = lim o7 (r,0) 27 (2) r>0 where r is the distance from the crack tip and normal stress component a, (r,O) is the average stress of the each ply. 3. Finite Element Analysis II : From the Jintegral calculated from finite element models. The relation of the energy release rate and stress intensity factor is given by the following equation [17]. / 1 2 1 12 1/2 2G aa2212 a22 2a12 +a66 (3 GI l + a (3) ( 2 ) alIj 2all where a, are elastic constants. When the remote applied stress is the failure stress, stress intensity factor KI becomes the fracture toughness KQ. The stress intensity factor of the L loadcarrying ply, KQ, can be estimated in two different ways. 1. From simple stress analysis using LEFM and lamination theory by calculating the portion of the applied load that is carried by the loadcarrying ply. The analytical derivation of the fracture toughness of the loadcarrying ply is presented in following section. 2. Same procedure as in Eq. (2) but using normal stress field of the loadcarrying ply extracted from a series of finite element models using the following equation. Kf = lim oL (r,0)4 2n (4) In order to estimate fracture toughness of various laminates configurations, the failure stresses of the notched laminates have to be determined from experiments. However, if the general fracture parameter, the fracture toughness of the loadcarrying ply which is layup independent, is determined from preliminary test, laminates fracture toughness can be obtained using simple analysis. Consequently, it will be discussed in the following section. 2.3 Derivation of Stress Intensity Factor in the LoadCarrying Ply In this section, we derive an analytical expression for the stress intensity factor for the loadcarrying ply in terms of the average laminate stress intensity factor obtained using three methods mentioned earlier. The derivation of stress intensity factor presented here is based on LEFM and classical lamination theory. The detailed derivation procedure can be found in Appendix A and B. J Jit.L 0 X a, (r,O) = Re[l] (5.a) (r,= K Re[ ss2] (5.b) where the parameters sl and s2 are related to the orthotropic elastic constants as explained in Appendix B. We will use a new laminate parameter 1 to denote the ratio between the two normal stresses shown in Eqs. (5.a) and (5.b). P0 = ,) Re [ ss2] (6) ( (r,O) We will use the classical lamination theory to extract the stresses in the loadcarrying ply from the force resultants acting in the entire laminate. According to the classical lamination theory presented in Appendix A, in the case of symmetric laminated plates without coupling under plane stress or plane strain conditions, the forcemidplane strain equations can be expressed in matrix form as Nx Al A12 A16 e SN = A12 A2 A o y 12 A22 A26 (7) N, _A6 A26 A66 yo where [A] is the laminate extensional stiffness matrix. The inverse relation is given by ex Al', Al A16 Nx 0 = 1 A, A* N, E y A 2 A22 26 y] (8) 7 A/16 A26 A66 xy where superscript denotes the component of inverse matrix of [A]. The stresses in the loadcarrying ply can be derived from the midplane strains as L L L SQ12 Q22 Q26 y (9) L L L L 0 xy Y16 Q26 Q66 xAY f where superscript L indicates the load carrying ply and the quantities Q, (ij = 1, 2, 6) are the stiffness coefficients of the principal load carrying plies. Substituting Eq. (8) into Eq. (9) yields the following equation. FL 0_ 1A2 Q6 A1 A2 (10) L L L L Q12 Q22 26 12 22 A6 (10) L L L L A* A* A Ar xT L 16 Q26 Q66 A1 6 A66 xy In the vicinity of the crack tip the force resultants can be written in terms of the average stresses in the orthotropic laminate N, = ta, = ta,, N, = tar, N = trX = o (11) where t is the total thickness of laminates. Substituting from Eq. (11) into Eq. (10) we obtain the stresses in the loadcarrying ply as L* P +A* L L L 2 22 126 A4+1A2 22 ) (12) L L L L +A} 16 Q26 Q66 6 +6 26 In particular we are interested in the stress component aJL responsible for fracture and it is obtained from Eq. (9) as L = A(1P+A12))+ Q (Ap+A2] (tt, (13) Then the stress intensity factor K in the loadcarrying ply can be expressed in terms of the laminate stress intensity factor K. as KA=t 12 (Aj +A1,)+Q 2(A,12 +A22) K (14) In deriving Eq. (14) we have used the assumption Cj l/ = K /K,. The layup independent fracture criteria assume that there is a critical value of K for each material system and is independent of laminate configuration as long as there is a loadcarrying ply in the laminate. In order to verify this concept we computed K. from the experimental failure loads [11] using Eq. (1). Then KL at the instant of fracture initiation was computed using Eq. (14). The resulting fracture toughness will be called KL Sun and Vaiday [11] used a similar approach to calculate the stress intensity factor in the loadcarrying ply, but they used the remote stresses applied to the laminate in order to determine the ratio of oa l/oy. Since the applied load is uniaxial, this ratio was equal to zero in their case. This is equivalent to taking the factor P as equal to zero. It should be emphasized that this stress ratio is not equal to zero in the vicinity of the crack tip, as there is a nonzero component of ox is present (see Eq. 5b). We denote this fracture toughness as K(B Thus the relation between K( and Kg can be obtained by setting p =0 in Eq. (14) and it takes the form L t .L L K (B) 12A12 Q22A22 KQ (15) 2.4 Results and Discussion The values of K (A)and KQn are shown in Figure 23 for the nine laminate configurations. The average values and corresponding standard deviation are 74.35 MPa m1/2 and 18.35 % for K(A), and 110.28 MPam1/2 and 9.8 % for KL(). Surprisingly the case B wherein the 3 value was taken as zero yielded consistent layer independent fracture toughness compared to the case A where the actual stress ratio (3>0) was used. 140 Q (B) 120 L K 100 E 80 LL 40 20  S1 S2 S3 S4 S5 S6 S7 S8 S9 Laminate configuration Figure 23 Comparison of the fracture toughness of the loadcarrying ply 16 Such analyses are approximate, since they do not consider any stress redistribution caused by physical fracture behavior such as local damage in the form of matrix cracking, delamination, etc. In order to fully understand the nature of crack tip stress field in finitewidth laminates a detailed finite element analysis was performed. The procedures and the results are discussed in Chapter 3. CHAPTER 3 FINITE ELEMENT ANALYSIS A detailed finite element analysis of fracture behavior of notched laminated composites was conducted in conjunction with the analytical failure models described earlier. The purpose of the finite element analysis was to develop a model that could predict the fracture parameters of notched laminated composites and investigate the effect of local damage and crack tip shape on the stress intensity factor for notched laminated composites. Another goal of the study was to determine the orthotropic finitewidth correction factor using Jintegral. One of the leading commercial FE packages, ABAQUS 6.2 [18] was used to analyze the various test specimens. Two types of analyses were performed. In the 2D model the specimens were modeled as orthotropic laminate. In the second model 3D solid elements were used to model the individual layers of the laminate. In both cases submodeling was performed to improve the accuracy of the calculated crack tip parameters such as stress intensity factor and JIntegral. The mode I stress intensity factor, KI, can be calculated in two different ways based on the finite element analysis as mentioned earlier. However, stress intensity factor of the loadcarrying ply can be calculated only one way using the Eq. (4) because ABAQUS can not calculate Jintegral for each ply level. aoL (r,0) is extracted from normal stress field in the loadcarrying ply of FE model. The stress intensity factors obtained using the above methods from two types of FE models are compared with the analytical models in the subsequent section. It is noted that overall analysis procedure and detailed results are presented for case of the [0/45]s laminate and the results of other eight laminate configurations are summarized. 3.1 2D Finite Element Global Model The purpose of the 2D analysis is to compare the results with the analytical model so that the effect of finitewidth of the specimen can be understood. Further FE models can be used to understand the effects of blunt crack tip and also other forms of damage such as delamination and fiber splitting. The various laminate configurations with center notch were modeled with eight node plane stress elements (CPS8R element). A quarter model was used, with symmetric boundary conditions. The width of the model, w, was 19.05 mm, and the length was 254 mm. The notch was modeled as a sharp crack with a half width, a= 5 mm. Since the layup is symmetric, it was only necessary to model half of the thickness. The main difference between global model and submodel is mesh refinement. The FE global model uses relatively coarse mesh compared to the FE sub models. A fixed element size with width of 1 mm was used in the FE global model. A relatively fine mesh was used adjacent to the notch. The geometry and the finite element models were created using ABAQUS/CAE modeling tool and ABAQUS keyword editor. Figure 32 shows the initial mesh of the upper left quadrant. Separate elements were used to represent each ply and common nodes were used for interface of plies. Figure 31 shows the scheme of twodimensional FE model of [0/+45]s laminate. Figure 33 shows the overall global modeling procedure in case of [0/45]s laminate. The material property of each ply was modeled as a homogeneous linear elastic orthotropic material throughout this FE analysis. In order to use a single global coordinate system, the material properties of angle plies were transformed using the transformation relation for engineering constants. Orthotropic properties for AS4/35016 graphite/epoxy unidirectional prepreg were defined as shown in Table 21. The material property of each angle ply was implemented in ABAQUS by means of user material subroutine (UMAT). The fixed grip loading condition was simulated by constraining the nodes along the edge of the plate to have the same displacement under an applied load. This was also implemented by using the EQUATION command. The failure load obtained from experiments (see Table 21) was applied. In global model analysis, Jintegral also was calculated using ten contour lines to determine the orthotropic finitewidth correction factors. It will be discussed in Chapter 3.4. 0O Loadcarrying ply +45ply Fiber orientation boundary condition 3 layers of orthorropic elements Elements attached at common nodes Figure 31 Scheme of twodimensional FE model of [0/+45]s laminate 21 M Mutti point constraint, v = Co1nstat' L> S..Crackt p. CracKtip Figure 32 Finite element global model mesh and boundary condition F com serial pSS S ABAQUS/CAE I S Keyword Editor Geometry modeling elcopy, element shift=10000, shift a nodes=0: duplicate element to model each ply and common nodes are used for interface of each ply w Material property modeling for each ply : User material subroutine [ Boundary and load condition Equaton : define grip condition : define symmetric boundary condition * History output and field output request : stress, displacement, contour integral, UVAR * contour integral, contours= 10,symm ,crack,1.0,0.0: request Jintergral and define crack tip normal vector * node file, U,S : request data for interpolation to submodel boudnary * Mesh modeling and Element definition a : CPS8R element SRequest to generate G_S42D.inp  *Submit input file to ABAQUS/standard A N L Y I S A B A______________________________________________________ Transform material property of OE ply to angle ply SDefine a user variable  G_S42D.inp : modeling input data G_S42D.fil : results for interpolation to submodel boundary G_S42D.dat : numerical data of FE analysis G_S42D.odb : visualization data  SG_S42D.cae/msg/res/com/...etc * Calculate K using Eqs. 2, 4 * Convert Jintegral to K using Eq. 3 Visualize the stress and displacement distribution Figure 33 Scheme of2D FE global model analysis procedure Figure 34 shows the distribution of normal stress of the loadcarrying ply and average normal stress of the laminate, respectively, for the global model. From the normal stress distribution in the Figure 34, it is clear that a, stress in the loadcarrying ply is much higher than the average and hence that of angle ply. In the Figure, oy was L +45 45 defined as ' ' A contour plot of a, distribution is shown in Figure 36. 3 Figure 35 shows the values of o 2ir and a F2r as a function of r in the [0/145]s laminate. The stress intensity factor of the loadcarrying ply and the laminate can be obtained by extrapolating measured normal stress distribution from the FE global model based on Eqs. (2) and (4). The stress intensity factor obtained from global FE model agree well with the analytical model case B (Eq. 15) which was calculated assuming ox 0 ahead of crack tip. 6.00E+09 5.00E+09  4.00E+09 1 3.00E+09  2.00E+09 i 1.OOE+09  I I i .A** L in the loadcarrying Y *. aL in the loadcarrying Sx ..... 7 in the laminate ...,... i in the laminate , A .A "Ni [. ',". ,..... A.., .. .. ........ ..... ..... .. . I I I S.... ..... ..... ..... ... ...... n i i , ALI , ~Cl4.. ...i. 0.OOE+00 .......... ............... ............. ..... ......... .. g.. 4 0.00E+00 2.00E03 4.00E03 6.00E03 8.00E03 1.00E02 1.20E02 1.40E02 Distance from crack tip, r (m) Figure 34 Normal stress distribution in the global model in the [0/45]s laminate 3.00E+08 2.50E+08  0 o 2.00E+08 SE 1.50E+08 0 '' 1.00E+08 M 5.00E+07 * O.OOE+00 0.00E+00 2.00E03 4.00E03 6.00E03 8.00E03 1.00E02 Distance from crack tip, r (m) 1.20E02 1.40E02 Figure 35 Stress intensity factor in the global model for [0/45]s laminate (a) Loadcarrying ply, OEdegree (b) +45E ply Figure 36 a, distribution in the [0/+45]s laminate under a load of 351.14 MPa ODB: 3o04 4.b 2AQUS/Sta2dard 6 2 19 ed FA OS 22 23:32 'I Clj. 2003 Step: Step1 Inree 1: Step Time 1 000 No=a V: efodmation cale Factor +1.000e+01 2 OD:B 3204450d AQ ITS/Stadard 2 1 Wed Pb 05 22 2: 329 712 lk i 2003 Increen 1: Step T..e = 1 000 Defo ed var: S Defomatn Scae Factor +1.00e+01 3.2 2D Finite Element SubModel The analysis was repeated with a very refined element size in order to investigate the local fracture behavior and the sensitivity of the results to mesh refinement. All other aspects of the analysis were kept the same as global model. For efficiency of computation, the finite element submodeling analysis technique was adopted. The sub modeling analysis is most useful when it is necessary to obtain an accurate, detailed solution in a local area region based on interpolation of the solution from an initial, relatively coarse, global model. The submodel is run as a separate analysis. The link between the submodel and the global model is the transfer of results saved in the global model to the relevant boundary nodes of the submodel. Thus, the response at the boundary of the submodel is defined by the solution of the global model. However, in order to adopt submodeling technique, the accuracy of submodel should be ensured by checking the comparing important parameter to determine reasonable submodeling size. Three different sizes of submodels were modeled to determine adequate submodel size to minimize the execution time and maximize accuracy. Figure 37 shows the different size of submodel and Figure 38 shows the results of comparison of stress intensity factor obtained from submodels with stress intensity factor obtained from Jintegral using Eq. (3). For efficiency of computation, submodel size B was chosen for subsequent studies. \ \ ABC Figure 37 Different sizes of submodel (A= 10 %, B= 20 %, C= 40 % of crack size, a) 60 SIF obtained from Jintegral in the 50 global model 40  0 40 aE 30 0a 20 10 submodel A submodel B submodel C Figure 38 Comparison of SIF obtained from different sizes of submodel A fixed mesh size, 0.01 mm was used in the submodel and overall size of submodel is 20% of crack size, a. Figure 39 shows submodel mesh and linking with global model. __ , Submodel _____ L  T boundary !i : ^ : 1. i l 1] 1 i li.l i11 ; i 40!?"" ! il i ..i! .l' li !!. !! !! i"' i! I nodes of global model ::. .I.'i: !; .I!! ;:I. ._ solution for interpolation ii i_,i';,:, i, i i .liii i i ii to submodel boundary crack tip Figure 39 Finite element submodel mesh and linking with global model Figure 310 shows the distribution of normal stress of the loadcarrying ply and average normal stress of the laminate, respectively, for the submodel. From the Figure for normal stress distribution (Fig. 310), it can be seen that the nature of stress variation is similar to that of global model shown in Figure 34. Same procedures of global model analysis were repeated to estimate the stress intensity factor of the loadcarrying ply and the average stress intensity factor for the laminate in the submodel. It is noted that the average stress intensity factor for the laminate obtained from the submodel, 45.7 MPa ml/2, is almost same as that of global model, 45.9 MPaml/2. However, the stress intensity factor of the loadcarrying ply, 59.2 MPam1/2, is much less compared to that of global model, 107 MPaml/2 1.00E+10 8.00E+09 6.00E+09 4.00E+09 2.00E+09 0 nnF4nn 0.00E+00 1.00E04 2.00E04 3.00E04 4.00E04 5.00E04 Distance from crack tip, r (m) Figure 310 Normal stress distribution in the submodel in case of [0/45]s laminate 1.20E+08 1.00E+08 S 8.00E+07 S '' 6.00E+07 _E 0 2 4.00E+07 2.E+07 2.00E+07 0.00E+00 0.00E+00 1.00E04 2.00E04 3.00E04 Distance from crack tip, r (m) 4.00E04 Figure 311 Stress intensity factor in the submodel in case of [0/45]s laminate S... aL in the loadcarrying ply a'  in the loadcarrying ply 7 *" in the laminate "'" in the laminate A 1  AAAAAAAAA A* AA  0666690l>..4 OO ..O,0 .. 0. 00115's 101g 1 ______ _. __ _. *._ @ *. .. .n ... W Jj iz o (~r LVJ in the loadcarrying ply  y2r in the laminate 000o000 D i \ i S\ Extrapolation line y = 9.73E+10x + 5.92E+07 Extrapolation line y = 7.52E+10x + 4.57E+07 *i  5.00E04  In wdFC5Z234 Yz?:rr iAa~isAZDo (a) Loadcarrying ply, OEdegree L ODE adsu04S4S_mddb ABkAUS/ ca dard61N Ted FebO Z u3434 w,* AS 0 D. Sc at +1.617 (a) +45 ply Figure 312 a, distribution in the [0/+45]s laminate under a load of 351.14 MPa 2 Step: Step Defoned var: Z DeZformao Scale Facor: 3.3 Comparison of FE Results and Analytical Model The stress intensity factor in the loadcarrying ply was calculated using both the global and submodel. The laminate stress intensity factors calculated from the two models were in good agreement for all laminate configurations indicating that the mesh refinement was sufficient. From the laminate stress intensity factor, we can calculate the finitewidth correction factor Yusing the relation K, = 'Y, (16) The values of Yfor various laminate configurations are shown as a function of alw and / in Figure 321. It has beenfound that the finitewidth correction factor is a strong function of the newly introduced lamination parameter / More on this effect and significance of P will be presented in Chapter 3.4. The results for the loadcarrying ply stress intensity factor yielded some interesting trends. The results of KQ estimated through the finite element analysis are shown in Figure 316 and compared with two analytical models. The KQ(,B calculated from Eq. (15) agrees well with the results of the global model, which has a relatively coarse mesh. On the other hand, the K(A) calculated from the exact LEFM solution, Eq. (14), shows good agreement with the results of submodel, which has a very fine mesh. It is obvious from the results that the ox stresses ahead of the crack tip play a significant role in the estimation of K The coarse mesh of global model does not have sufficient nodes to capture the ox effect, although it is good enough for determining ay,. The global model is not able to present a complete picture of stresses in the vicinity of the crack tip. 1.00E+10 8.00E+09 6.00E+09 4.00E+09 2.00E+09 O.OOE+00 0.00E+00 1.00E04 2.00E04 3.00E04 4.0 Distance from crack tip, r (m) IOE04 5.00E04 Figure 313 Comparisonof normal stress distribution in the submodel and global model in case of [0/45]s laminate 3.00E+08 2.50E+08 2.00E+08 1.50E+08 1.00E+08 5.00E+07 O.OOE+00 0.00E+00 2.00E03 4.00E03 6.00E03 8.00E03 1.00E02 1.20E02 1.40E02 Distance from crack tip, r (m) Figure 314 Stress intensity factor in the global model in case of [0/45]s laminate *L in global model Y  . L in global model ~ o in submodel ... a in submodel , ...............*. ... f , , P .^ ........ ........ 0  _ C  __T   A oL2inr in global model 0 Tr'l // in global model ..... .  Extrapolation line Sy=1.17E+10x + 1.07E+08  Extrapolation line y = 5.44E+08x + 1.35E+07 u 0 t 5_E  A U) 0) a L 0.00E+00 1.00E04 2.00E04 3.00E04 4.00E04 5.00E04 Distance from crack tip, r (m) Figure 315 Stress intensity factor in the submodel in case of [0/+45]s laminate o Eq. (15) o,Eq. (14) o global model w submodel 0 li S1 S2 S3 S4 S5 S6 S7 S8 S9 Laminate configuration Figure 316 Comparisons of the fracture toughness of principal loadcarrying ply 0) oc U TO ^~ 1.1 160 140 120 100 80 60 40 20 0  KL KQ(B) K KL Q(A) 3.4 FiniteWidth Correction (FWC) Factor 3.4.1 Isotropic FiniteWidth Correction Factor The layup independent failure model presented in Ref. [11] enables the prediction of the notched strength of composite laminates. This model was formulated assuming that the plates are of infinite width, thus, the infinite width notched strength, a', is predicted. However, experimental data provide notched strength data on finite width specimens, ao To account for finite width of the specimen, the finitewidth correction (FWC) factor Y is required to estimate the stress intensity factor accurately. According to the definition, the finitewidth correction factor is a scale factor, which is applied to multiply the notched infinite solution to obtain the notched finite plate result. A common method used extensively in the literature is to relate experimental notched strength, aN, for plates of finite width to the notched strength of plates of infinite width is to simply multiply oN by the finitewidth correction factor Y, where KI is the mode I stress intensity factor. K, ON = N (17) KI The mode I critical stress intensity factor for isotropic plate of finite width is calculated from following equation. K, = YJsN a (18) Due to the lack of an analytical expression for orthotropic or anisotropic finitewidth correction factors, the existing layup independent failure models used isotropic finite width correction factor, YIso to evaluate stress intensity factor of notched composite laminates. Usually YIso is a 3rd order polynomial in a/w, which was developed empirically for the case of center cracks in isotropic panels. For example Yso = 1 + 0.1282(a /w) 0.2881(a/w)2 + 1.5254(a /w)3 (19) can be found in Ref [11] However, the isotropic finitewidth correction factor does not properly account for the anisotropy exhibited by different laminate configuration. In some cases, the application of the isotropic finitewidth correction factors to estimate the anisotropic or orthotropic finitewidth correction factors can cause significant error. 3.4.2 Orthotropic FiniteWidth Correction Factor 3.4.2.1 Developing procedure for FWC solution To improve the accuracy of notched strength predictions for finitewidth notched composite laminates, orthotropic finitewidth correction factor is obviously required. There are a couple of methods to determine the orthotropic finitewidth correction factor [1315]. However, no closed form solution is available. For this purpose, closed form of orthotropic finitewidth correction factor is developed empirically based on the results of finite element analysis. It is found that Ydepends onfi also. The finitewidth correction factor for orthotropic plate can be obtained from following equation, where a( (x,0) is the normal stress distribution in an infinite plate. K YOT = (20) where a' is the remote uniaxial stress, and a is the half crack length. However, a closed form expression for K, does not exist. To estimate values of K, for various laminate configurations, the commercial finite element code, ABAQUS 6.2/Standard, was used. The stress intensity factor, Ki, can be calculated in two different ways based on the finite element analysis as mentioned earlier. The accuracy of these methods is investigated by comparison between a known closed form solution Eq. (19) for notched isotropic plate and the solution obtained from ABAQUS finite element models. In the finite element analysis, eight node, plane stress elements were used to model notched plates made from isotropic material with material property E = 100 GPa and v = 0.25. Geometry and mesh size of the global model were used to evaluate the Jintegral for the range of crack sizes given by a/w = 0.05, 0.1, 0.2, 0.2625 (specimen), 0.3, 0.4, 0.5. Ten contour lines were used to evaluate the Jintegral in the finite element models. Each contour is a ring of elements completely surrounding the crack tip starting from one crack face and ending at the opposite crack face. These rings of elements are defined recursively to surround all previous contours. ABAQUS 6.2 automatically finds the elements that form each ring from the node sets given as the cracktip or crackfront definition. Each contour provides an evaluation of the Jintegral Figure 317 shows the estimated finitewidth correction factor determined by finite element methods and a closed form solution for an isotropic notched plate with various notch sizes. Excellent agreement between these analysis methods is noted over the whole notch size of the plate. 1.20 1.15 0 C) 1.05 E 1.00 LL 0.95 1 1 1 0.00 0.05 0.10 0.15 0.20 0.25 a/w Figure 317 Comparison of the finitewidth correction factors in an isotropic plate computed by the finite element methods with the closed form solution The most accurate method, finite element Jintegral, was adopted to develop the finitewidth correction factor for orthotropic plates. Figure 318 shows the calculated J integral value from FE global model for the material and layups in Table 21. Ten contour lines were used to evaluate J interagl and these value s are reasonably independent of the path as expected. The first four Jintegral values were taken to calculate average Jintegral and it was used to estimate K1 using Eq. (3). The orthotropic finitewidth correction factor obtained using Eq. (20) are shown in Figure 319. As expected, the finitewidth correction factors increase with the crack size to panel width ratio. Additionally, the finitewidth correction factor strongly depends on the anisotropy of laminate characterized by f3. Figure 320 shows the finitewidth correction factor for different values of anisotropy parameter 3 and the / values of laminated composite panel using Eq. (6) are shown in Table 32. * Finite Element Jlntegral I A Finite Element Stress Intensity Factor  Closed form solution Standard Deviation ( __0.107% A 0.461 % i i I I 1.20E+03 1.00E+03 i. .a. ..a .... [. [ .[  .  8.00E+02 6.00E+02 4.00E+02 2.00E+02 0.00E+00 ,xxxxxx x  X X     X ... . ..    K A    .    . . )  '" Crack size *) .* a/w=10%   a/w=20% * .x** a/w=26%   a/w=30%  o a/w=40%   a/w=50% 1 2 3 4 5 6 7 8 9 10 Contour line Figure 318 Jintegral vs. contour line calculated from FE model of the [0/45]s laminate Table 31 Jintegral value calculated from FE model of the [0/45]s laminate Crack size 0.1 0.2 0.26 0.3 0.4 0.5 (a/w) Contour 1 1.31E+02 2.72E+02 3.75E+02 4.44E+02 6.71E+02 9.63E+02 Contour 2 1.31E+02 2.81E+02 3.88E+02 4.59E+02 6.82E+02 9.75E+02 Contour 3 1.31E+02 2.81E+02 3.88E+02 4.59E+02 6.82E+02 9.75E+02 Contour 4 1.26E+02 2.81E+02 3.88E+02 4.59E+02 6.82E+02 9.75E+02 Contour 5 1.29E+02 2.81E+02 3.88E+02 4.59E+02 6.82E+02 9.75E+02 Contour 6 1.37E+02 2.70E+02 3.88E+02 4.59E+02 6.82E+02 9.75E+02 Contour 7 1.48E+02 2.76E+02 3.73E+02 4.59E+02 6.82E+02 9.75E+02 Contour 8 1.64E+02 2.87E+02 3.81E+02 4.41E+02 6.82E+02 9.75E+02 Contour 9 1.75E+02 3.16E+02 3.96E+02 4.44E+02 6.55E+02 9.75E+02 Contour 10 1.97E+02 3.38E+02 4.04E+02 4.59E+02 6.60E+02 9.75E+02 1.50 1.40 1.30 1.20 1.10 1.00  0.00 0.10 0.20 0.30 0.40 0.50 Figure 319 The finitewidth correction factors vs. ratio of crack size to panel width 1 0.3 0.5 0.7 0.9 Anisotropy parameter, Figure 320 The finitewidth correction factors vs. anisotropy parameter, P A a/w=0.2 S4 a/w=0.26 $5 <> a/w=0.3 [S a/w=0.4  a/w=0.5 S6 1 S8 S9 S1 ___ ... AA A ,, 3,7 Table 32 Values of the anisotropy parameter, f, for the test specimens specimen S1 S2 S3 S4 S5 S6 S7 S8 S9 0/ 1 1 1 0.285 0.396 0.656 1 0.767 0.823 The above results indicated that there is a definite relation between the finite width correction factors, geometric parameter, a/w, and anisotropy parameter, P Thus, the general semiempirical solution of orthotropic finitewidth correction factor was developed using multiple least square regressions to fit measured data in the finite element analysis. It can be expressed in terms offi and ratio of crack size, a/w, in the following form Yo,= +b, a 1+cB+c2B )+b (1+c3B+c4B2)+b3 a (1+cB+c6B2) (21) where B is defined as 1 P, and the coefficients of least square fit are shown in Table 33. A 3D plot of the finitewidth correction factors is given in Figure 321 and estimated values of the factor are summarized in Table 34. Note that YOT increases with a/w and B. Table 33 The coefficients of semiempirical solution of orthotropic finitewidth correction factor bl c1 c2 b2 C3 C4 b3 C5 C6 0.1091 5.0461 2.1324 0.2319 2.9103 4.4927 1.4727 1.5124 0.0375 41 1.  an 1t o c s t p t  3 12. "* "* 0. solution (Eq. 21) Orthotropic, YOT Isotropic, a/ a/w0 SS2 S3 S4 S5 S6 S7 S8 S9 Yparameter 0.10 1.011 1.011 1.011 1.046 1.041 1.029 1.010 1.023 1.020 1.01 0.20 1.024 1.024 1.024 1.107 1.094 1.065 1.024 1.052 1.045 1.02 0.26 1.040 1.040 1.040 1.152 1.134 1.092 1.040 1.075 1.066 1.04 0.30 1.051 1.051 1.051 1.182 1.160 1.111 1.052 1.091 1.082 1.05 0.40 1.101 1.101 1.101 1.269 1.239 1.173 1.101 1.148 1.136 1.10 0.50 1.181 1.181 1.181 1.369 1.331 1.254 1.181 1.227 1.214 1.18 Figure 321 The finitewidth correction factors as a function of anisotropy parameter, fi, and ratio of crack size to panel width, a/w. Table 34 The finitewidth correction factors obtained from the semiempirical solution (Eq. 21) Orthotropic, Yon Isotropic, a/w Si $2 $3 $4 S5 $6 $7 S8 $9 Yzso 0.10 1.011 1.011 1.011 1.046 1.041 1.029 1.010 1.023 1.020 1.01 0.20 1.024 1.024 1.024 1.107 1.094 1.065 1.024 1.052 1.045 1.02 0.26 1.040 1.040 1.040 1.152 1.134 1.092 1.040 1.075 1.066 1.04 0.30 1.051 1.051 1.051 1.182 1.160 1.111 1.052 1.091 1.082 1.05 0.40 1.101 1.101 1.101 1.269 1.239 1.173 1.101 1.148 1.136 1.10 0.50 1.181 1.181 1.181 1.369 1.331 1.254 1.181 1.227 1.214 1.18 3.4.2.2 Anisotropv parameter, If The anisotropy parameter, f, has been calculated using classical lamination theory for a variety of AS4/35016 graphite/epoxy composite laminates. The values of 3 depend on both material property and laminate configuration. It is equal to 1 when the laminate is isotropic or quasiisotropic. As the anisotropy increases, the value of 3 increases to certain finite value. The finitewidth correction factors for orthotropic laminates depend on the ratio of crack length to panel width, a/w, and also the anisotropy parameter, 3. In general, Yincreases with a/w and 3. A plot of variation of the anisotropy parameter as a function of 0 for [ 0]s and [0/ 0 ]s is given in Figure 322. Notice that the anisotropy parameter, / attains a maximum value at 0 =900 in [ 0 ]s laminates. 4 3.5  3 [0]s layup 2.5 1.5 [0/ ]s layup 1  0.5 0 10 20 30 40 50 60 70 80 90 Fiber angle, 0 (degree) Figure 322 Anisotropy parameter, f, as a function of lamination angle for graphite/epoxy [ 0 ]s and [0/ 0 ]s laminates Anisotropy parameter, 3, can be 0 < P3 < +c. However, the range of the values is quite limited in most existing laminate materials. For example, the ranges of computed values of anisotropy parameter for a few laminate materials, which are widely used in composite structures, are shown in Table 35. Table 35 Ranges of anisotropy parameter, /, various composite materials Material property o Ranges of B Er (GPa) EL (GPa) VLT GLT (GPa) AS4/35016 /3 9.65 138 0.3 5.24 0.264 < <3.781 graphite/epoxy EGlass/Epoxy 19.5 52 0.28 3.24 0.612/ < 1.633 SGlass/Epoxy 8.9 43 0.27 4.5 0.454 / 2.198 Kevlar 149/Epoxy 5.5 87 0.34 2.2 0.251 CFS003/LTM25 CFS3/L 25 54.0 54.7 0.065 4.05 0.993 Carbon/Epoxy The semiempirical solution (Eq. 21) for the finitewidth correction factor for an orthotropic material was developed using a curve fitting method from anisotropy parameter, 0.285 < /3 <1. According to the characteristic of a curve fitting method, if data for prediction is far beyond the limits of the observed data, a prediction can cause a relatively large error. However, the ranges of practical values are within small bounds. Therefore, the methodology described above can be used to develop the finitewidth correction factor solution for most widely used composite materials. 3.5 Effect of Blunt Crack Tip Shape 3.5.1 2D FE Modeling Procedure The effect of the shape of the crack tip is not considered in any of the fracture models when calculating stress intensity factor that controls the laminate failure. The effects of different initial crack tip shape were analyzed. According to the specimen preparation process [11 ], to make the initial cracks, a starter hole was first drilled in the laminate to minimize any delamination caused by the waterjet. The crack was made by 44 waterjet cut and further extended with a 0.2mm thick jeweler's saw blade. Thus, in the present analysis, the crack tip thickness was assumed less than 0.2mm and three different crack tip shapes, elliptical, triangle, and rectangular, were considered. These assumptions were carefully investigated through the series of FE submodels and the results are discussed below for the case of [0/+45]s laminate. IP t t rectangle triangle semi cirde Y crack tip LX I I I Figure 323 Crack tip shape profiles 3.5.2 Results and Discussion Figures 324 and 325 compare the effect of crack tip profile and crack thickness on the stress intensity factor of principal loadcarrying ply using global and submodel. In this analysis, thickness of the crack was assumed to be less than 0.2 mm. The global nature of FE models requires that the details of the crack tip shape are not explicitly modeled due to the mesh size. Therefore, the global model may not accurately capture the details near the crack tip and is not reliable in capturing the behavior of the crack tip shape. From the results of the global model, it is evident that stress intensity factors of global model are not greatly influenced by crack tip shape. 120 110 100 U 90 U) =c6 80 ns 70 60 LI_ 50 40 0 0.05 0.1 0.15 Crack thickness, 2h 0.2 0.25 Figure 324 Effect of crack tip shape on predicted fracture toughness of [0/45]s laminate; 2D FE global model results 120 1 20 ... 110 100 90 80 70 I l 60   S50a rectangle S50 o semicircle 40 S 40 triangle h/c=2 30 triangle h/c=1 20 0.1 0.15 Crack thickness, 2h 0.2 0.25 Figure 325 Effect of crack tip shape on predicted fracture toughness of [0/45]s laminate ; 2D FE submodel results  .. * I I I I However, the results of the submodel indicate that the crack tip shape has a significant effect on the stress intensity factor at the crack tip. Furthermore, the large variation was observed in the results shown in Figure 325. It indicates that the stress intensity factor is very sensitive to the crack tip slope ahead of the crack tip. The results shows that the crack tip slope near the crack tip is a more critical parameter than the crack tip thickness. The results clearly indicate that the stress intensity factors at the crack tip are strongly affected by the behavior of the crack tip shape. 3.6 Effect of Local Damage The complicated nature of fracture behavior in notched composite laminates makes it difficult to predict the exact position and size of every fiber break, matrix crack, and delamination even in a relatively simple notched panel studied here. The philosophy is not to model exactly, but to make approximations that agree well with actual behavior. The objective of this section is to study the effect of damage in the vicinity of crack tip such as delamination and fiber splitting by comparing stress intensity factor computed using the threedimensional FE analysis. It has been noted that for a tension loaded laminate, local damage is produced ahead of the crack tip in the form of the matrix cracks in offaxis plies, splitting in OE plies and some delamination [11]. This local damage acts as a stressrelieving mechanism and relieves some portion of the high stress concentrated around the crack tip. This damage, such as matrix cracking and delamination, may significantly affect the structural integrity of the structures when they become sufficiently severe. However, failure models discussed in the previous section do not represent any effects of local damage. Axial splitting and delamination are the most common types of damage in laminated fiber reinforced composites due to their relatively weak interlaminar strengths. Previous finite element analysis used plane stress element in twodimensional modeling. Therefore, the effect of delamination between plies in the thickness direction can not be modeled. In present finite element models, threedimensional analysis was adopted to investigate the effect of delamination and axial splitting using twenty node, solid element (C3D20R element). Typically, solid element with reduced integration, is used to form the element stiffness for more accurate results and reduce running time. Submodeling analysis can be applied to shelltosolid submodel, however, the z direction (thickness direction) stress and strain field can not be interpolated to 3D solid submodel boundary from 2D global model. Thus, 3D global models were modeled to analyze 3D submodel with local damage. Figure 326 shows the VonMises stress distribution in the [0/45]s laminate of 3D global model and 3D submodel. It should be noted that before interpolating the results of the 3D global model, the comparison between the results of 2D global model and 3D global model should be checked for consistent analysis. In Figure 327, the estimated stress intensity factors in the load carrying ply from 3D global model are compared with those of 2D global model. The stress intensity factors obtained from 3D global model are slightly underestimated compared to those of the 2D globalmodel, however, these estimated values are fairly consistent for the nine laminate configurations. Consequently, two types of damage were modeled using 3D submodel to predict the effect of local damage on the stress intensity factor of the loadcarrying ply. (a) 3D global model (b) 3D submodel Figure 326 VonMises stress distribution in [0/" 45]s laminate SStep: Stepi 3 f Itiremeti 1: S1ep Tune 1.000 Primary Var: H f I : I 140 B120 2D global model E 3D global model 0 c2 100 20 0 I i lllll S1 S2 S3 S4 S5 S6 S7 S8 S9 Laminate configuration Figure 3 27 Comparison of stress intensity factor of the loadcarrying ply between 2D and 3D global model 3.6.1 3D FE Modeling Procedure for Delamination All aspects of the finite element model were kept the same as 3D submodel except interface of ply. Each ply was modeled by four elements through the thickness as shown in Figure 326(b). Elements of layers are connected through either side of any ply interfaces at common nodes except where delamination is expected. The delamination was modeled as separate, unconnected nodes with identical coordinates. Figure 328 show the schematic of ply interface mesh where delamination is expected. To estimate delamination area, a simple delamination criterion was implemented in ABAQUS by means of a UVAR user subroutine using the following equation. Delamination area : 2J + T + < S (22 ) where ST is critical stress of matrix which is 75 MPa based on typical epoxy yield strength. Figure 3 29 shows the estimated delamination area using the above 50 delamination criterion in the ply between the loadcarrying ply and +45E ply of [0/+45]s laminate. (a) no damage (b) delamination Figure 328 Scheme of ply interface mesh in the [0/45]s laminate liWAR, UL tR1 ili ll lii lliiigii IHIQIIIHIII ilillillP crack line, arms % p,1 1 t cra ck line uu uu uu ou uu Defarea Var; V Dfra .an 3aI PacT..ti .ifit a00 Figure 329 Estimated delamination area on interface between OE and +45E ply in the [0/c45]s laminate 3.6.2 3D FE Modeling Procedure for Axial Splitting When a notched composite panel is subjected to tension loading axial splitting occurs in the crack tip area due to high stress concentration and low matrix tensile strength. Apparently, this failure mode causes severe stiffness reduction in transverse direction. Thus, transverse stiffness of axial splitting area can be modeled by taking E, & 0. This failure mode and assumption are illustrated in Figure 330. Similar method described in the previous section was used to estimate axial splitting area where reduced stiffness property was implemented, ET=10 Pa that is extremely small compared to initial modulus, ET =9.65 GPa. A critical stress of 75 MPa was used, based on a typical epoxy yield stress. Figure 331 shows the estimated axial splitting area in the interface between OE and +45E ply for a tensile loading 351.14 MPa. tL t.t r t _ Axial splitting area (E T 0) Figure 330 Axial splitting failure mode in the loadcarrying ply .... IHHHHHHUHN. uuuurnuuuuuiKi iuuiiiiuuniiuiiuiiiniiuuieinuiuiiiuni ' crack line 4 ..646 Mop: 4 limr Ti.:: 1.000m Figure 331 Estimated axial splitting area on interface between OE and +45E ply 3.6.3 Results and Discussion Typical damage in notched laminated composites occurs in the form of axial splitting in the loadcarrying ply and delamination. This damage was modeled to study effect of local damage on the normal stress distribution in the vicinity of crack tip in the loadcarrying ply. The estimated delamination area and axial splitting area are modeled by having duplicate nodes and reduced stiffness, respectively. Results of normal stress distribution are shown in Figure 332. From the results it is observed that s, distribution in the FE model without damage shows the typical square root singularity. Axial splitting and delamination removes the singularity, though there still is a region of high stress concentration near the crack tip. It also can be seen that o, distribution increases away from the crack tip and ax distribution in the model with damage decreases than in the model with no damage. Thus, the stress field is strongly affected by the presence of local damage. From the results, it is obvious that the local damage serve as main mechanism to increase a, distribution and decreases ox distribution in the loadcarrying ply. This simulation is similar to the redistribution of stress in the presence of small scale yielding in homogenous materials. 1.00E+10 T A Sy with no damage . Sx with no damage 8.00E+09 T   a Sy with damage n 0 Sx with damage 6.00E+09  oI 2.00E+09 U, I I ! 0.00E+00 1.00E04 2.00E04 3.00E04 4.00E04 5.00E04 Distance from crack tip, r (m) Figure 332 Effect of the local damage on normal stress distribution in the load carrying ply in the [0/45], laminate 3.7 Summary of FE Results and Discussion It is well established that the fracture behavior of composite laminates depends on a variety of variables. All may affect to varying degrees the fracture behavior of the notched laminates. A comprehensive evaluation is still lacking regarding the effects of all variables on the notch sensitivity of composite laminates. Through the detailed finite element analysis, we investigated effect of distinct factors such as crack tip shape and local damage. O.OOE+OOe0 6eot :44. "^ The presence of blunt crack tip has a significant effect on the behavior of notched composites, and leads to stress redistribution. This relives the stresses in the nonload carrying plies and increases the SIF of the loadcarrying plies. Similar effects are observed when local damage such as matrix cracking and delamination are introduced. Figure 333 shows the results of finite element submodel, which include blunt crack tip and local damage. From the comparison of the results between models with and without damage, it is obvious that these effects also increase the stress intensity factor of the load carrying ply as a result of the reduction in ox stress concentration in the laminate. By comparing the results of finite element analysis with analytical failure model, it can be reasonably assumed that the effect of initial damage is accounted by reducing the o" component ahead of crack tip to zero. 140 M no damage in 3D submodel IM damage in 3D submodel 120 U) S100 ' 0 E, 80 60 U 40 20 0 I I + S1 S2 S3 S4 S5 S6 S7 S8 S9 Laminate configuration Figure 333 Effect of the local damage on stress intensity factor in the load carrying ply CHAPTER 4 CONCLUSIONS AND FUTURE WORK CONCLUSIONS The SIF of the loadcarrying ply is a critical parameter for predicting the failure of notched composite laminates. The SIF of the loadcarrying ply, K can be estimated by using a detailed 3D FE analysis which can model local damage modes such as fiber splitting and delamination that occurs prior to fracture. Then the SIF of the loadcarrying ply can be calculated accurately and compared with the critical value for the material system to predict fracture. The results from this model are presented in Column 2 of Table 41. On the other hand, a simpler analytical model could be used. In this model finite width correction factor for orthotropic laminates should be used. The effects of local damage at the crack tip is accounted for by setting the parameter /=0 in Eq. (15) for calculating the SIF of the loadcarrying ply. Results from such analysis are shown in Column 3 of Table 41. The results from calculations performed by Sun and Vaiday [11] are given in Column 4 for comparison. Using a mean value of 113.81 MPam'2 for K in the analytical model, the predictions of the laminate fracture toughness are compared with the experimental results in Figure 42 for various value of il. 140 10 Analytical model 120  100 T I 100 , 3D FE analysis 80 T Q(A) 60 P K : 40 20 7 l 20 S1 S2 S3 S4 S5 S6 S7 S8 S9 Laminate configuration Figure 41 Comparison of the fracture toughness of loadcarrying ply Table 41 Comparison of the fracture toughness of loadcarrying ply obtained from different methods Fracture toughness of loadcarrying ply, K MPam'l2 Analytical Specimen 3D FE analysis l Ref [11] model S1 S2 S3 S4 S5 S6 S7 S8 S9 average S.D. 116.65 114.23 115.71 118.16 115.21 113.53 121.40 116.01 110.51 115.71 3.03 % 115.66 108.82 106.43 112.62 109.97 111.52 109.04 123.66 126.58 113.81 6.95 % 115.66 108.82 106.43 101.81 101.00 106.32 109.04 119.81 123.64 110.28 9.80 % T standard deviation v, U, 1) UE TO u g_ The variable 17 was defined as ratio of stress intensity factor of the loadcarrying ply to that of laminate and can be obtained from Eq. (15). It should be noted that ?, likef3, depends entirely on the laminate properties. Good agreement is observed between the experiments and predictions. By comparison with experimental results, it is concluded that the proposed layup independent model with orthotropic finitewidth correction factor is capable of predicting fracture toughness of notched laminated composites with reasonable accuracy for mode I loading. U) U) 0) co OCN E co 0 ... I I Predicted curve , S\J/ (analytical model) _ kSI I I I I I I I  I I I I I  I   eE     S4   1 S S9 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 1 is defined as n = t A, + 2A] from Eq. (15) e Figure 42 Comparison of experimental results with failure model predictions 2.4  experiment prediction 2.2 2 S4 S5 S7 S1 S8 S9 1.6 S6 S6 S2S3 S 1.4 1.2 1 0 0.1 0.2 0.3 0.4 0.5 0.6 log(r7) Figure 43 Comparison of experimental results with failure model predictions (log scale) Table 42 Comparison of experimental results with failure model predictions Specimen Fracture toughness of laminates, K. (MPam12) Notation Layup ExperimentT Prediction* Relative error (%) S1 [0/90/45]s 2.5791 44.67768 44.1295 1.23 S2 [45/90/0]s 2.5791 40.22554 44.1295 9.71 S3 [90/0/45]s 2.5791 41.16729 44.1295 7.20 S4 [0/+15]s 1.1209 100.4675 101.5384 1.07 S5 [0/30]s 1.6619 66.24291 68.48451 3.38 S6 [0/45]s 2.3215 48.07307 49.02623 1.98 S7 [", J_ 1.8706 58.19245 60.84379 4.56 S8 [45/0/+45]s 3.1979 38.68474 35.59036 7.99 S9 [452/0/+45]s 3.8168 33.19171 29.81933 10.16 t Fracture toughness of laminates obtained using Eq. (1) with orthotropic finitewidth correction factor in Table 34. SPredicted fracture toughness of laminates calculated using the mean value, 113.81 MPam 2, for KL in the analytical model. FUTURE WORK There seems to be ample room for further investigation of this problem. Fracture problem studied in the current study was limited to AS4/35016 graphite/epoxy laminated panel containing a center straight crack subjected to tension loading. For general application, validity of currently proposed layup independent criterion can be further verified for the case of laminated composite panels containing double edge notch, single edge notch, circular hole, etc. with different material systems using experimental and numerical analyses for mode II and mixed loading conditions. Additionally, this failure model can be compared with other popular failure models such as Point Stress Criterion, Average Stress Criterion, MarLin failure model, etc. so that composite structure designers can select an appropriate failure model in diverse practical situations. APPENDIX A LAMINATION THEORY For laminated plates with bendingextension coupling under plane stress, the complete set of forcemidplane deformation equations can be expressed in matrix form as {IN} LA lB]{El (Al) M L BI D where N and M are the inplane forces and moments, respectively, and 0 is the mid plane strains, and K is the curvature. In the case of symmetric laminates without coupling, Eq. (A1) is reduced to Nx All A12 A16 E N, = 412 A22 A 26 e; (A2) Nz A16 A 26 A 66 where the laminate extensional stiffness are given by St/2  4 =Jt/2 (Q,)kdz (A3) where t is the thickness of the laminate. If there are N layers in the layup, we can rewrite the above equations as a summation of integrals over the laminate. The material coefficients will then take the form N A, = (QJ)k(zk Zk 1) (A4) k=1 LX Figure Ai Laminated plate geometry where the Zk and zk 1 in these equation indicate that the k th lamina is bounded by surfaces Zk and k1 Thus, the Q, depend on the material properties and fiber orientation of the k th layer. It can be obtained using the following equation. =T l[] [R] [T] [R (A5) where nm n mn 1 0 0 [T]= n2 m mn [R]= 0 1 0 (A6) 2mn 2mn m2 n2 0 0 2 m=cosO and n=sin where the Q, are the components of the transformed lamina stiffness matrix which are defined as follows Q, = Q cos 4 0 + 22 Sin 4 + 2(Q12 + 2Q66) in 2 coS 2 0 Q12 = (Q01 + Q22 4Q66)sin2 0 cos2 0 + Q12 (sin 4 + cos4 0) 022 = Q sin 4 0 +022 coS4 0 +2(Q12 + 266)sin 2 0COS2 0 6 (11 Q12 2Q66)Sin 0 COS3 + (012 22 +266 )sin 0 coS 026 =(11 12 2Q66)sin3 coS + (Q12 22 +2Q66)sn 0cos30 Q66 = (11 + Q2 2Q2 2Qs66) s2in 0 cs2 0 +Q6(in4 0 + cos4 ) (A7) 0 A* A22 A26 N (A8) 0 N xy A16 26 66 zy where superscript denotes the component of inverse matrix of [A] The stresses in each ply can be recovered form the midplane strains. In particular, the stresses in the loadcarrying ply can be expressed as where superscript L means the component of principal load carrying ply The normal stress field applied in the loadcarrying ply can be obtained by calculating the portion of the applied load that is carried by the loadcarrying ply using lamination theory using Eqs. (A8) and (A9). S ~QQ Q 1 Q6 A A2 A166 E I I= Q Q2 2 A2 922 1 26 (A10) 7 Q Q2 6 26 66 A x XY 1626 6 A16 A26 A66 XY APPENDIX B MATHEMATICAL THEORIES OF BRITTLE FRACTURE The objective of this Appendix was to provide the brief review of mathematical formulation of crack problems for derivation of stress intensity factor in homogeneous orthotropic materials [17]. For orthotropic materials in the case of plane stress, the generalized Hooke law can be expressed as E = axx + alz1 + al Jx eY = a 120 +a22Cyy +a26TZ, (B1) y a, = a16x +a26y + a66z G' = = o = 0 The equilibrium equations under plane stress conditions are S =0, + =0 (B2) ax ay ax ay The equilibrium equations will be satisfied if the stress function U(x,y) is expressed as 02U 02U 02U =y2' ax2' y x (B3) Substituting for ao, oayy,, r from Eq. (B3) in the compatibility equation (B4) ay2 x2 8x8y The governing characteristic equation of plane stress of orthotropic materials can be expressed as a4U _4U _4U a4U a4U a22 2a6 + (2a, +a66) 2a16 = 0 (B5) a22 a aa a2 a2 1611 0 8x 8x ox 3y axcy 4y Defining the operators D, (= 1, 2, 3, 4) as D =  ( = 1, 2, 3, 4) (B6) ay ax The governing equation in U(x,y) becomes D12 D 3 D4 U (x,y) = 0 (B7) and p, are the roots of the characteristic equation. 4 3 2 al 2al6p + (2a12 + a66)P 2a26P + a22 = 0 (B8) The roots are either complex or purely imaginary and cannot be real and can be expressed as s1 = 1 = al + i(5, 2s = 22 = a2 +i52, u3 = 1, 14 = U2 (B9) where a,, j (j= 1, 2) are real constants. The stress function U(x,y) can be expressed in the form U(x,y) 2 Re [Ui(z)+ U2(z2)] (B10) where Ui(zi) and U2(z2) are the arbitrary functions of the complex variables z, = x + sy and z2 = x + s2y, respectively. Let new functions o(z,)=dU,/dz,, (Z2) = dUgldz2 (B11) Substituting the stress function from Eq. (B10) into the Eq. (B3) and taking into account the relations in Eq. (B11), the stress components can be expressed as 65 o x = 2Re[s, 20'(z1)+s2 2y/'(z )] o, = 2Re[o'(z, )+ /'(z2 )] (B12) r. = 2Re[sO'(z,) + s2y'(z )] For pure mode I case, the stress components in the vicinity of crack tip can be expressed in terms of stress intensity factor, Ki, and the roots of characteristic Eq. (B8) K 1 s a, = Re 1 2r sL s cos6+ssin K, a = 4 Re T = Re v n& S s2 s s s1 2 cos 0 + S2 sin 0 S1 2 , S1 S2 1 cos 0 1 sin 0 S2 s1 ,rcos 0 + s1 sin 0 1 4cos0+ssin j U x IIr;7>;V1 Figure B Stress components in the vicinity of crack tip when 0 =0, the new parameterfi introduced in the Chapter 2.3 can be expressed as ( Sa(r,O) a,= (r,0) Re[ ss,2] (B14) Consequently, the model I stress intensity factor can be expressed as K, = lim oy (r,0) 2 7 r>0 (B13) (B15) Ki can also be expressed in terms of ax as lim o (r,0),2/7 K, = r> (B16) The increase in strain energy due to the presence of the crack can be calculated using following equation for mode I case. 1la AW = I (x,0)Au dY (B17) a The derivative of AW with respect to crack size, a, yield OAW 2 S +S2 18) da s, s The energy release rate can be expressed in terms of stress intensity factor K 2s, + s G,= 2 RRe i SS2 (B19) 2 ss J where s, ( = 1,2) are the roots of the characteristic equation (B8) which can be derived from the elastic constants as S1S a221 S S, 1/ a22 )1/ (2a12 +a66 )1 (20) sS2 = 2, s, + S2 = i F 222a!a!a6 12 (B20) Then, the relation equation between Gi and KI for an orthotropic material can be expressed as 1 /2 1/ 2 a 1/2 a21/2 2a12 +a66 11/2 G =a22 + 2a+a (B21) 2 a11) 2a I LIST OF REFERENCES [1] Awerbuch J, Madhukar M.S., "Notched strength of composite laminates: prediction and experimentsa review." Journal of Reinforced Plastics and Composites 1985;4:3159 [2] Waddoups M.E., Eisenmann J.R., Kaminski B.E., "Macroscopic fracture mechanics of advanced composites materials." Journal of Composites Materials 1971;5:44654 [3] Whitney J.M., Nuismer R.J., "Stress fracture criteria for laminated composites containing concentrations." Journal of Composite Materials 1974;8:25365. [4] Tan S.C., "Effective stress fracture models for unnotched and notched multidirectional laminates." Journal of Composites Material 1988;22:32240. [5] Mar L.W., Lin K.Y., "Fracture mechanics correlation for tensile failure of filamentary composites with holes." Journal of Aircraft 1977;14:703707 [6] Chang F.K., Chang K.Y., "Progressive damage model for laminated composites containing stress concentration" Journal of Composites Materials 1987;21:834 855 [7] Tan S.C., "A progressive failure model for composite laminates containing openings." Journal of Composites Materials 1991;25:556577 [8] Poe C.C. Jr., "An unified strain criterion of fracture of fibrous composite laminates." Engineering Fracture Mechanics 1983;17:153171 [9] Poe C.C. Jr., Sova J.A., "Fracture toughness of boron/aluminum laminates with various proportions of OE and +45Eplies." NASA Technical Paper 1707. Nov 1980 [10] Kageyama K., "Fracture mechanics of notched composite laminates." Application of Fracture Mechanics to Composites Materials 1989;327396 [11] Sun C.T., Vaiday R.S., "Fracture criterion for notched thin composite laminates." AIAA Journal 1998;36:818. [12] Sun C.T., Vaiday R.S., Klug J.C., "Effect of ply thickness on fracture of notched composite laminates." AIAA Journal 1988;36:818. [13] Tan S.C., "Finitewidth correction factors for anisotropic plate containing a central opening." Journal of Composite Materials 1998;22:10801098 [14] Naik N.K., Shembekar P.S., "Notched strength of fabric laminates." Composites Science and Technology 1992;44:112. [15] Gillespie, J.W. Jr., Carlsson L.A., "Influence of finite width on notched laminate strength predictions." Composites Science and Technology 1988;32:1530 [16] WisnomM.R., Chang F.K., "Modeling of splitting and delamination in notched crossply laminates." Composites Science and Technology 2000;60:28492856 [17] Sih G.C., Liebowitz H., Fracture and an Advanced Treatise. Volume II Mathematical Fundamentals edited by Liebowitz H. 1968, Academic Press, New York and London [18] Hibbit, Karlsson, Sorensen, ABAQUS version 6.2 Theory Manual published by Hibbit, Karlsson & Sorensen, Inc. 1998, U.S.A. BIOGRAPHICAL SKETCH Bokwon Lee was born on July 15, 1973, in Kyungki province, Republic of Korea, as the second son of Jonghan Lee and Jongok Woo. He graduated from Korea Airforce Academy with a B.S. degree in aeronautical engineering in March 1996. After graduation, He worked at the 19th Fighter Wing, Jungwon, Republic of Korea, for 3 years. After promotion to Captain, he worked for Airforce Logistics Command as a technical assistant for tactical combat aircraft. He came to the University of Florida for his graduate study in August 2001 and began his research with Dr. Bhavani V. Sankar in the Department of Mechanical and Aerospace Engineering. He completed his master's program in August 2003. 