The members of the Committee approve the thesis of Kedar A. Malusare presented on 3/13/2014.

Transcription

1 To the University of Wyoing: The ebers of the Coittee approve the thesis of Kedar A. Malusare presented on 3/13/014. Dr. Ray S. Fertig, Chairperson Dr. Victor E. Ginting, External Departent Meber Dr. Mark. R. Garnich APPROVED: Paul A. Dellenback, Head, Departent of Mechanical Engineering. Dr. Khaled A.M. Gase, Dean, College of Engineering and Applied Science.

2 Malusare, Kedar A., Energy Conserving Stresses to Predict Coposite Failure, M.S., Departent of Mechanical Engineering, May,014. The coplexity of the ultiple failure echaniss exhibited by unidirectional fibrous coposites akes failure prediction a daunting challenge for the analyst. One approach to better identify failure echaniss has been the use of volue average constituent stresses of the coposite and directly predicting constituent failure. However, the total strain energy is not conserved by volue average constituent quantities (stresses and strains). To address this, this issing energy, tered interaction energy, was quantified as a function of loading, aterial properties, and fiber volue fraction. It was concluded that the stress/strain fluctuations of the atrix constituent were the ajor cause of interaction energy. Consequently, an energy conserving atrix stress etric was developed to attept to iprove failure load predictions. This energy consistent atrix stress was then used with an existing failure theory to predict failure loads of unidirectional fiber reinforced lainae subjected to biaxial and tri-axial load states. 1

3 ENERGY CONSERVING STRESSES TO PREDICT COMPOSITE FAILURE by Kedar A. Malusare A thesis subitted to the Departent of Mechanical Engineering and the University of Wyoing in partial fulfillent of the requireents for the degree of MASTER OF SCIENCE in MECHANICAL ENGINEERING Laraie, Wyoing May, 014

4 UMI Nuber: All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy subitted. In the unlikely event that the author did not send a coplete anuscript and there are issing pages, these will be noted. Also, if aterial had to be reoved, a note will indicate the deletion. UMI Published by ProQuest LLC (014). Copyright in the Dissertation held by the Author. Microfor Edition ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI

5 014, Kedar A. Malusare ii

6 DEDICATION PAGE I would like to dedicate this thesis to y parents Anant and Jyoti Malusare and y younger brother Okar Malusare. iii

7 ACKNOWLEDGEMENTS I would like to thank y advisor Dr. Ray Fertig for believing in y abilities and giving e the opportunity of pursuing research in the field of coposites aterials. I would also like to thank Dr. Mark Garnich and Dr. Victor Ginting for providing their valuable inputs to this thesis. Last but not the least, I would like to thank y copanion Aditi Anand. Without her support, y journey through research would have indeed been very difficult. iv

8 TABLE OF CONTENTS List of Figures..ix List of Tables.xiii Chapter 1. Introduction to Coposite Materials What are Coposite Materials? Advantages of Fibrous Coposite Materials Unidirectional Fiber Reinforced Polyers Failure of Unidirectional Fiber Reinforced Polyers A Representative Volue Eleent (RVE) Distribution of Stresses and Strain in the Constituents of a Coposite Material Objective...11 Chapter. Literature Review..1.1 Early Approaches for Predicting Failure Loads of Coposite Materials World Wide Failure Exercise I 17.3 World Wide Failure Exercise II...3 v

9 Chapter 3. Interaction Energy Overview of Volue Average Quantities Used for Coposite Failure Load Predictions Theoretical Motivation Finite Eleent Modeling Results and Discussion Effect of Fiber Volue Fraction on Interaction Energy Effect of Relative Matrix Modulus on Interaction Energy Effect of Cobined Loading on Interaction Energy.5 Suary and Conclusions...54 Chapter 4. Bencharking of laina failure tests fro WWFE-I and WWFE-II Introduction Failure Modeling Constituent Level Failure Modeling The Fertig Failure Theory Matrix Failure Theory Based on von Mises-Maxiu Principal Stress..61 vi

10 4.3 Results Failure predictions of the Fertig Failure Theory Conclusions...76 Chapter 5. Matrix Failure Modeling Augenting the Volue Average Matrix Level Quantities Approxiating the Volue Average Stress Fluctuations of the Matrix Constituent Coputing the Fluctuation Energy Constant The Modified Fertig Failure Theory Results Discussion..103 Chapter 6. Conclusions and Recoendations Suary and Conclusions Recoendations for Future Work 107 References vii

11 Appendix A: Periodic boundary conditions Appendix B: Influence coefficients for the fiber and atrix constituents of coposites of the World Wide Failure Exercises..116 viii

12 LIST OF FIGURES Figure 1: A unidirectional laina..3 Figure : Synthesis of a lainate fro individual laiae..4 Figure 3: Difference in volue average laina stresses and constituent level stresses.6 Figure 4: Idea of a Representative Volue Eleent (RVE)...7 Figure 5: Coparison of volue average stresses in a hoogeneous and a coposite aterial. 8 Figure 6: Coparison of volue average stresses and the actual distribution of stresses in the RVE 9 Figure 7: Fluctuation of shear stress in the through thickness direction...10 Figure 8: RVE with hexagonal fiber packing for fiber volue fraction of Figure 9: Decoposition of biaxial load into its coponents..37 Figure 10: Variation of interaction energy with fiber volue fraction.40 Figure 11: Stress plots for various loading configurations Figure 1: Probability distribution function for strains for different load cases...43 Figure 13: Variation of contribution of atrix to interaction energy with fiber volue fraction...44 Figure 14: Variation of contribution of C11 ~1 ter to interaction energy of the atrix Figure 15: Variation of contribution of C ~ ter to interaction energy of the atrix 46 Figure 16: Variation of contribution of C ~ ter to interaction energy 33 3 of the atrix 46 ix

13 Figure 17: Variation of contribution of C1 ~1 ~ ter to interaction energy of the atrix 47 Figure 18: Variation of contribution of C13 ~1 ~3 ter to interaction energy of the atrix 47 ~ ~ Figure 19: Variation of contribution of C3 3 ter to interaction energy of the atrix 48 Figure 0: Variation of contribution of C44 ~4 ter to interaction energy of the atrix 48 Figure 1: Variation of contribution of C55 ~5 ter to interaction energy of the atrix 49 Figure : Variation of contribution of C66 ~6 ter to interaction energy of the atrix Figure 3: Variation of interaction energy with atrix odulus for various loading configurations Figure 4: Variation of interaction energy with biaxial loads..53 Figure 5: The RVE with hexagonal fiber packing..57 Figure 6: Fluctuations in axiu principal stress in the atrix constituent...61 Figure 7: Biaxial failure envelopes for 0 laina ade of GRP (E-glass/LY556) 66 Figure 8: Biaxial failure envelopes for 0 laina ade of CFRP aterial (T300/BSL914C)..67 Figure 9: Biaxial failure envelopes for 0 laina ade of GRP aterial (E-glass/MY750).. 68 Figure 30: Biaxial failure envelopes for 0 laina ade of GRP aterial (E-glass/MY750) predicted by Von Mises Max. principal stress theory...69 x

14 Figure 31: Biaxial failure envelopes for 0 laina ade of GRP aterial (E-glass/MY750) predicted by Von Mises Max. principal stress theory with progressive atrix daage.71 Figure 3: Triaxial failure envelopes for 0 laina ade of CFRP aterial (T300/PR319) Figure 33: Triaxial failure envelopes for 0 laina ade of GRP aterial (E-glass/MY750)..73 Figure 34: Triaxial failure envelopes for 0 laina ade of GRP aterial (S-glass/Epoxy)...74 Figure 35: Triaxial failure envelopes for 0 laina ade of CFRP aterial (Carbon/Epoxy)...75 Figure 36: Decoposition of biaxial load into its coponents Figure 37: Coparison of volue average stresses and volue average stress fluctuations and the energy conserving stresses of the atrix constituent for GRP laina Figure 38: Interaction energy for test case Figure 39: Interaction energy for test case Figure 40: Interaction energy for test case Figure 41: Interaction energy for test case Figure 4: Interaction energy for test case Figure 43: Interaction energy for test case Figure 44: Interaction energy for test case Figure 45: Coparison of failure load predictions for GRP laina under cobined noral and shear loading...94 xi

15 Figure 46: Coparison of stress distributions..96 Figure 47: Coparison of failure load predictions for CFRP laina under cobined longitudinal and shear loading. 97 Figure 48: Coparison of failure load predictions for GRP laina under cobined longitudinal and shear loading Figure 49: Coparison of failure load predictions for CFRP laina under cobined hydrostatic and shear loading.99 Figure 50: Coparison of failure load predictions for GRP laina under cobined transverse and through thickness loading Figure 51: Coparison of failure load predictions for GRP laina under cobined through thickness and longitudinal loading Figure 5: Coparison of failure load predictions for CFRP laina under cobined through thickness and longitudinal loading Figure 53: Naing convention for node sets of the Representative Volue Eleent xii

16 LIST OF TABLES Table 1: Baseline aterial properties of the fiber..37 Table : Laina failure tests fro WWFE-I and WWFE-II.64 Table 3: Type of fluctuations observed due to different loading configurations in the WWFEs..83 Table 4: Coparison of values of Fij for various test cases of the WWFEs.84 Table 5: Coparison of values of Aij for various test cases of the WWFEs.84 Table 6: Coparison of values of Bij for various test cases of the WWFEs. 85 Table 7: Values of Ψ for different test cases of the WWFEs Table 8: Values of Ψ for different aterials subjected to sae loading configuration. 87 Table 9: Coparison of Bi paraeters obtained by using volue average and energy conserving atrix level stresses...93 Table 10: Coparison of ratios of the Bi paraeters obtained by using volue average atrix level stresses and energy conserving atrix level stresses...93 xiii

17 1. Introduction to coposite aterials 1.1 What are coposite aterials? Coposite aterials are defined as heterogeneous ixtures that are obtained by bonding two or ore hoogeneous phases. A careful observation would reveal that coposite aterials have always been present in nature. Perhaps the ost coonly used coposite aterial that is present in nature is wood, which has long fibers of cellulose that provide strength that are held together by a uch weaker substance called lignin which acts as the glue that bonds and holds the fibers in place (Rowell, 01). Other natural coposites include bones, teeth and plant leaves. Recent studies on spider silk have shown that it also is a coposite consisting of a gel core enfolded by a solid hollow structure (Mukhopadhyay and Sakthivel, 005; Vollrath et al., 1996). This suggests that Mother Nature has been far ahead of huans in exploiting the benefits of coposites. The beginning of anade coposite aterials can be traced back to the early Egyptian civilization c B.C. (Herakovich, 01). They ade papyrus paper by laying strips fro the fibrous papyrus plant into two layers with each layer at right angles to the other. In addition, they ade bricks fro ud and then reinforced the with straw to provide strength (Nicholson and Shaw, 000). Over tie other anade coposite aterials like concrete and papier-âché were produced and used all over the world. In 1939, continuous glass fibers were produced coercially (Knox, 198) for high teperature electrical applications. After two ore decades fibers fro boron (Talley, 004) and carbon (Tiot, 1961) were produced. The history of odern coposite aterials thus begins in the second half of the 0th century when lightweight fibers were ebedded in polyeric resins to obtain novel aterials having superior aterial properties and durability. 1

18 1. Advantages of fibrous coposite aterials Copared to conventional hoogeneous etals, fibrous coposites offer any advantages that ake the attractive alternatives for designers to consider. Unquestionably, the ost cited advantage of fibrous coposites is their high specific stiffness and high specific strength as copared to conventional engineering aterials. These properties translate into significant iproveent in perforance and reduction in consuption of energy. Unlike isotropic engineering aterials like etals, properties of coposites are often anisotropic. As a result coposite aterials can be designed to have desired properties in one direction without having to overdesign in other directions, which leads to significant weight savings. Use of coposites also increases fatigue life and corrosion resistance of final products (Herakovich, 01). Due to these advantages coposite aterials have found applications in various industries like aerospace, autootive, sports, wind energy, and athletic and recreational equipent. 1.3 Unidirectional fiber reinforced polyers Unidirectional fiber reinforced polyers (UD FRPs) are a special class of coposite aterials that are produced by ebedding long, lightweight fibers in a polyeric resin called the atrix. The ost coonly used fibers include glass and carbon/graphite which are inlaid in theroset epoxies. A unidirectional laina, the ost basic for of continuous fiber reinforced polyers is shown in Fig. 1. The laina or ply is a flat arrangeent of unidirectional fibers in a atrix. The fibers are the principal load carrying entities and are generally very strong and stiff as copared to the atrix. The atrix provides support and protection to the fibers. Another iportant function of the atrix is to provide a eans of distributing load aong and transitting load between the fibers. The stiffness and strength of the laina in the fiber

19 Figure 1: Unidirectional laina direction is typically uch greater than in the transverse directions. Generally, the properties of a unidirectional laina are orthotropic, with different properties in the aterial principal directions. The three-diensional elastic constants for an orthotropic unidirectional laina consist of the following independent in-plane and through thickness properties: E, E, E, G, G, G,, and 3 where 1, and 3 represent the three utually independent aterial directions. It is usually assued that a unidirectional fiber reinforced laina can be treated as being transversely isotropic which reduces the independent elastic constants to five since E E 3, G G 1 3, 13 and G E Transverse isotropy also reduces the independent strength values to six as copared to nine for fully orthotropic aterials. These are longitudinal and copressive strengths S S 1 T 1C,, transverse tensile and copressive strengths,, longitudinal shear strength S 1 and transverse shear strength S 3. In practical S S T C applications, lainates with fibers oriented in only one direction are a rarity. Even though unidirectional lainates (fibers oriented only in one direction) are extreely strong and stiff in 3

20 the fiber direction, they are very weak in the transverse direction and under shear loads because the load is carried by the uch weaker polyer atrix. In order to ake the final product stronger in the non-fiber directions, unidirectional lainae with fibers oriented in different directions are stacked on top of one another and the final layup is called a coposite lainate as shown in Fig.. The resulting lainates are uch stronger in the non-fiber directions and also have a higher shear load carrying capacity than their constituent lainae considered individually. Figure : Synthesis of a lainate fro individual lainae 4

21 1.4 Failure of unidirectional fiber reinforced polyers Failure in unidirectional (UD) fiber reinforced polyers ay occur due a variety of echaniss at the constituent level including: fiber fracture, fiber buckling (kinking), fiber splitting, fiber pullout, fiber/atrix debonding, atrix cracking and atrix yielding. Fiber fracture involves breaking of the continuous fibers into two or ore distinct segents and is the ost catastrophic of failure echanics since the fibers are the priary load-carrying agents in UD fiber reinforced polyers. This type of fiber failure is typically due to tensile stress in the fiber direction. Fiber fracture accopanied by fiber/atrix debonding causes fiber pullout. Axial copressive loading causes buckling of the fibers. Matrix cracking results fro transverse or shear loads that exceed the strength of the atrix. Since UD fiber reinforced polyers are widely used in any applications it is very iportant to understand coposite failure echaniss to iprove safety and reliability of coposite products. The ajority of failure theories analyze a coposite lainate on a ply-by-ply basis. These theories try to predict the failure of one of the plies of the lainate which is tered as first ply or initial failure after which, the properties of the failed laina are degraded such that its load carrying capacity is significantly decreased and the reaining un-failed lainae carry ore loads. Final or catastrophic failure occurs when all the lainae of the lainate have failed and the lainate can no longer support any load. Based on the type of quantities that the failure ethodologies use, they can be classified into esoechanics based or icroechanics based. Mesoechanics based failure techniques use quantities (stresses/strains) that are averaged over the entire volue of the laina to assess failure. A ajor drawback of using volue average laina quantities is that they do not accurately represent the true stress/strain state in the constituents of the laina. As shown in Fig. 3 a coposite laina was subjected to a transverse load of 40 MPa, which generated stresses in 5

22 the fiber and atrix constituents. The volue average stress in the transverse direction in the fiber constituent is about 46 MPa and is about 3 MPa in the atrix constituent. Moreover, the stresses in the longitudinal and the through thickness directions in the constituents was non-zero Figure 3: Difference in volue average laina stresses and constituent level stresses even in the absence of such volue average stresses in the laina. These stresses are selfequilibrating and so do not appear on the laina scale. This suggests that volue average constituent stresses as opposed to volue average laina stresses are uch ore accurate in describing the true stress/strain state in the constituent. Since failure of a laina initiates at the constituent level, it is iportant to predict failure of the constituent to assess failure of a laina. Moreover, since constituent level stresses ore accurately describe the stress/strain state in the 6

23 constituents, volue average constituent level stresses are a better choice than laina-averaged stresses for failure prediction. 1.5 A Representative Volue Eleent (RVE) In real icrostructures of unidirectional fiber reinforced coposites, fibers are distributed randoly in the atrix. Figure 4(a) shows an SEM iage (De Oliveira et al., 009) with the fibers colored gray that are distributed randoly in the atrix which is colored black. In the theoretical analysis of coposite aterials, a Representative Volue Eleent (RVE), also called (a) (b) (c) Figure 4: Idea of a Representative Volue Eleent a unit cell, can be defined as the sallest volue over which the easureents of average stress and strain and properties like fiber volue fraction, oduli etc. will yield a values that are representative of the whole of the coposite (Hashin, 1983; Hill, 1963). For the work in this thesis, it was assued that the fibers were packed in a hexagonal array in the atrix as shown in Fig. 4(b) and a Representative Volue Eleent (RVE) of this idealized icrostructure as shown in Fig. (c) was used to study the behavior of unidirectional fiber reinforced coposites. To produce hoogeneous fields, the RVE was subjected to periodic boundary conditions (Appendix 7

24 A). The use of an RVE with hexagonal fiber packing not only offered significant reduction in coputational costs but provided siilar volue average quantities as copared to an RVE with a rando distribution of fibers. 1.6 Distribution of stresses and strain in the constituents of a coposite aterial Figure 5 shows coparison of volue average stresses in the RVEs of a hoogeneous aterial like steel and an in-hoogeneous coposite aterial consisting of fibers ade of carbon that are ebedded in an epoxy atrix. Both the RVEs were subjected to a transverse load of 10 MPa and volue average stresses were coputed. The volue average stress in the (a) Hoogeneous aterial (b) Coposite aterial Figure 5: Coparison of volue average stresses in a hoogeneous and a coposite aterial 8

25 load direction in the hoogeneous aterial equals the applied load but for the in-hoogeneous aterial, the volue average stress in the load direction is different for the two constituents of the coposite aterial. A ore careful observation of the stress state in icrostructure as shown in Fig. 6 reveals that even in the constituents, the actual stresses in the load direction are not constant as suggested by the volue average constituent stresses. The volue average stresses Figure 6: Coparison of constituent volue average stresses and the actual distribution of stresses in the RVE are also not able to capture the fluctuations in the through thickness direction as shown in Fig. 7 which ay play a role in prooting failure. Thus we can conclude that volue average 9

26 constituent quantities do not acknowledge these fluctuations and as a result, the strain energy coputed fro the constituent volue average quantities ay not account for all the strain energy of the coposite which raises a question of their accuracy in failure load predictions. Figure 7: Fluctuation of shear stress in the through thickness direction 10

27 1.7 Objective The thesis of this work is that an energy consistent constituent -level stress etric exists that accounts for stress fluctuations within each constituent and can be used for failure prediction. The goal of this work is to test this hypothesis and, if the stress etric exists, to develop a ethod for coputing it and using it to predict constituent-level failure in coposites. 11

28 . Literature review.1 Early approaches for predicting failure loads of coposite aterials A failure theory is a ethodology that is used to predict the conditions under which aterials fail when subjected to various types of loads. In atheatical ters, a failure theory is expressed in the for of an inequality and failure of the aterial is said to occur when one equality is satisfied. The concept of a failure theory is better explained by considering the exaple of von Mises failure criterion which can be expressed as I I II III I III 1 Sy (.1) where I, II and III are principal stresses and S y is the uniaxial yield strength of the aterial. According to the von Mises failure criterion, a aterial fails when Eq. (.1) is satisfied. Since it assues the aterials to be hoogeneous and isotropic, it can be applied to conventional engineering aterials like etals. However, since coposites are in-hoogeneous and highly anisotropic, the von Mises failure criterion cannot be directly applied to these aterials. Fibrous coposites exhibit a range of coplex failure behaviors and a variety of theories have been proposed to predict their failure. The axiu stress failure criterion is the earliest and one of the siplest failure theories that was applied to predict the strength of coposites (Burke, 1983; Gerstle and Reedy, 1985; Nahas, 1986). It can be expressed as i Si 1 (.) 1

29 where i =11,, 3, i is the stress in i - direction and Si is the strength in i - direction. It is a non-interactive failure theory that is used to assess failure for a coposite ply. According to this theory, a coposite ply fails when the agnitude of stress in any direction exceeds its corresponding liit in that direction. This theory evaluates the failure state of the coposite laina based on a single value of stress in that laina and does not take into consideration the state of ulti-axial loading or the effects of the cobinations of different stress coponents on the onset of failure. Although, the axiu stress failure criterion is capable of distinguishing between various laina failure odes, it cannot explicitly separate fiber and atrix constituent failure. The non-interactive nature of the theory underines its accuracy for predicting failure loads of coposite aterials under a ulti-axial state of stress. The axiu strain failure criterion is identical to the axiu stress failure criterion except that the forer accounts for soe of the interactions between the stresses that are attributable to the Poisson s effects in the aterial (i.e., stresses in the fiber and throughthickness directions will affect the strain in the transverse-direction). It can be expressed as i 1 ax i (.3) where i =11,, 3, i is the strain in i - direction and ax i is the axiu allowable strain in i - direction. The axiu strain theory, as the nae suggests, evaluates the state of failure of a laina based on strains in that coposite laina. Even though both the failure criteria are siple and ay sound priitive, they are often used either in their original or odified fors by any structural analysts today. 13

30 Non-interactive failure criteria like the axiu stress and axiu strain assess failure based on a single coponent of stress and strain respectively. However, under a biaxial/triaxial load state, the stress coponents ay interact in which case the non-interactive failure theories ay overpredict the failure loads. Gol denblat and Kopnov (Gol denblat and Kopnov, 1965) were one of the first to suggest the use of a quadratic interactive or tensor polynoial failure criteria with the use of stress tensors for glass reinforced plastics. For the first tie a failure theory considered interactions of various stress coponents of the laina to evaluate daage in coposites. Tsai-Wu (Tsai and Wu, 1971) adopted a siilar approach and cae up with a siplified equation siilar to Gol denblat and Kopnov's which can be expressed as F F 1 (.4) i i ij i j where, j, k 1,,..6; i i F and F ij are strength tensors. Their quadratic, interactive, stress-based failure criteria used laina level stress to deterine the loads at which a coposite ply fails. Laina strengths were used to deterine coefficients of stress coponents. However, neither of the two theories could distinguish between laina failure odes or between fiber or atrix constituent failure. Moreover, neither theory was based on actual physical phenoena. With the availability of new experiental test data, there arose a need for failure theories that were not copletely epirical but also had soe physical significance. Hashin (Hashin, 1980) proposed a stress-based failure criteria for coposite aterials that considered a ulti-axial stress state for atrix doinated failure odes and a uniaxial stress state in the fiber direction for fiber doinated failure odes which can be expressed as 14

31 11 1 Fiber failure in tension 1 S11 S1 (.5) 11 Fiber failure in copression 1 S11 (.6) 1 Matrix failure in tension 1 S S1 (.7) 3 S 1 Matrix failure in copression S 1 S S S (.8) where ij is the stress coponent on the i -coordinate surface in the j -direction, Sij and tensile and copressive strengths respectively and is a user-specified paraeter. Hashin, Sij are siilar to his predecessors, used laina level stresses to predict failure in a coposite ply. The theory identified four different odes of failure of a coposite ply: tensile fiber failure, copressive fiber failure, tensile atrix failure and copressive atrix failure. A ajor drawback of this theory is the need to specify an independent paraeter to deterine fiber or atrix tensile failure in the ply which has to be deterined fro experiental data. Thus in the absence of any experiental evidence, this value ay be arbitrarily chosen between an allowable range of 0 to 1 which akes accuracy of blind predictions ade by the theory questionable. Christensen (Christensen, 1997) also forulated a failure criteria which used laina level stresses to identify failure in coposite plies which can be expressed as 15

32 Fiber failure criterion 11 1 (.9) S S SS Matrix failure criterion 1 S 1 S 3 1 S S 1 1 S S (.10) The theory recognized failure as either fiber or atrix doinated. Unlike Hashin, Christensen considered ulti-axial stress state for both fiber and atrix constituent failure. His theory did not involve an independent paraeter that required correlation with test data and yet was general since it provided failure ode predictions for both the fiber and atrix constituents. Over tie nuerous other failure theories were proposed but none of the existing failure theories could provide accurate and eaningful predictions for the entire spectru of loadings encountered in a coposite structure. To confir the then current state-of-the of predicting failure and daage in fiber reinforced polyers, Hinton and Soden launched a study called the 'World Wide Failure Exercise' (WWFE) (Hinton and Soden, 1998). This failure benchark was aied at coparing failure predicting capabilities of various leading coposite failure theories fro around the world for a variety of experiental test data. The first failure exercise was deeed a success and was then followed by two other exercises (Hinton and Kaddour, 01; Kaddour et al., 013). The text in the following sections contains a brief discussion of soe of the iportant failure ethodologies fro the First and Second World Wide Failure Exercises. 16

33 . World Wide Failure Exercise I The first World Wide Failure Exercise (WWFE-I) contains a detailed assessent of nineteen failure ethodologies for predicting the deforation and failure response of polyer coposite lainates when subjected to coplex states of stress. Of all the nineteen failure techniques, the theories of Zinoviev(Zinoviev et al., 1998, 00), Bogetti (Bogetti et al., 004a, 004b), Puck (Puck and Schürann, 1998, 00), Cuntze (Cuntze, 004; Cuntze and Freund, 004) and Tsai (Kuraishi et al., 00; Liu and Tsai, 1998) were ranked highest with regards to accuracy of failure prediction (Soden et al., 004). These leading theories along with soe other icroechanical failure techniques fro WWFE-1 are discussed in the following text. Zinoviev (Zinoviev et al., 1998) used a non-interactive axiu stress failure criterion to deterine failure echaniss and took appropriate post-initial failure action. He used ply level stresses to predict failure loads on the laina. Since the theory is based on the axiu stress criterion, it is not able to distinguish between various constituent-failure odes. The postinitial failure degradation odel involved a gradual drop in the properties of the laina. The final elastic and the shear oduli of the daaged laina were degraded to a fraction of the original oduli depending on the strains on the lainate. The non-interactive nature of the failure criteria led to over-prediction of ultiate loads for certain loading cobinations. In spite of assuing linear-elastic aterial properties, the stress-strain predictions of the theory were in close agreeent with test data. The theory failed to predict the observed large deforations in one of the test cases and predicted an unrealistic open failure envelope in another test case. The Zinoviev theory used a siple axiu stress failure criterion but due to a well-controlled degradation odel, the theory perfored very well overall and appeared to be one of the best theories of the exercise. 17

34 Bogetti (Bogetti et al., 004a) used a non-interactive axiu strain failure criterion to obtain failure envelopes for the available test data but unlike Zinoviev, Bogetti considered the non-linear behavior of the laina under shear loads. To siulate progressive failure, the elastic/shear odulus of the failed laina was set to zero and resultant loads were transferred to the un-failed lainae. Like Zinoviev's theory, this theory is also not capable of separating the various constituent-failure odes. The theory over-predicted the transverse copressive strength of one of the unidirectional lainae. For the sae test case, the predictions in the biaxial tension and copression quadrants were uch higher as copared to other theories. The theory gave good initial predictions for the non-linear stress-strain curves of ulti-directional lainates. Owing to its siplicity and capability of predicting the wide range of test data features, it was ranked higher than other theories. Puck's failure theory (Puck and Schürann, 1998) is one of the ost sophisticated theories of the failure exercise. It is an interactive failure theory which is able to differentiate between fiber and atrix constituent failures. Puck used both stresses and strains of the laina as well as constituent properties of the laina to assess failure. This failure theory has two fiber failure odes, the first being longitudinal tensile failure and second being longitudinal copressive failure. For the atrix constituent, Puck's theory provides three different kinds of odes, also known as 'inter- fiber failure odes'. The three odes which are tered A, B and C are distinguished by the orientation of fracture planes relative to the unidirectional fibers. It ust be noted that laina level properties were used by Puck to evaluate failure in individual coposite plies. Progressive daage was achieved by a gradual degradation of the properties of the failed laina. The final elastic and shear oduli and the Poisson s ratio of the failed laina were assued to be a fraction of the original laina properties depending on the stress on the 18

35 lainate. The theoretical failure envelopes and stress-strain curves predicted by the theory were in very close agreeent with the experients except where large non-linear deforations were observed. Puck s failure theory has a large nuber of paraeters to be deterined which akes calibration very difficult. Cuntze's (Cuntze, 004) failure theory bears a strong reseblance to the failure odeling used by Puck but coparatively has fewer paraeters which akes calibration easier. He used laina level stresses to predict failure of coposite plies. The theory recognized five failure echaniss: Longitudinal tensile and copressive fiber-doinated failures and three atrix doinated failure odes also called 'inter-fiber failure odes'. The axiu stress criterion was used to assess failure state of the fibers. Unlike Puck, Cuntze assued interaction between various failure echaniss due to soe probabilistic effects. The degradation of properties of the atrix constituent was based on experientally obtained non-linear stress-strain curves of the laina. The theory perfored very well overall and provided quite accurate predictions for a variety of test data. Liu and Tsai (Liu and Tsai, 1998) used Tsai s well know interactive failure criterion (Tsai and Wu, 1971) to predict strength of coposites in the first World Wide Failure Exercise. After initial failure had occurred, they degraded the properties of the isotropic atrix by 85% and the effective properties of the failed laina were coputed using icroechanics. Their theory described the failure envelopes for unidirectional laina better than any other theory. However, the theory predicted an enhanceent in strength under copression-copression biaxial loading which was not verified due to lack of experiental test data. The predicted initial failure stresses for ulti-directional lainates were in poor agreeent with experiental data but the predicted shapes of the final failure envelopes agreed quite well with the experients. 19

36 Owing to the assuption of linear elasticity, their theory could not predict the large non-linear strains observed in the test cases where high laina shear was involved. Overall, due to its siplicity and good failure predicting capabilities, the Liu and Tsai failure theory was one of the best theories of the exercise. Overall, approxiately half of the theories of the First World Wide Failure Exercise relied on icroechanics in their forulation. Chais s failure theory (Gotsis et al., 1998) used relations based on icroechanics to predict properties of the laina fro properties of its constituents using a coputer code called Integrated Coposite Analyzer (ICAN). ICAN required constituent level properties to copute several laina level properties which were then used to assess failure using a odified distortion energy criterion. The coputed laina level properties like ply-stress coponents, ply-strengths, noral oduli of the ply and the Poisson s ratio of the ply were used to predict failure of the laina. Another coputer code Coposite Durability Structural Analysis (CODSTRAN) was used to degrade constituent properties after first ply failure and eventually predict catastrophic failure in ulti-ply lainates. It ust be noted that even though a siple odified distortion energy theory was used to assess failure at the laina level, the process of obtaining the laina properties using constituent properties was very coplicated. For exaple, the ply-stress coponents were coputed using an expression that contains ore than fifteen ters. Moreover, the failure criteria used laina level stresses as opposed to constituent level stresses to assess failure and so this technique failed to take full advantage of the power of icroechanical odeling. The Chais failure theory did not explicitly provide the ode of failure but a close exaination of the ters in the criterion can reveal if the daage is due to atrix or fiber constituent failure. The post initial degradation odel replaced the odulus of the atrix to a negligible value after initial failure and the 0

37 resultant properties of the failed laina were coputed using icroechanics relations. The failure predictions for unidirectional lainae agreed with the experiental evidence but were very conservative for ulti-directional lainates indicating a proble in post-initial failure odeling. The theory was not able to predict the observed nonlinear stress-strain curves. Chais s failure theory is very coplicated and involves a large nuber of variables aking it difficult to work with. Moreover, the predictions were not as accurate as other theories dues to which it was ranked lower than the leading failure theories. Huang s theory (Huang, 004), is also icroechanics based and cobines ideas like bridging odel, plasticity and generalized axiu stress theory into a single failure odeling technique. The odel consists of bridging atrices that relate constituent level stresses to laina level stresses. The various ters of these atrices were coputed using both laina and constituent level properties: fiber volue fraction, elastic and shear oduli, and Poisson s ratio. Fiber and atrix level stresses were then used to copute axiu and iniu principal stresses in the constituents and an equivalent stress was deterined for each of the constituent using these principal stresses. The coposite laina was assued to have failed if any of the two (fiber or atrix) equivalent stresses reached an ultiate stress value that corresponded to the strength of the constituent. For post initial failure odeling, Huang replaced all the constituent properties with a zero and assued that the failed laina did not take carry any load after it had failed. The theory predicted large values for uniaxial transverse tensile and shear strengths for the unidirectional lainae which differed significantly fro the experiental data. The predictions of final strengths and deforation of ultidirectional lainates was very low for soe loading configurations. A unique feature of this ethodology was the use of plasticity flow theory (Prandtl Reuss flow theory) while calculating deforation and loads on the atrix 1

38 constituent enabling Haung to show atrix yielding while predicting failure loads. Overall, Haung s theory sees to be quite ature but needs iproveent to enhance the assessent of failure. Mayes and Hansen (Mayes and Hansen, 004) used a failure theory that expressed the failure state of the coposite aterial in ters of transversely isotropic stress invariants of the volue average stresses of the constituents. Their theory assued that daage in a coposite laina begins at the constituent level which is why Mayes and Hansen used constituent level stress and strain data to predict failure of a constituent (fiber or atrix). A finite eleent code based on the Multicontinuu theory (Garnich and Hansen, 1997) was used to extract stress and strain field data for constituents of the coposite. The theory had two separate criteria, one for fiber and the other for atrix constituent failure. The failure criteria are coposed of stress invariants with appropriate coefficients that can be deterined fro coposite strength data. Since it is very difficult to obtain soe of constituent strengths experientally, Mayes and Hansen used a nuerical technique to obtain the sae. Their degradation odel eployed a discrete reduction in the properties of the failed laina by reducing all the oduli of the constituents to near zero values. The theory perfored oderately well in predicting unidirectional laina failure enveloped but failed to predict the axiu shear stresses observed in experients under cobined direct and shear loading. In spite of being an interactive stress based theory, the predicted failure envelope for biaxial tension and copression test was siilar to that of non-interactive theories. This was due to the assuption that coposite failure in the fiber direction is fiber doinated which led to neglecting the atrix constituent stresses in the fiber direction. Even though the final failure stresses predicted by their theory for ulti-directional lainates were soeties very low as copared to experients, the

39 initial failure strengths of the sae lainates were in better agreeent with the test data. The stress-strain curves were also truncated at uch lower strains that the final strains observed in experients. Thus Mayes and Hansen tried to solve the coplicated proble of coposite failure prediction using a novel approach and have seeed to have paved the path for other researchers to look at constituent level behavior for assessing coposite failure. Soe theories fro WWFE-I were not copletely based on icroechanics but used soe properties of the constituents to predict final failure loads. Hart-Sith s theory (Hart- Sith, 00) used constituent properties (fiber volue fraction, elastic oduli and Poisson s ratios of the fiber and atrix constituents) to copute final failure strains required for obtaining failure envelopes in the strain plane. These strain envelopes were then converted into appropriate failure stress envelopes. Rote's failure technique (Rote, 1998) used elastic odulus and strengths of the atrix constituent in his atrix failure criterion..3 World Wide Failure Exercise II The second world wide failure exercise was set up to assess the aturity of failure criteria for predicting the failure strengths and deforation of unidirectional fiber reinforced coposite aterials subjected to tri-axial load states. A total of twelve failure theories were eployed to solve twelve test cases. Of the twelve different ethodologies, the failure theories of Carrere (Carrere et al., 01, 013), Cuntze (Cuntze, 01, 013), Pinho (Pinho et al., 01, 013) and Puck (Deuschle and Kröplin, 01; Deuschle and Puck, 013) were ranked highest due to their accurate predictions of failure strength and stress-strain curves of unidirectional 3

40 lainae and ultidirectional lainates (Kaddour and Hinton, 013). These failure theories and soe iportant icroechanics based theories are discussed in the following text. Carrere (Carrere et al., 01) eployed a icroechanical-based hybrid esoscopic (MHM) 3D approach where a failure echanis was forulated in ters of physical principles and introduced icroechanical aspects at the esoscopic scale. His failure theory considered non-linear aterial behavior of the laina and distinguished between two fiber failure odes and two atrix failure (called Inter Fiber Failure) odes. He used various paraeters which described daage at the icro-scale along with laina based quantities to predict first ply failure. Post initial failure odeling eployed a gradual reduction in the properties of the failed laina again with the use of paraeters which describe the effect of daage at fiber/atrix constituent level. The failure odel was able to capture the iportant features over the wide range of test data. It was one of the two theories that were able to predict fiber failure due to fiber kinking under copressive loads. Carrere s failure theory is very coplicated and deands a lot of effort on the part of the analyst to provide eaningful results. However, Carrere was able to forulate a physically sound failure technique by taking the advantage of icroechanics and laina based quantities which can prove to be a useful tool in understanding the failure behavior of fiber reinforced polyers. Cuntze s failure theory (Cuntze, 01) was established on the odeling capability called Failure Mode Concept (FMC). Based on nonlinear analysis, Cuntze s used an in-house MathCad code to explain the behavior of coposite aterials under triaxial load states. The theory incorporated five FMC odes of failure: two fiber failure odes and three atrix failure odes. The fiber failure odes eployed the axiu stress failure criterion and the atrix failure odes used the stress invariants of the transversely isotropic laina to asses and predict 4

41 failure of the laina. After initial failure, the theory assued softening behavior of the failed laina and the effective oduli of the failed laina were coputed using stresses and available stress-strain data of the lainate. Unlike Carrere, Cuntze eployed a uch less rigorous approach to describe the failure behavior of coposite aterials. The predictions of the theory were in good agreeent with the experiental evidence. Deuschle and Kröplin (Deuschle and Kröplin, 01) used the sae ature failure theory which was eployed by Puck (Puck and Schürann, 1998) in the first World Wide Failure exercise with slight odifications to predict failure loads and deforation of fiber reinforced coposites under tri-axial stress states. Originally, the theory was forulated based on the assuption that atrix failure can occur on planes parallel to the fibers which is indeed the case in UD coposites. However for pure atrix, fracture ay occur in a plane at any orientation and to capture this behavior, two ore atrix failure odes were introduced. Overall, the predictions of their failure theory were in good agreeent with the test data fro the twelve test cases. Pinho (Pinho et al., 01) used physically-based constitutive odeling to forulate his theory and like all the other participants considered non-linear aterial properties of the laina for predicting failure in fiber reinforced polyers. His failure theory used laina level quantities to assess failure and distinguished between fiber and atrix failure odes. The fiber failure odes incorporated failure due to longitudinal tensile loads and fiber kinking/splitting due to copressive loads. A axiu stress failure criterion was used to predict fiber failure in longitudinal tension. Pinho concluded fro various experients on FRPs under copressive loads that atrix failure next to isaligned fibers was one of the driving forces behind fiber kinking. His theory used transverse tensile stresses as well as in-plane and through thickness 5

42 shear stresses of the laina to predict buckling of fibers. Pinho coputed the noral, transverse and longitudinal coponents of the traction in the fracture plane and used the sae to predict atrix failure. For propagating failure, Pinho reduced the coponents of the traction vector acting on the fracture plane to zero such that the all the load is carried by un-failed lainae in the coposite. Overall, Pinho s failure theory perfored very well as copared to other theories. Unlike, soe theories of the exercise which ranked very high, Pinho s theory is coparatively siple thus aking it one of the best failure theories of the exercise. Tsai-Ha s theory (Huang et al., 01) in the second failure exercise is copletely different than the failure ethodology eployed by Liu and Tsai (Liu and Tsai, 1998) in the first World Wide Failure Exercise. Huang et al. developed a new icroechanics based failure theory that used constituent level stresses to predict failure of a constituent. His odel used a unit cell with hexagonally arranged fibers to extract Stress Aplification Factors which relate laina level stresses to constituent level stresses. The new approach was able to distinguish between two fiber failure odes, one atrix failure ode and one interface failure ode using constituent level stresses. The axiu stress failure criterion was used for predicting failure of fibers under longitudinal tensile and copressive loads. A odified distortion energy failure criterion was used to predict failure of the atrix constituent. A quadratic failure criterion was proposed which used tractions on the interface to predict initiation of daage at the fiber-atrix interface. They never used the interface failure criterion assuing that the bonding between the fiber and the atrix was sufficiently strong such that separation of the fiber-atrix interface did not occur. After initial failure of a ply, the atrix properties of the failed ply were degraded gradually depending on the stresses in the ply. The predictions of the theory were conservative for ost of the test case and need several iproveents. Even though Huang et al. explained the 6

43 use of the Stress Aplification Factors to copute constituent level stresses fro laina level stresses, they have not explained the procedure in detail. Conservative predictions for the test case indicate that the theory ay need a ore robust progressive failure ethodology. In spite of all these drawbacks, their theory is very siple and appears to have potential to grow into a uch ore robust failure odeling technique. Nelson, Hansen and Mayes (Nelson et al., 01) used a siilar approach like the one in the first World Wide Failure exercise, in the second failure exercise with soe ajor odifications to the atrix failure criterion and post initial failure odeling technique. They continued the use of constituent level stresses to predict failure loads on coposites which were extracted fro coposite level stresses using a FE approach based on the Multicontinuu theory. A new atrix failure criterion was proposed which now incorporated the atrix constituent stresses in the longitudinal (fiber) direction. A coupling between shear and transverse copressive stresses was also added. The post initial failure approach adopted in the second failure exercise allowed gradual degradation of the properties of the atrix constituent. The odification of the atrix failure criterion has ade calibration ore difficult, as one of the invariants used in the criterion has no physical eaning. The odel predicted large and unbounded strengths in various test cases that were analyzed. Contrary to the predictions of other theories, the odel predicted an open envelope under tension-tension case for isotropic atrix. Although the failure theory is not as coplicated as other failure ethodologies, it has a lot of roo for growth since overall its predictions were not as highly ranked as the best theory of the exercise. In the second failure exercise, Haung (Zhou and Huang, 01) used 3D lainate theory along with the original bridging odel used in the first failure exercise to predict failure loads 7

44 and deforation of fiber reinforced polyers. Bridging atrices were used to copute constituent level quantities fro laina level quantities. The tensile failure criterion of the fiber and the atrix constituents was not changed but a new copressive failure criterion for the constituents was introduced. This new failure criterion, which took into account the enhanceent of laina strength due to tri-axial copressive loading, used principal stresses of the constituents to predict their copressive failure. In the new post initial degradation odel, the odulus of the resin aterial in the failed laina was reduced to one percent of its original agnitude. The odifications introduced by Haung in his failure theory did not iprove its overall failure predicting capability. The ajority of failure theories in the WWFEs used laina level quantities to assess failure state of the coposite. Soe of the theories were icroechanics based and used constituent level quantities to predict failure loads of the coposite. The work in this thesis is priarily concerned with iproving the easureent of volue average constituent level quantities that are used in failure odeling by icroechanical theories. The new stress etric that was obtained could be used in conjunction with any of the icroechanical failure ethodologies that use volue average stresses in evaluating failure. 8

45 3. Interaction Energy (The ajority of this chapter has been accepted for publication in the AIAA Journal). Coplex interactions in fiber reinforced coposites between ultiple failure echaniss have ade accurate failure prediction a daunting challenge. One approach to better identify failure echaniss has been the use of volue average constituent stresses in the coposite to predict the onset and outcoe of failure in individual constituents. However, this approach is shown here to not conserve strain energy in the coposite, which could potentially affect the accuracy of failure prediction under certain loading conditions. The focus of this chapter is to develop an expression for the discrepancy in strain energy, tered the interaction energy, and to nuerically evaluate the influence of constituent properties, fiber volue fraction, and load cobinations on the agnitude of this energy Overview of volue average quantities used for coposite failure load predictions The use of coposite aterials in the wind and aerospace industries has grown draatically in recent years, with coposites coprising 50% or ore of Boeing s 787 and Airbus s A380. With this increase in use has coe a greater deand for design tools that can accurately predict daage initiation and propagation, durability, and reaining life. This has proven to be quite challenging due the range of coplex failure behaviors exhibited by coposites. Unlike conventional hoogeneous aterials like etals, the constituent undergoing failure ay switch rapidly and the failure echaniss within each constituent can vary widely depending on the loading. At the laina level, tensile loads parallel to the fiber direction ay 9

46 cause both fiber and atrix fracture. Under transverse loading, Mode I atrix cracking or fiber/atrix debonding ay occur in tension, with a shear atrix failure occurring in copression. At the lainate level, delaination ay occur, which will substantially affect the distribution of stresses in each laina. The result of this coplexity has been a large nuber of proposed coposite failure theories with no clearly superior choice. The First and the Second World-Wide Failure Exercises (WWFE I and WWFE II, respectively) have provided a uch needed benchark against which to copare the effectiveness of various coposite failure theories for both glass- and carbon-fiber coposites subject to coplex loading configurations (Hinton and Kaddour, 01; Hinton and Soden, 1998). One distinction aong failure theories was the use of icroechanics in the prediction of failure, such that soe easure of fiber and atrix constituent stresses were used to predict laina failure rather than using laina-level failure criteria to predict failure. Although none of the icroechanics based theories were judged in the benchark exercises to exceed the perforance of ore established theories, they were assessed as evolving towards aturity (Kaddour and Hinton, 013). The unique appeal of constituent-level failure theories is not siply in static failure prediction, but their potential for physics-based odeling and aterials design efforts, such as the Materials Genoe Initiative and Integrated Coputational Materials Engineering. The ethods used to extract constituent stresses and strains fro coposite stresses typically utilize soe for of a localization tensor to ap coposite stresses to constituent stresses, which ay be evaluated at specific points (like peak stresses) or as volue average quantities. The use of constituent stresses at specific points is strongly dependent on the choice of representative volue eleent (RVE) for the coposite icrostructure. Any deviation fro 30

47 the idealized structure ay substantially alter the stress at a particular location. The use of volue average stresses is appealing because it is less sensitive to details of the icrostructure; rando fiber packing and idealized hexagonal fiber packing will yield nearly the sae average constituent stresses and strains. However, the use of volue average constituent stresses and strains ay not account for the total strain energy in a coposite. For exaple, a careful inspection of the stress fields in a coposite icrostructure reveals the presence of shear stresses in the constituents even when the coposite loading is purely transverse noral these stresses, however, average to zero and therefore cannot be accounted for in any failure or aterial nonlinearity odel. As a result, failure prediction under particular loading configurations ay not be consistent with bulk constituent properties because a significant portion of the strain energy ay be unaccounted for. The energy not accounted for when using volue average constituent stresses and strains is herein tered the interaction energy. To enhance physics-based coposite failure prediction, all strain energy should be accounted for. The goal of this chapter is to provide a quantitative foundation for the developent of constituent-level coposite failure theories based on the conservation of strain energy across length scales. As such, the focus of this research is to quantify the agnitude of the interaction energy relative to the total strain energy for a variety of fiber volue fractions, atrix elastic properties, and coplex ulti-axial loading configurations. The reported results clearly indicate that a significant fraction of energy is unaccounted for under a variety of loading conditions in typical aerospace coposite aterials, but also suggest that this interaction energy could be readily incorporated into constituent failure criteria. 31

48 3.. Theoretical Motivation This section focuses on establishing the equations that can be used to quantify the interaction energy. Consider a RVE of a coposite aterial consisting of fiber and atrix phases. Let U denote the strain energy of this coposite under an arbitrary load state. Assuing a linear elastic aterial, this energy can be represented as U 1 V C dv C ij C ij C (3.1) where C ij and C ij are the coposite stresses and strains, respectively; repeated ij indices are sued, and integration perfored over the entire volue of the copositev c. For a periodic RVE subject to hoogeneous boundary conditions as defined by Hashin (Hashin, 1983), Equation (3.1) can be readily written in ters of average coposite properties (Sun and Vaidya, 1996) U 1 C C ij ij VC (3.) where the ters in the brackets denote volue average quantities. Strain energy ay be separated into contributions fro the fiber and the atrix constituents U U f U (3.3) such that U f 1 V f f ij f ij dv f (3.4) 3

49 U 1 ij ij dv V (3.5) where the superscripts f and denote quantities for fiber and atrix, respectively. In contrast to using average laina stresses, suing the strain energies of the constituents coputed using constituent volue average stresses does not account for all of the strain energy in the coposite. To quantify this interaction energy the strains in each constituent are written in ters of a ean value ij plus a fluctuation ~ij ; siilarly, the stresses are also written as a ean value ij plus a fluctuation ~ij. ij ij ~ij (3.6) ij ~ij ~ij (3.7) For the fiber, Eqs. (3.6) and (3.7) are substituted into Eq. (3.4) to obtain Uf 1 ijf ~ijf V f f ij ~ijf dv f (3.8) Expanding Eq. (3.8) yields Uf ijf ijf dv f ~ijf ~ijf dv f ijf ~ijf dv f ~ijf ijf dv f Vf Vf Vf Vf (3.9) which on further siplification gives Uf 1 f 1 ij ijf V f ~ijf ~ijf dv f Vf 33 (3.10)

50 Because volue average quantities are constant and ~ and ~ are local fluctuations with averages that are identically zero. The atrix contribution is derived in a siilar anner to give f ij f ij U 1 1 ij ij V ~ ~ ij ij dv V (3.11) The strain energies in the fiber and atrix can then be written as U f 1 f ij f ij V f f V f (3.1) U 1 ij ij V V, (3.13) where and are the fiber and atrix contributions to the interaction energy density, f respectively given by f f 1 V f V f ~ ~ f ij f ij dv f (3.14) 1 V V ~ ~ dv ij ij (3.15) The interaction energy U is thus related to the total strain energy and the energy coputed fro volue average constituent quantities via U U f U 1 ij ij V 1 f ij f ij V f U (3.16) where U V V. (3.17) f f 34

51 Substituting Hooke s law into Eq. (3.14) allows the interaction energy density to be written entirely in ters of strain fluctuations f 1 1 Cijkl ~ijf ~klf dv f C n ~f ~nf dv f Vf Vf (3.18) where Cn is the,n coponent of the 6x6 fiber stiffness atrix. Assuing transverse isotropy and expanding Eq. (3.18) yields C f ~ f C f ~ f C f ~ f C f ~ f ~ f C f ~ f ~ f f f ~f f ~f ~f C 44f ~5f C66f ~6f C 3 3 C 44 4 (3.19) The interaction energy density for the atrix can be siilarly shown to be C ~ C ~ C ~ C ~ ~ C ~ ~ ~ ~ ~ C 44 ~5 C66 ~6 C 3 3 C 44 4 (3.0) Equations (3.19) and (3.0) quantify the interaction energy between fiber and atrix, but they contain a total of 18 ters. The focus of the subsequent odeling effort is to deterine the significance of the interaction energy relative to the total strain energy, to investigate how the interaction energy ay change as a function of loading and aterial properties, to evaluate the contribution of each of the 18 ters to interaction energy and to attept to establish relative contribution of each constituent to the interaction energy. 35

52 3.3. Finite Eleent Modeling A finite eleent odel was developed and studied using Abaqus to investigate the agnitude of the interaction energy as a function of fiber volue fraction, elastic oduli of the constituents, and varying loading conditions. For the purpose of analysis, a representative volue eleent (RVE) of a hexagonal fiber packing was odeled, as shown in Fig. 8. Periodic boundary conditions were applied on all RVE edges, faces, and corners. This was achieved by extracting nodes fro the RVE after eshing, reordering the properly, and using equation constraints for Figure 8: Representative volue eleent (RVE) with hexagonal fiber packing for fiber volue fraction of

53 the node sets. The axial direction of the fibers is defined as the 1-direction (longitudinal). Carbon fiber properties (AS4) reported by Sun (Sun and Vaidya, 1996) and shown in Table 1 were used for all siulations. The atrix was assued to be isotropic with a Poisson s ratio of Table 1: Baseline aterial properties of the fiber Material Type AS4 Transversely isotropic E1 (GPa) 35 E (GPa) G1 (GPa) Three paraetric studies were conducted. First, the fiber volue fraction was varied fro 0.05 to 0.85 with the atrix odulus was fixed at 1% of the fiber-direction fiber odulus, E.35GPa. In the second study, the atrix odulus was varied fro 1% to 10% of the fiber odulus in the fiber direction. (A atrix odulus of E11fiber roughly corresponds to epoxy (E = 4.8 GPa).) Results for the odulus study were obtained for four different fiber volue fractions of 0.05, 0.5, 0.6 and In both of these paraetric studies, four unique coposite loading states were exained: and. These loads were achieved Figure 9: Decoposition of biaxial load into its coponents 37

54 by fixing displaceents of the RVE control nodes such that a strain of 0.01 was applied in each load case. In the third study, five types of biaxial loads were applied:, ),, ), ( 33 ( 1, ),, ), and, ). These loads were applied such that they corresponded to ( 1 3 ( 1 13 ( 3 the x- and y-coponents of the biaxial load represented as the radius of a circle as shown in Fig. 9, with varying fro 0 to 180. The two uniaxial loads required to generate the biaxial load state can be expressed as I cos sin II (3.1) where 10MPa was the resultant biaxial load vector agnitude, and I and II are the corresponding uniaxial loads. For this third study, the atrix odulus and fiber volue fraction were held constant at E11fiber (E = 4.0 GPa) and 0.6, respectively Results and Discussion As discussed above, three series of siulations were carried out to quantify the dependencies of the interaction energy on aterial properties, fiber volue fraction, and loading conditions: (i) constant atrix and fiber odulus with varying fiber volue fraction, (ii) varying atrix odulus for four different fiber volue fractions, and (iii) constant aterial and icrostructure properties with varying biaxial loading conditions. 38

55 Effect of Fiber Volue Fraction on Interaction Energy Volue average quantities (stresses, strains and stiffness) of the coposite were extracted fro the odel and then the total strain energy for the coposite was calculated using Eq. (3.). The volue average quantities (stresses, strains and stiffness) of the constituents were also extracted. Constituent strain fluctuations and average stresses and strains were coputed and used in Eqs. (3.19) and (3.0) to give f and. The total interaction energy was coputed using Eq. (3.17). The interaction energy fraction ( U / U ) was coputed for all the fiber volue fractions and each load case. The properties of the RVE were transversely isotropic, thus only four (and not all six) load cases are unique. Figure 10 shows U / U as a function of fiber volue fraction for each load case. Three features of these data are of particular interest. First, the interaction energy is strongly dependent on the load case and it can be a significant fraction of the total energy. It can be ore than 30% of the total strain energy for longitudinal shear loading (shear-1), although it was negligible for unidirectional loading in the fiber direction (tension-11). Second, interaction energy increases with increasing fiber volue fraction, up to typical volue fractions. Finally, transverse tension (tension-) and transverse shear (shear-3) show a peak at a fiber volue fraction of about 0.65, very close to typical fiber volue fractions in aerospace-grade coposites. 39

56 Figure 10: Variation of interaction energy with fiber volue fraction To qualitatively understand these features consider stresses in an RVE with a fiber volue fraction of 0.6 subject to the four load cases. In case of longitudinal tension, shown in Fig. 11a, the stress distribution is unifor in each constituent throughout the structure, thus strain fluctuations are inial. Consequently, as shown in Equations (3.19) and (3.0), f and, and correspondingly U, are nearly zero for this load case. In the case of longitudinal shear, shown in Fig. 11b, the stress fluctuation is largest. Consequently, the interaction energy is 40

57 axiized for this case, as seen in Fig. 10. This result is critically iportant because it eans that at a typical fiber volue fraction for aerospace grade coposites (60%), volue average constituent stresses neglect nearly 30% of the distortion energy in the coposite in shear. The transverse tension and transverse shear load cases are shown in Fig. 11c and Fig. 11d, respectively. For both cases the stress fluctuations are significant, so interaction energy will not Figure 11: (a) Stress plot for load case tension-11 (b) Stress plot for load case shear-1 (c) Stress plot for load case tension- (d) Stress plot for load case shear-3 41

58 be negligible. Stress distributions for longitudinal shear, transverse shear, and transverse tensionare quantitatively shown in Figure 1. The high probability densities at low strain values result fro fiber strains, thus the fluctuations in fiber strain are sall. The atrix strain distribution consists of the larger strain values. The distinction is shown in Fig.1 by enclosing fiber and atrix distribution in dashed lines. The atrix strain distribution is widest in longitudinal shear and sallest in transverse tension. Consequently, the interaction energy should be the least for transverse tension and the greatest for longitudinal shear in agreeent with Fig

59 Figure 1: Probability distribution function for strains for different load cases 43

60 An iportant observation to be ade fro these data is that the atrix strain distribution, and consequently the atrix strain fluctuation, is uch broader and at uch higher strains than the fiber strain distribution. This suggests that the sall strain values of the fiber in conjunction with the even saller strain fluctuations in fiber render the fiber a inor contributor to the interaction energy in the coposite. To confir this, the relative contribution of each constituent was exained. Figure 13 shows the relative atrix contribution to interaction energy plotted against fiber volue fraction for the four different load cases. The results are notable: up to typical fiber Figure 13: Variation of contribution of atrix to interaction energy with fiber volue fraction 44

61 volue fractions, the atrix accounts for vast ajority of interaction energy. For a fiber volue fraction of 0.6, the atrix contributes alost entirety (ore than 99%) of the interaction energy of the coposite for longitudinal shear loading, the ost significant load case. Even for transverse tension and transverse shear, atrix contribution to interaction is about 90%. This result is iportant because it perits the interaction energy contribution of the fiber to be neglected, such that augenting atrix failure theories with interaction energies can be the focus of future failure prediction efforts. Figures 14- show the contribution of the various ters in Eq. (3.0) to interaction energy of the atrix constituent. It can be seen that the contribution of ters 11 ~ 1 C, ~ 1 Figure 14: Variation of contribution of interaction energy of the atrix C ter to 11 45

62 Figure 15: Variation of contribution of C ~ ter to interaction energy of the atrix ~ Figure 16: Variation of contribution of C33 3 ter to interaction energy of the atrix 46

63 Figure 17: Variation of contribution of C1 ~1 ~ ter to interaction energy of the atrix Figure 18: Variation of contribution of C13 ~1 ~3 ter to interaction energy of the atrix 47

64 ~ ~ Figure 19: Variation of contribution of C3 3 ter to interaction energy of the atrix Figure 0: Variation of contribution of C44 ~4 ter to interaction energy of the atrix 48

65 Figure 1: Variation of contribution of C55 ~5 ter to interaction energy of the atrix Figure : Variation of contribution of C66 ~6 ter to interaction energy of the atrix 49

66 C1 ~1 ~ and C13 ~1 ~3 is always zero irrespective of fiber volue fraction and load case. The contribution of ters C ~ and C33 ~3 to interaction energy of the atrix is always zero irrespective of the fiber volue fraction for shear loading in the longitudinal directions Effect of Relative Matrix Modulus on Interaction Energy In order to investigate the variation of the interaction energy with atrix odulus, the atrix odulus was varied fro 1% to 10% of the fiber-direction fiber odulus. Results were obtained for four fiber volue fractions: 0.05, 0.5, 0.60, and Figure 3 shows the variation of relative interaction energy with atrix odulus for the four unique load cases. Fig. 3a shows the variation of interaction energy with atrix odulus in longitudinal tension. For this load case, the interaction energy for all volue fractions is negligible. For the other three load cases, shown in Fig. 3b-d, a general trend is observed: an initial increase in atrix odulus results in a decrease in interaction energy until a iniu is reached, after which additional increase in atrix odulus results in an increase in interaction energy. The iniu of the interaction energy occurs when the relevant fiber and atrix stiffnesses are the closest. For the load case in the transverse direction, shown in Fig. 3b, the interaction energy reaches a iniu when the atrix odulus is 5% of the fiber-direction fiber odulus, corresponding to a atrix odulus of GPa, which is close to the fiber odulus in the transverse direction of 50

67 Figure 3: (a) Variation of interaction energy with atrix odulus in load case tension-11 (b) Variation of interaction energy with atrix odulus in load case tension- (c) Variation of interaction energy with atrix odulus in load case shear-1 (d) Variation of interaction energy with atrix odulus in load case shear-3 51

68 14 GPa. For a load case of shear-1, shown in Fig. 3c, the interaction energy reaches a iniu when the atrix odulus is 30% of the fiber-direction fiber odulus, corresponding to a atrix shear odulus of 6.31 GPa, which is close to the fiber shear odulus in 1-direction of 8 GPa. For a load case of shear-3, shown in Fig. 3d, the interaction energy reaches a iniu when the atrix odulus is 5% of the fiber-direction fiber odulus, corresponding to a atrix shear odulus of 4.38 GPa, which is close to the fiber shear odulus in 3-direction of 5.6 GPa. These results indicate that for near iso-stress loadings the interaction energy increases with increasing difference in the fiber and atrix odulus in the loading directions Effect of Cobined Loading on Interaction Energy In actual application, a coposite is rarely subjected to uniaxial stresses and so it is iportant to study the behavior of interaction energy under ulti-axial load states. As discussed in sections and 3.4., the interaction energy for tensile loading in the longitudinal is negligible. Thus it was not included in the study of cobined loads. The fiber volue fraction and the atrix odulus were held constant at 0.6 and 4.0 GPa, respectively. The biaxial load is represented by the radius of a circle as shown in Fig 9, where is the angle ade by the radius of the circle with the x-axis which is varied fro 0 to 180. Figure 4 shows the variation of interaction energy under five types of biaxial loadings:, ),, ),, ),, ), and, ). Under a biaxial longitudinal shear ( 33 ( 1 ( 1 3 ( 1 13 ( 3 loading 1 13 the interaction energy does not change, reaining constant at 7% of the total strain energy. This gives us an iportant etric for evaluating interaction energy under 5

69 cobined longitudinal shear loading. For a particular fiber-atrix cobination and fiber volue fraction, interaction energy for biaxial shear loading is always the sae as that for pure Figure 4: Variation of interaction energy with biaxial loads longitudinal shear loading. Varying the ratio of 1 to 13 essentially rotates the stress distributions in the RVE, but does not substantially change their relative quantitative values. When the RVE was subjected to a transverse noral, ) biaxial loading the interaction ( 33 energy reached a iniu at 45, corresponding to equal biaxial transverse loading. At this angle a nearly unifor distribution of strains and stresses in the constituents is observed, siilar 53

70 to Fig. 11a, producing negligible interaction energy. In the reaining three cases,, ), ( 1 3, ), and, ), the axiu interaction energy occurs at 90, which corresponds to ( 1 ( 3 a pure shear loading. This result is rearkable: any deviation fro a pure shear loading reduces the interaction energy that would be coputed. Thus, the coposite shear loading in particular causes interaction energies to becoe significant, such that the interaction energy could be bounded by evaluating only a sall nuber of load states. 3.5 Suary and Conclusions Accurate failure prediction is critical in efforts to axiize the proising advantages offered by coposite aterials. This requires physics-based failure prediction odels, which necessarily require accurate understanding of constituent stress and strain. The use of volue average constituent stresses and strains to predict failure is a coputationally efficient first step toward this effort, but does not account for the entire strain energy in the coposite. In this study, this issing energy, tered the interaction energy, was introduced and expressions to define it were derived. A series of paraetric finite eleent studies were conducted to quantify the relative agnitude of the interaction energy for varying fiber volue fractions, atrix odulus, and loading conditions. Our results showed that for typical carbon-epoxy coposites used in the aerospace industry the interaction energy ay be as high as 30% of the total strain energy in the coposite under shear loading, which is the load state yielding the highest relative interaction energy. Furtherore, for such systes the atrix constituent is the ajor contributor to interaction energy. This result is iportant for future efforts to enhance coposite failure prediction because it suggests that focus be placed on atrix failure criteria augentation. 54

71 4. Bencharking of laina failure tests fro WWFE-I and WWFE-II with a three paraeter icroechanics based atrix failure theory This chapter presents an existing three paraeter icroechanics based atrix failure theory and its bencharking with laina test data fro well-known coposite failure bencharks World Wide Failure Exercise-I (WWFE-I) and World Wide Failure Exercise-II (WWFE-II) (Hinton and Soden, 1998; Hinton and Kaddour, 01). The WWFEs contain a detailed assessent of various leading theoretical approaches for predicting deforation and failure responses of polyer coposite lainates under coplex states of stress. The theoretical predictions were copared with one another and with experiental tests data. This chapter contains a detailed discussion of the Fertig failure theory which is followed by a brief discussion of the results and bencharking against the available experiental test data fro the WWFEs. A von Mises-axiu principal stress based atrix failure theory is also presented and is applied to one of the laina failure test cases of the first World Wide Failure Exercise. 4.1 Introduction For alost fifty years, accurate failure prediction of coposites has been an overarching goal for both the scientists and engineers alike. This large span of tie saw an abundance of failure odeling techniques introduced, which usually satisfy a particular set of data but fail to explain the generalized failure behavior of coposites. This can be attributed to the coplicated behavior of coposites owing to their heterogeneous aterial properties and the large nuber of unknowns encountered during failure prediction. Then, there is the question of the scale at which to characterize failure. It is an ongoing debate whether constituent level failure prediction is 55

72 better than laina or lainate level failure prediction. Finally, failure of coposites is very subjective and end product oriented. What ight constitute failure for an aerospace grade coposite aterial ight be totally different to failure of coposite aterial used in an autoobile. Considering the difficulty of the topic, it is not surprising that no theory is universally recognized as being a unified, verified and a coplete failure ethodology. 4. Failure odeling 4..1 Constituent level failure odeling Coposite structures are built fro coposite lainates, which are fabricated by stacking several coposite lainae together. The behavior of each laina is governed by its constituents, i.e. the properties of fibers, the surrounding atrix, the volue fraction of fibers, and fiber orphology. Most coposite failure theories operate at the laina scale by predicting laina level failure using laina level hoogenized stresses and strains. Daage and failure at the laina level is then used to predict lainate level failure. A ajor disadvantage of using laina level quantities is that it does not allow the use of constituent level physics to explain the failure behavior of coposites. Laina level stress/strains are poor at describing the actual stress/strain state in the constituents. In reality, coposite failure is a result of failure of one of its constituents, either the atrix or fiber. Thus it is iportant to characterize failure at the constituent level as opposed to the laina level. This kind of failure ethodology is called icroechanical failure odeling, which uses volue average quantities of the constituents to predict failure. For any type of coposite loading, constituent level, volue average stresses are better at describing the stress/strain state in the constituents. They enable developent of 56

73 physics-based odels which can further expand our understanding of behavior of coposites. It is due to these advantages that the failure techniques studied and developed in this work eploy constituent level stresses in contrast to laina level stresses. These constituent level stresses were extracted fro a representative volue eleent (RVE) of an idealized icrostructure of a hexagonally packed fiber reinforced coposite, shown in Fig. 5. The RVE has periodic boundary conditions enforced on all its sides and was subjected to 6 types of load 11,, 33, 1, 13, 3, which generated stresses in the fiber and the atrix regions. After Figure 5: The RVE with hexagonal fiber packing extracting stresses fro the fiber and the atrix regions, volue average constituent stresses were coputed. This was accoplished by coputing the apping between the coposite and constituent stresses, which can be used to copute constituent level stresses for any type of coposite load state. 57

74 This apping can be coputed as shown below f f i i L i and c L i c L L (4.1) where the subscript L denotes the load case, i ( i = 11,,,3) denotes the six coponents of the stress vector, f denotes the fiber, denotes the atrix and c denotes the coposite. For exaple the apping functions for atrix constituent under a pure 11 coposite load state are 1 i (4.) i c 1 Thus for any coposite load state, the stress i in a constituent a is given by a i 6 j1 j a i c j (4.3) 4... The Fertig Failure theory The Fertig failure theory (Fertig III, 01) is a icroechanical atrix failure theory that requires three paraeters, which all have physical eaning, and utilizes volue-average constituent level stresses to predict failure of a constituent (atrix or fiber here) and thereby of the coposite. One particularly attractive feature of this theory lies in its siple calibration to obtain three paraeters, which can then be used to predict failure load under any coposite state of stress. According to this theory, atrix failure occurs when 58

75 B t I t B 1 s1i s1 BsI s I h (4.4) where B,, t Bs 1 Bs represent the coefficients of atrix average stress invariants; I t, I s1, I s, I h represent the invariants of atrix stress tensor, 0 represents the shear strength of the atrix, and represents pressure strengthening due to copressive loading (~ 0.35) (Hoppel et al., 1995a). The values of B i are deterined fro three coposite static failure tests: transverse tension, transverse copression, and longitudinal shear, all of which involve failure of the atrix constituent. The invariants are coputed fro the volue average atrix stresses as follows I t (4.5) I s (4.6) I s (4.7) 4 I h 33 (4.8) where It corresponds to the axiu tensile stress noral to the fiber, Is1 is related to the shear in the longitudinal direction, Is is related to the transverse shear, and Ih represents the pressure on the axiu transverse shear plane. 59

76 In order to predict the failure envelopes for laina failure test cases of the World Wide Failure Exercises, the Bi ters were calibrated for each case by using available laina strengths which correspond to failure loads. The RVE was first subjected to a transverse failure load and the resultant atrix stresses were used to copute atrix stress invariants using Eqs. (4.5)- (4.8) which were then substituted in Eq. (4.4) to obtain three values of the three unknown co-efficients B t, Bs Bs 1,. Siilarly, the RVE was subjected to transverse copressive failure load and longitudinal shear failure load to obtain three ore values for each load case. The three equations with three unknowns were then expressed as V V V V V V 1 3 V V V Bt Bs Bs (4.9) where V ij are the values obtained by subjecting the RVE to failure loads. The three coefficients for each test case were coputed by inverting the Vij atrix as shown below Bt Bs Bs 1 V V V V V V 1 3 V V V (4.10) The fiber failure criterion uses axiu stress failure criterion given by S f 11 f 11 1 f or 11 1 f S 11 (4.11) where f S 11 is the longitudinal tensile strength of the fiber and f S 11 is the longitudinal copressive strength of the fiber. It ust be noted that the longitudinal strength of the fiber is used here to 60

77 express the in situ failure of fibers due various failure echanis like fiber fracture, fiber pullout, fiber buckling etc Matrix failure theory based on von Mises-axiu principal stress The second atrix failure theory used in this chapter is based on von Mises-axiu principal stress. The failure theory predicts atrix failure when the von Mises stress or the axiu principal stress in the atrix exceeds a critical value. This approach first requires the calculation of atrix stress concentration factor in an ideal icrostructure at the point of atrix failure Figure 6: Fluctuations in axiu principal stress in the atrix under transverse tension. Figure 6 shows the fluctuations in the axiu principal stress in the atrix constituent when an RVE was subjected to transverse failure load. The atrix stress 61

78 concentration factor ( ) can be coputed using the axiu principal stress in the atrix constituent and the noinal (or volue average) stress in the atrix as follows (4.1) ax no where is the atrix stress concentration factor and and are the axiu principal stress and noinal atrix stress respectively that ust be extracted fro the RVE. Since uses peak stress in the atrix it is strongly dependent on the esh size of the RVE. A coarse esh in the RVE ay not capture the peak stress in the atrix constituent accurately and lead to incorrect coputation of and this error in coputation of ay lead to incorrect prediction failure loads. A esh convergence study needs to be done to ake sure that the esh size of the RVE is refined enough to capture the correct axiu principal stress in the atrix constituent. ax no The atrix failure criterion is taken to be ax principal St 1 VM or 1 S VM (4.13) where is the axiu principal atrix stress, is the von Mises stress in the ax principal VM atrix, S t is the transverse tensile strength of the atrix, S VM is the von Mises strength of the atrix, and is the atrix stress concentration factor. The fiber failure criterion reains unchanged. In order to predict the failure envelopes for laina failure test cases of the World Wide Failure Exercises, three quantities were required to be pre-coputed for each of the test cases. S was coputed by using Eq. (4.3) and the transverse strength of the laina. was t 6

79 coputed by using Eq. (4.1). The axiu and the noinal stresses used in Eq. (4.1) were extracted by subjecting the RVEs to transverse failure loads. Finally, the lainae were subjected to longitudinal shear failure loads (or transverse copressive failure loads) and the atrix stresses were coputed using Eq. (4.3). These stresses were then used to copute principal stresses in the atrix constituent and the von Mises strength of the atrix constituent then obtained as follows SVM was S VM I I II III I III (4.14) where I, II and III are the principal stresses of the atrix constituent Results The focus of this chapter is to use the Fertig failure theory to predict failure of coposites and benchark it against laina failure test data fro well-known coposite failure bencharks World Wide Failure Exercise-I (WWFE-I) and World Wide Failure Exercise-II (WWFE-II). The failure exercises contain in total twenty-six carefully selected test cases which include strength envelopes and stress-strain curves for a range of unidirectional and ultidirectional lainates. The Fertig failure theory is bencharked against seven strength envelopes for unidirectional lainae. These are ideal for evaluating failure criteria, whereas lainate level tests are appropriate for evaluating the cobination of failure criteria with progressive daage ethodology. The details of the test cases are included in Table. 63

80 Table : Laina failure tests fro WWFE-I and WWFE-II Test Laina Material Loading case layup 1 0 E-glass/LY556 epoxy vs. 1 0 T300/BSL914C carbon/epoxy 1 vs E-glass/MY750 epoxy vs T300/PR319 1 vs. ( 1 3 ) 5 90 E-glass/MY750 epoxy vs. 3 ( 1 3 ) 6 0 S-glass/epoxy 1 vs. 3 ( 3 ) 7 0 Carbon/epoxy 1 vs. 3 ( 3 ) 64

81 Failure predictions of the Fertig failure theory The bencharking of the Fertig failure theory and its coparison with laina test data and leading failure theories fro the WWFEs is shown below. Test case 1: GRP laina under cobined transverse and shear loading The failure envelopes predicted by the Fertig failure theory for test case 1 are shown in Fig. 7. Figure 7(a) shows the failure envelope when the UD values provided by the WWFE-I authors were used as odel inputs. Figure 7(b) shows the failure envelope obtained when different UD values were used as inputs to the odel which yielded slightly better results. In both the cases (a) and (b), the theory fits the shape of the test data very well especially in the (,1) quadrant.in case (a), the Fertig failure theory is conservative, especially in the (,1) quadrant. By choosing a different transverse copressive strength than the one given by the originators of the exercise, the theory predicts a failure envelope which is a little less conservative. The test data shows an increase in shear strength with an increase in the transverse copressive load. This feature of the experiental data is not captured by the theory. It is known that aterial inhoogeneity in a coposite gives rise to stress and strain fluctuations in the constituents. We have already shown that the volue average atrix stresses do not capture these stress/strain fluctuations in the constituents of the coposite aterial and thus all the strain energy of a constituent is not accounted for. The bulk of this issing energy (about 30%) is due to fluctuations in the atrix constituent when the coposite is a under shear state of stress. Because our failure theory uses volue average constituent level stresses to predict failure of the 65

82 constituents of the coposite aterial, the atrix failure in this high shear region is not captured well since the issing strain energy in the atrix constituent is ignored. The atrix failure theory needs to be augented with this issing energy to iprove the predictions for this test case. Figure 7: (a) Biaxial failure envelopes for 0 laina ade of GRP (E-glass/LY556) aterial with UD values provided as odel inputs (b) Biaxial failure envelopes for 0 laina ade of GRP (E-glass/LY556) aterial with different UD values used as odel inputs 66

83 Test case : CFRP laina under cobined longitudinal and shear loading The failure envelope predicted by the Fertig failure theory for test case is shown in Fig.8. It can be seen that the theory captures the general shape of the experients very well except in the high-shear region. As in case of test case 1, a atrix failure theory augented with the issing energy ay iprove the predictions for this test case. Figure 8: Biaxial failure envelopes for 0 laina ade of CFRP aterial (T300/BSL914C). 67

84 Test case 3: GRP laina under cobined transverse and longitudinal loading The failure prediction of the Fertig failure theory is shown in Fig. 9. It can be seen that like ost of leading theories, the Fertig failure theory is very non conservative in the ( 1, ) quadrant. The discrepancy between the observed and predicted odes of failure is due to the inability of theory to incorporate atrix stresses ( 11 ) in the fiber direction. Figure 9: Biaxial failure envelopes for 0 laina ade of GRP aterial (E-glass/MY750). 68

85 The failure envelopes predicted by the von Mises-axiu principal stress failure theory for test case 3 are shown in Fig. 30. It can be seen that the theory captures the shape of the data well, especially in the ( 1, ) quadrant upto a point after which it is very conservative. Figure 30: Biaxial failure envelopes for 0 laina ade of GRP aterial (Eglass/MY750) predicted by Von Mises Max. principal stress theory In the absence of the stress concentration factor, the atrix fails uch before actual failure as suggested by the experiental data. This failure is captured by the theory when the appropriate stress concentration factor was used. The test data suggests that catastrophic failure of the laina 69

86 occurred just before the load reached the tensile strength of the fiber. This was because even though the atrix had failed, the fiber could hold the coposite together but now it was carrying a larger load. Von Mises axiu principal stress based failure theory was used with a progressive daage schee to capture this behavior in the fourth quadrant. After atrix cracking, the load carrying capacity of the atrix reduces significantly. To calculate the resultant fiber stresses after atrix failure, the atrix properties were degraded as follows Enew 15%Eoriginal (4.15) new 0.01% original (4.16) where Enew and Eoriginal are the new and original Elastic oduli of the atrix respectively and new and original are the new and original poisons ratio of the atrix respectively. The new resultant fiber stresses were coputed fro the RVE with hexagonal fiber packing using the procedure discussed in section 4.. The original fiber failure criterion and new fiber stresses were then used to copute failure load which represents catastrophic coposite failure. The failure envelope predicted is shown in Fig. 31 along with the failure loads predicted previously without using any progressive daage schee. It can be seen that after a particular load, the atrix fails uch before what is seen in the experients and all the load is now transferred to the fibers, the failure of which results in catastrophic coposite failure. 70

87 Figure 31: Biaxial failure envelopes for 0 laina ade of GRP aterial (Eglass/MY750) predicted by Von Mises Max. principal stress theory with progressive atrix daage 71

88 Test case 4: CFRP laina under cobined hydrostatic and shear loading The failure envelopes predicted by the Fertig failure theory for test case 4 are shown in Fig. 3. Figure 3(a) shows the failure envelope when the UD values provided by the authors were used as odel inputs and =0.1 and Fig. 3(b) shows the failure envelope when different UD values were used as inputs to the odel and =0.35. In both the cases (a) and (b), the theory fits the shape of the test data very well. When the shear strength of the 0 tubes is used as one of the odel inputs, the theory captures the test data of the 0 very well. When the UD values provided by the Figure 3: (a) Triaxial failure envelopes for 0 laina ade of CFRP aterial (T300/PR319) with β=0.01 (b) Triaxial failure envelopes for 0 laina ade of CFRP aterial (T300/PR319) with β=0.35 7

89 originators of the exercise are used, the pressure strengthening ter has to be reduced to 0.1 fro 0.35 to capture the failure of 90 tubes. Test case 5: GRP laina with cobined transverse and through thickness loading The failure envelope predicted by the Fertig failure theory for test case 5 is shown in Fig. 33. It can be seen that the predictions of the theory captures failure very well. Figure 33: Triaxial failure envelopes for 0 laina ade of GRP aterial (E-glass/MY750) 73

90 Test case 6: GRP laina with cobined through thickness and longitudinal loading The failure envelope predicted by the Fertig failure theory for test case 6 is shown in Fig. 34. There is a large scatter of data in the ( 1, / 3) quadrant in which the theory does a good job predicting failure. In the ( 1, / 3 ) failure sees to be occurring due to fiber kinking which was concluded by Pinho (Pinho et al., 013) - For atrix failure under considerable superposed hydrostatic pressure, the stress state in the specien after atrix failure can in principle lead to subsequent fibre kinking. Since the fiber failure criterion does not account for fiber kinking, the Figure 34: Triaxial failure envelopes for 0 laina ade of GRP aterial (S-glass/Epoxy) 74

91 theory gives ore conservative results. It can be seen that two theories Pinho (Pinho et al., 01) and Carrerre (Carrere et al., 01) capture fiber kinking very well. Test case 7: CFRP laina with cobined through thickness and longitudinal loading A coparison of failure theories for test case 7 is shown in Fig. 35. The Fertig failure theory captures the overall shape of the data. Siilar to case 6, in the ( 1, / 3 ) copressive failure was not captured by the theory. Since the fiber failure criterion does not account for fiber Figure 35: Triaxial failure envelopes for 0 laina ade of CFRP aterial (Carbon/Epoxy) 75

92 kinking, the predictions of the theory are again conservative. It can be seen again, that the Pinho failure theory captures fiber kinking very well Conclusions An existing three paraeter icroechanics based atrix failure theory was presented and its failure load predictions were copared with experiental data fro the WWFE exercises. It was concluded for ost of the test cases, the predictions of the Fertig failure theory were in close agreeent with the test data. The Fertig failure theory over-predicted failure loads for GRP laina under cobined noral and longitudinal copressive loading (test case 3) where the failure was caused by atrix cracking perpendicular to the fibers. The Fertig failure theory could not predict these cracks in the atrix due to the inability of the theory to incorporate the contribution of the longitudinal stress in the atrix constituent that had caused the cracking. The theory was conservative in predicting the fiber failures under copressive loads in test case 6 and 7 since a siple axiu stress failure criterion was used to predict the strength of fibers under copression. The Fertig failure theory needs to incorporate the effect of fiber buckling to iprove failure prediction of the fiber constituent under copressive loads. In general, the predictions of the theory were also siilar to the leading failure theories fro the WWFE exercises. However, the leading theories fro WWFE-I and WWFE-II require substantially ore input paraeters, which akes calibration very difficult. The advantage of the Fertig failure theory lies in its siplistic calibration, which requires just three paraeters that can be obtained fro three standard coposite failure tests (transverse tension, transverse copression and longitudinal shear). The theory uses volue average atrix stresses to predict 11 76

93 failure loads. It has been shown that these volue average constituent quantities do not capture all the energy. If the failure theory is augented with this issing energy, the failure predicting capabilities ight be iproved especially for shear loads. The second failure ethodology presented in this work is based on von Mises axiu principal stress which uses constituent stresses to predict failure loads. A failure envelope was obtained using this theory for test case 3. The atrix appeared to fail uch before catastrophic failure of the laina that was observed in the test data. After atrix cracking noral to the fiber direction, the fibers were the only load carrying agents left and the laina was then held together only by the fibers. This behavior of the laina was predicted by using von Mises axiu principal stress theory along with a progressive daage schee for test case 3 where the properties of the atrix were degraded substantially and the failure envelope obtained was better than the predictions of any of the leading failure theories. 77

94 5. Matrix failure odeling accounting for interaction energy In Chapter it was shown that volue average constituent level stresses that are used to predict strengths using a icroechanics-based failure theory do not account for all the energy of the coposite. This issing energy, tered interaction energy, is due to the inability of the volue average constituent level stresses to capture the stress/strain fluctuations produced in the constituents due to aterial inhoogeneity. Since these fluctuations are greatest in the atrix constituent and negligible in the fiber constituent, the bulk of the interaction energy contribution is due to fluctuations within the softer atrix constituent of the coposite. Thus only the atrix failure theory was augented with the interaction energy to iprove coposite failure load predictions. This chapter presents a ethodology to augent the volue average atrix level stresses with appropriate fluctuations of stress in the atrix constituent. These atrix level stresses are energy consistent and are used with the Fertig failure theory to predict failure envelopes for laina failure test cases of the first and second World Wide Failure Exercise. The results are copared with failure envelopes generated previously with the Fertig failure theory but with volue average atrix level stresses. 5.1 Augenting the volue average atrix level quantities The odelling technique presented in this chapter ais at developing a constituent-level stress easure that conserves energy by using respective stress/strain fluctuations in the atrix constituent as shown below ij ij sign ij ij 78 (5.1)

95 +sign (5.) ij ij ij ij where ij and ij are volue average atrix level stresses and strains respectively, ~ ij and ij are stress and strain fluctuations in the atrix respectively, ij and ij are energy consistent atrix level stresses and strains respectively and is fluctuation energy constant that iniizes the average error in the total energy. The fiber level stresses are not augented since the stress/strain fluctuations in the fiber constituent are negligible. A RVE of hexagonal fiber packing ust be subjected to unit biaxial/tri-axial loading according to the load configuration being analyzed. The volue average fiber and atrix level quantities and volue average stress/strain fluctuation squared quantities of the atrix constituent are extracted fro this Representative Volue Eleent (RVE). Average stress and strain fluctuations cannot be directly coputed by conventional averaging technique as that would yield zero. To obtain consistent quantities for augentation, volue average squared quantities are extracted and their square roots are added to the respective volue average atrix quantities. Direct use of the energy consistent stresses gives rise to excess strain energy and so the fluctuation energy constant needs to be incorporated such that the new quantities do not over predict the strain energy. A detailed procedure for approxiating the fluctuations is presented in the following text for Test case-1 of the first World Wide Failure Exercise. The fluctuations of the atrix constituent for all the other test cases were obtained siilarly. 79

96 5.1.1 Approxiating the volue average stress fluctuations of the atrix constituent for a GRP laina under cobined noral and shear loading (Test case-1 of WWFE-I) An RVE with hexagonal fiber packing was subjected to cobined noral and shear loading. For extracting fluctuations and calibration, a unit biaxial load was applied such that it corresponded to the x- and y-coponents of the biaxial load represented as the radius of a circle as shown in Fig. 36 and is varied fro 0 to 360. The two uniaxial loads required to generate the noral and shear loading can be expressed as I cos (5.3) II sin (5.4) where is the resultant bi-axial load and, I and II are the corresponding uniaxial loads. Volue average quantities (stresses and strains) of the fiber and atrix constituents and volue average fluctuation squared quantities (stresses and strains) of the atrix constituent were extracted fro this RVE. Figure 37 shows plots of the stress fluctuations, the volue average stresses and the energy consistent stresses of the atrix constituent. 80 Figure 36: Decoposition of biaxial load into its coponents

97 Figure 37: Coparison of volue average stresses and volue average stress fluctuations and the energy conserving stresses of the atrix constituent for GRP laina under cobined transverse noral and shear loading 81

98 The stress fluctuations observed in the atrix constituent can be categorized into two types: TYPE-I and TYPE-II. TYPE-I fluctuations can be deterined fro volue average stresses of the atrix constituent directly using the expression ~ij (no suation over repeated indices) where Fij ij ~ij (5.5) is volue average stress fluctuation of the atrix constituent, Fij is a constant and ij is the volue average stress of the atrix constituent. The constant Fij was deterined using the volue average stress fluctuations and the volue average stresses of the atrix constituent that were extracted fro the RVE under unit biaxial load. In Fig. 37, volue average stress fluctuations in plots (a), (b) and (c) are TYPE-I fluctuations. TYPE-II fluctuations cannot be directly deterined fro the corresponding volue average atrix stresses and because the volue average stress is zero. These fluctuations are coputed using either of the expressions where ~ij ~ij L Aij cos Bij (5.6) ~ij L Aij sin Bij (5.7) is the volue average stress fluctuation of the atrix constituent, L is agnitude of the biaxial/tri-axial load, is the angle at which load is applied and Aij, B ij are constants. Again, Aij and Bij were calibrated using the volue average stress fluctuations of the atrix constituent of the RVE under unit biaxial load. For the load case under consideration, 8

99 volue average stress fluctuations in plots (d) and (e) are TYPE-II fluctuations. Table 3 shows the types of fluctuations that were observed in the various stress coponents when an RVE was subjected to biaxial/tri-axial states of stress that corresponded to the loading configurations of the laina failure tests of the World Wide Failure Exercises. The fluctuation in the longitudinal noral stress coponent of the atrix constituent was not analyzed because it was concluded in Chapter 3 that these fluctuations do not give rise to any interaction energy. Table 3: Type of fluctuations observed due to different loading configurations in the WWFEs 1 E-glass/LY556 epoxy vs. 1 Stress coponents TYPE-I fluctuations, 33, 1 1 vs. 1 1, 33, 13 *, 3 3 T300/BSL914C carbon/epoxy E-glass/MY750 epoxy 1 vs., 1, 13, 3 4 T300/PR319 1 vs., 33, 1 Test case Material Loading ( 1 3 ) Stress coponents TYPE-II fluctuations 13 *, 3 13 *, 3 5 E-glass/MY750 epoxy vs. 3 ( 1 3 ) -, 33, 1, 13, 3 6 S-glass/epoxy 1 vs. 3 ( 3 ), 33 1, 13, 3 7 Carbon/epoxy 1 vs. 3 ( 3 ), 33 1, 13, 3 *Although the fluctuations in the 13 coponent of the atrix stress is of TYPE-II, it was approxiated as TYPE-I fluctuation using fluctuations in the 1 coponent of the atrix constituent. 83

100 A coparison of the constants Fij and Aij, B ij for various laina failure test cases of the World Wide Failure Exercises is shown in Table 4, Table 5 and Table 6 respectively. Table 4: Coparison of values of Fij for various test cases of the WWFEs Constants F F33 F1 F Test case Table 5: Coparison of values of Aij for various test cases of the WWFEs Constants 1 3 Test case A A A A33 A1 84

101 Table 6: Coparison of values of Bij for various test cases of the WWFEs Constants 1 3 Test case B B B B33 B The value fro Tables 4, 5 and 6 suggest that all the constants Fij, Aij and Bij are strongly dependent on the type of loading and the aterial properties of the coposite aterial Coputing the fluctuation energy constant Strain energy of the coposite was coputed using volue average quantities of the fiber constituent and energy consistent quantities of the atrix constituent. In all the load cases it was seen that direct augentation of the atrix level quantities resulted in an over prediction of the strain energy of the coposite. The fluctuation energy constant was introduced not only to prevent this over prediction of energy of the coposite but also to iniize the average error in the interaction energy. The interaction energy was coputed using the volue average coposite quantities, volue average fiber level quantities and energy consistent quantities of the atrix constituent which can be shown as 85

102 U 1 C C 1 1 ij ij VC ijf ijf V f ij sign ij ~ij ij sign ij ~ij V (5.8) An iterative process was used to copute such that the ean error in the interaction energy was zero. The different values of that were coputed for the various laina failure test cases are provided in Table 7. Table 7: Values of Ψ for different test cases of the WWFEs Test case Material E-glass/LY556 epoxy T300/BSL914C carbon/epoxy E-glass/MY750 epoxy T300/PR319 1 Loading vs. 1 1 vs. 1 1 vs. vs. ( 1 3 ) E-glass/MY750 epoxy vs. 3 ( 1 3 ) S-glass/epoxy 1 vs. 3 ( 3 ) Carbon/epoxy 1 vs. 3 ( 3 ) It can be seen fro table 7 that is a function of loading type, aterial properties of the coposite, and the type of fiber packing in the RVE. Table 8 shows the values of that were obtained for coposites with different aterial properties and sae fiber volue fraction subjected to cobined noral and longitudinal shear loading. 86

103 Table 8: Values of Ψ for different aterials subjected to sae loading configuration Material Ψ E-glass/LY556 epoxy T300/BSL914C carbon/epoxy E-glass/MY750 epoxy T300/PR319 E-glass/MY750 epoxy S-glass/epoxy Carbon/epoxy A coparison of interaction energy coputed using volue average atrix stresses and energy consistent atrix level stresses fro Eqs. (5.1) and (5.) is shown in Figures Figure 38: Interaction energy for test case-1 87

104 Figure 39: Interaction energy for test case- Figure 40: Interaction energy for test case-3 88

105 Figure 41: Interaction energy for test case-4 Figure 4: Interaction energy for test case-5 89

106 Figure 43: Interaction energy for test case-6 Figure 44: Interaction energy for test case-7 90

107 5. Modified Fertig failure theory After calibration, Eqs. (5.5), (5.6) and (5.7) can be used to copute stress fluctuations of the atrix constituent for any type and agnitude of loading, aterial properties and type of distribution of fibers in the atrix. Using these stress fluctuations, augented stresses of the atrix constituent can be deterined using Eqs. (5.1) and (5.). The odified Fertig failure theory uses energy consistent constituent stresses of the atrix constituent to predict atrix level failure. For the sake of clarity and continuity the odified approach is outlined below. Failure in the atrix constituent occurs when B t I t B 1 s1i s1 BsI s I h (5.9) where B,, t Bs 1 Bs represent the coefficients of the stress invariants of the atrix constituent; I, t, I s1, I s I h represent the invariants of atrix stress tensor, 0 represents the shear strength of the atrix, and represents pressure strengthening due to copressive loading, the value of which is chosen to be 0.35 (Hoppel et al., 1995b). The values of B i are deterined fro three coposite static failure tests: transverse tension, transverse copression, and in-plane shear, all of which involve failure of the atrix constituent. The invariants are coputed fro the augented volue average atrix stresses as follows 91

108 I t (5.10) I s (5.11) I s (5.1) 4 I h 33 (5.13) where ij are energy consistent stresses of the atrix constituent, It corresponds to the axiu tensile stress noral to the fiber, Is1 is related to the in-plane shear, Is is related to the transverse shear, and Ih represents the pressure on the axiu transverse shear plane. The fiber failure criterion is outlined below S f 11 f 11 1 f or 11 1 f S 11 (5.1) f where is the volue average longitudinal stress of the fiber constituent, 11 longitudinal tensile strength of the fiber and f S 11 f S 11 is the is the longitudinal copressive strength of the fiber. The different values of B,, t Bs 1 Bs that were obtained by using volue average stresses of the atrix constituent and energy consistent atrix level stresses are listed in Table 9. 9

109 Table 9: Coparison of Bi paraeters obtained by using volue average and energy consistent atrix level stresses Test cases Bt N Bs1 N Bs N ij ij ij ij ij ij Table 10 contains the ratios of the paraeters obtained by using volue average atrix level stresses and energy consistent atrix level stresses. Table 10: Coparison of ratios of the paraeters obtained by using volue average atrix level stresses and energy consistent atrix level stresses. Test cases Bt N Bs1 N Bs1 N

110 5.3 Results The failure load predictions obtained for laina failure test cases of the World Wide Failure Exercises by using the Fertig failure theory with energy consistent stresses of the atrix constituent are presented in this section. The new failure plots are copared with available test data and the original predictions obtained with volue average atrix stresses. Test case 1: GRP laina under cobined noral and shear loading The failure predictions obtained by using energy consistent and volue average atrix stresses in the Fertig failure theory for a GRP laina under cobined noral and shear loadi Figure 45: Coparison of failure load predictions for GRP laina under cobined noral and shear loading 94

111 ng are shown in Fig. 45. The failure envelopes were the sae in the (, ) quadrant. 1 However, in the (, ) quadrant, the failure envelope obtained by using energy consistent 1 atrix stresses was slightly better in capturing the high shear failure loads that the volue average stresses failed to capture. Figure 46 shows the distribution of stresses in the RVE under unit noral and shear load for two loading angles: 45 (first quadrant) and 135 (second quadrant). Results show that the spread of the stress distribution in the transverse direction is greater for the RVEs loaded at an angle of 45 in the first quadrant as copared to the one loaded at an angle of 135 in the second quadrant. However, the shear stress distribution is larger for the RVE loaded at an angle of 135 in the first quadrant as copared to the one loadedat 45 in the second quadrant. Since axiu interaction energy occurs due to fluctuations in the shear stress, augented stresses are ore accurate in the second quadrant in this test case. 95

112 Figure 46: Coparison of stress distributions 96

113 Test case : CFRP laina under cobined longitudinal and shear loading The failure predictions obtained by using volue average and energy consistent atrix stresses in the Fertig failure theory for a CFRP laina under cobined longitudinal and shear loading are shown in Fig. 47. The results suggest that the energy consistent atrix level stresses produced a no iproveent in predicting failure loads for this load case. There is a large scatter of data especially in case of high shear failure loads which not captured by any failure theory of the first World Wide Failure Exercise. Figure 47: Coparison of failure load predictions for CFRP laina under cobined longitudinal and shear loading 97

114 Test case 3: GRP laina under cobined noral and longitudinal loading The failure predictions obtained by using volue average and energy consistent atrix stresses in the Fertig failure theory for a GRP laina under cobined noral and longitudinal loading are shown in Fig. 48. Like test case, the energy consistent atrix level stresses failed to produce any significant iproveent in the failure load predictions. Figure 48: Coparison of failure load predictions for GRP laina under cobined longitudinal and shear loading 98

115 Test case 4: CFRP laina under cobined hydrostatic and shear loading The failure predictions obtained by using volue average and energy consistent atrix level stresses in the Fertig failure theory for a CFRP laina under cobined hydrostatic and shear loading are shown in Fig. 49. Overall the use of energy consistent atrix level stresses only slightly changed the failure envelope. Figure 49: Coparison of failure load predictions for CFRP laina under cobined hydrostatic and shear loading 99

116 Test case 5: GRP laina with cobined noral and through thickness loading The failure predictions obtained by using volue average and energy consistent atrix stresses in the Fertig failure theory for a GRP laina under cobined noral and through thickness loading are shown in Fig. 50. The use of energy consistent stresses produced a ore conservative failure envelope as copared to that produced by volue average stresses of the atrix constituent. Figure 50: Coparison of failure load predictions for GRP laina under cobined transverse and through thickness loading 100

117 Test case 6: GRP laina with cobined through thickness and longitudinal loading The failure predictions obtained by using volue average and energy consistent atrix stresses in the Fertig failure theory for a GRP laina under cobined through thickness and longitudinal loading are shown in Fig. 51. The Fertig failure theory with energy consistent stresses for the atrix constituent predicted slightly higher atrix strengths under transverse and through thickness tension as copared to that predicted by using volue average stresses. Figure 51: Coparison of failure load predictions for GRP laina under cobined through thickness and longitudinal loading 101

118 Test case 7: CFRP laina with cobined through thickness and longitudinal loading The failure predictions obtained by using volue average and energy consistent atrix level stresses in the Fertig failure theory for a CFRP laina under cobined through thickness and longitudinal loading are shown in Fig. 5. Like test 6, the Fertig failure theory with energy consistent stresses for the atrix constituent did not lead to any iproveent in the failure load predictions for this test case. Figure 5: Coparison of failure load predictions for CFRP laina for cobined through thickness and longitudinal loading 10

119 5.4 Discussion Volue average atrix constituent quantities do not capture the stress/strain fluctuations that arise in the constituents of a coposite under different types of loading thereby giving rise to interaction energy. These fluctuations are greatest in the atrix constituent and are negligible in the fiber constituent of the coposite aterial. Consequently, bulk of the Interaction energy contribution is due to the atrix constituent. Thus only the volue average stresses and strains of the atrix constituent need to be augented with the fluctuations in stresses and strains respectively to iprove failure load predictions. In this chapter a ethod of capturing the fluctuations in the atrix constituent was presented and the volue average quantities of the atrix quantities were augented with these fluctuations. The new energy consistent stresses of the atrix constituent were then used with the Fertig failure theory to obtain failure load predictions for various laina failure tests of the first and second World Wide Failure Exercise. The results were then copared with experiental test data and with the failure envelopes that were previously obtained using volue average atrix stresses in the Fertig failure theory. The results show that for ost of the test cases the energy consistent atrix stresses did not iprove failure load predictions. In test case 5, the energy consistent atrix stresses led to prediction of a ore conservative failure envelope as copared to that which was predicted using volue average atrix stresses. Even though the iproveent in the strength predictions was by a very sall aount in a couple of test cases (test cases and 6), the atrix stresses used in failure load predictions were energy consistent and hence ore accurate than their volue average counterparts. Lastly, now that the new stress etric is energy consistent, we know that any discrepancy in the failure load predictions and the experiental test data is not due to inaccurate stresses but because of the deficiencies in the failure theory itself. Thus one proble 103

120 has been solved and now we ay direct our efforts towards other odifications to the failure theory to iprove strength predictions in unidirectional fiber reinforced coposite aterials. 104

121 6. Conclusions and Recoendations 6.1. Suary and Conclusions This study has detailed the otivation for creation of robust and reliable tools to predict failure in unidirectional fiber reinforced polyers based on icroechanical odeling using energy consistent volue average constituent level stresses. It was shown that volue average quantities (stresses/strains) of the constituents do not account for all the energy of the coposite due to their inability to capture the stress/strain fluctuations of the constituents. This study showed that the issing energy, tered interaction energy, is dependent on the fiber volue fraction, aterial properties of the coposite and the type of loading on the coposite. The interaction energy was nearly 30% for coposites with carbon fibers and epoxy atrix due to longitudinal shear loads and negligible for longitudinal tensile loads. Under transverse loads the interaction energy was about 1% and under transverse shear loads it was 15%. A three paraeter icroechanics based Fertig failure theory was used to asses failure loads for laina failure tests of the first and the second World Wide Failure Exercises. This theory used volue average constituent level stresses to predict failure of the coposite and thereby predict failure of the laina. For alost all the test cases, the predictions of the theory were in close agreeent with the test data. The theory assues that the longitudinal stresses in the atrix constituent do not contribute in any failure echanis which led to over-prediction of strengths under cobined tensile longitudinal and copressive transverse loads for glass/epoxy coposites. A axiu stress failure criterion was used to assess fiber failure due to which fiber failure due to fiber kinking echanis was not predicted in two of the test cases. Soe test cases were identified where failure loads could be iproved by augenting the atrix failure 105

122 theory with interaction energy. A von Mises axiu principal stress failure criteria was also used to obtain failure envelope for one of the test cases where the Fertig failure theory over predicted the failure loads. A progressive daage schee was used in which the properties of the atrix constituent in the failed laina was degraded significantly thereby reducing the load carrying capacity of the atrix. The resultant loads carried by fibers were calculated using icroechanics. Results with this progressive odeling schee were copared with one of the test cases where both the Fertig failure theory and the von Mises-Maxiu principal stress failure theory. The failure envelope predicted by the von Mises-axiu principal stress theory described the echanis of failure accurately and it was better than any of the leading failure theories. It was concluded that this theory with a siilar progressive daage schee would yield good results in the reaining test cases. A new ethodology was forulated to capture the stress-strain fluctuations of the atrix constituent and the volue average quantities of the atrix constituent were augented with respective fluctuations. The energy consistent stresses were used along with the Fertig failure theory to predict failure envelopes for the laina failure test cases of the first and second World Wide Failure Exercise. The results were then copared with experiental test data and the failure enveloped predicted by the sae theory using un-augented volue average stresses. The coparisons showed that the failure enveloped iproved failure load predictions slightly for soe of the test cases. Even though the iproveent of the failure load predictions was not by a significant aount, the proble of interaction energy has been solved. It is well known that volue average constituent quantities do not capture the stress/fluctuations in the constituents of the coposite aterial. Soe of these fluctuations, which ay play a role in failure echaniss, self-equilibrate and produce no corresponding volue average stress/strain in the 106

123 constituent. The ethodology that was developed in this thesis to copute energy consistent quantities (stresses and strains) provides a eans to capture these stress/strain fluctuations and thereby iprove the assessent of failure in coposite. 6.. Recoendations for future work The failure theories that were ranked highest by the authors of the World Wide Failure Exercises can be categorized into two types. First theories which were very siple in ipleentation and use and yet provided reasonably good results and second theories which were coplicated and involved a large nuber of paraeters but gave very accurate predictions. The Fertig failure theory is icroechanics based theory which requires just three paraeters that can be calibrated fro three siple failure tests on unidirectional lainae. Moreover it incorporates physics to assess failure of the constituents of the coposite. It is not as coplicated as soe of the leading failure theories and can be calibrated very easily. Coparisons showed that the predictions of the theory were in good agreeent not only with the experiental data but also with the predictions of the leading failure theories of the world wide failure exercises. One of the test cases revealed that the theory needs to consider the effects of longitudinal stress of the atrix coponent to assess failure of the atrix constituent. Currently, failure of the fiber constituent is assessed using a axiu stress failure criterion which is why the theory could not predict fiber failure due to fiber kinking in a couple of test cases. The theory assues linear elastic behavior of the constituents of the coposite aterial and it would be interesting to look at the results when non-linear atrix behavior is incorporated. Only laina failure test cases were analyzed since a progressive daage schee for predicting cascading 107

124 lainate failure was not adopted. The use of energy consistent stresses coupled with non-linear behavior of the atrix and a progressive schee will ake the Fertig failure theory a powerful approach in the field of failure load predictions of FRPs. Only unidirectional fiber reinforced coposites were analyzed in this work but the odeling technique presented in the thesis can be extended to any kind of coposite aterial. Laina failure for only a select nuber of loading configurations was considered in this study exercise and the energy consistent stresses ay yield iproveents in soe of the untested configurations. 108

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129 Appendix A: Periodic boundary conditions The naing convention as shown in Fig. X was adopted to derive the periodic boundary conditions for the Representative Volue Eleent (RVE) which is a cuboid where CN1, CN,, CN8- are node sets that contain the nodes on the eight corners (vertices) of the cuboid, ES1, ES, ES3, ES4, LS1, LS, LS3 and LS4 are node sets that contain the nodes on the edges of the cuboid. These node sets do not contain the nodes on the corners of the cuboid. FS1, FS,, FS6 are node sets that contain the nodes on the six faces of the cuboid. These node sets do not contain the nodes on the edges and the corners of the cuboid. 113

130 Figure 53: Naing convention for node sets of the Representative Volue Eleent The equation constraints for nodes on the corners are u CN6 i u CN i u CN7 i 0 u CN3 i u CN4 i u CN7 i 0 u CN5 i u CN6 i u CN7 i 0 u CN i u CN6 i u CN4 i 0 where i 1,, 3 denotes the directions and u denotes the displaceents of corner nodes in the i-direction. The equation constraints for nodes on the edges are 114