A REGULARIZED EXTENDED FINITE ELEMENT METHOD FOR MODELING THE COUPLED CRACKING AND DELAMINATION OF COMPOSITE MATERIALS

Transcription

1 A REGULARIZED EXTENDED FINITE ELEMENT METHOD FOR MODELING THE COUPLED CRACKING AND DELAMINATION OF COMPOSITE MATERIALS Dissertation Submitted to The School of Engineering of the UNIVERSITY OF DAYTON In Partial Fulfillment of the Requirements for The Degree Doctor of Philosophy in Mechanical Engineering By Michael James Swindeman UNIVERSITY OF DAYTON Dayton, Ohio December 2011

2 A REGULARIZED EXTENDED FINITE ELEMENT METHOD FOR MODELING THE COUPLED CRACKING AND DELAMINATION OF COMPOSITE MATERIALS Name: Swindeman, Michael James APPROVED BY: Robert A. Brockman, Ph.D. Advisory Committee Chairman Professor of Civil and Environmental Engineering and Engineering Mechanics Steven L. Donaldson, Ph.D. Committee Member Assistant Professor of Civil and Environmental Engineering and Engineering Mechanics Endel V. Iarve, Ph.D. Committee Member Distinguished Research Scientist University of Dayton Research Institute James M. Whitney, Ph.D. Committee Member Professor Emeritus of Civil and Environmental Engineering and Engineering Mechanics John G. Weber, Ph.D. Associate Dean School of Engineering Tony E. Saliba, Ph.D. Dean, School of Engineering & Wilke Distinguished Professor ii

3 Copyright by Michael James Swindeman All rights reserved 2011 iii

4 ABSTRACT A REGULARIZED EXTENDED FINITE ELEMENT METHOD FOR MODELING THE COUPLED CRACKING AND DELAMINATION OF COMPOSITE MATERIALS Name: Swindeman, Michael James University of Dayton Advisor: Dr. Robert A. Brockman As the use of composite materials in aerospace structures continues to increase, the need to properly characterize these materials, especially in terms of damage tolerance, takes on additional importance. The world wide failure exercises (WWFE) are an example of the international interest in this issue. But though there has been a great deal of progress in understanding the initiation of damage and modeling damage propagation along known interfaces, methods that can capture the effects of interactions among various failure modes accurately remain elusive. A method of modeling coupled matrix cracks and delamination in laminated composite materials based on the finite element method has been developed and experimentally validated. Damage initiation is determined using the LARC03 failure criterion. Delamination along ply interfaces is modeled using cohesive zones. Matrix cracks are incorporated into the discretization of the iv

5 problem domain through a robust Mesh-Independent Cracking (MIC) technique. The matrix cracking technique, termed the Regularized Extended Finite Element Method (Rx-FEM), uses regularized forms of the Heaviside and Dirac Delta generalized functions to transform the crack surface into a volumetric crack zone. The Regularized Extended Finite Element method is compared to benchmark cases. The sensitivity of the solution to mesh size and parameters within the cohesive zone model is studied. Finally, the full method with delamination is employed to study a set of experimental tests performed on open-hole quasi-isotropic laminates. The trends of hole-size and ply thickness are well predicted for the laminates. Rx-FEM is also able to simulate the pattern of damage, as demonstrated by comparisons to x-ray images. From the results of this series of analyses it can be concluded that failures occur when delamination originating at the hole links up with delamination originating at the edge along the path of matrix cracks. v

6 To Lisa, my wife vi

7 ACKNOWLEDGEMENTS This work represents a small part of a much larger effort and as such there are quite a few people that I believe deserve acknowledgement. First and foremost, I would like to acknowledge the support of NASA Langley, especially Carlos DaVilla and Cheryl Rose. The LARC failure criterion was critical to the successful outcome of the model. In addition, Dr. Rose provided the CT images of damage and gave me permission to use them in this dissertation. I am also grateful to Stephen Hallet at the University of Bristol for providing the experimental results and X-Ray images of damage which were the target of the analysis. Additionally, David Mollenhauer and Timothy Breitzman at the Air Force Research Labs (AFRL) at Wright Patterson Air Force Base have provided me much useful advice. My colleagues at the University of Dayton Research Institute and the University of Dayton deserve thanks as well, especially Endel Iarve who has overseen this work from the beginning, my advisor Bob Brockman, and the other members of my committee, Stephen Donaldson and James Whitney. I am also very grateful to Tim Fry, my group leader, and Tom Held, who provided computing support. Finally, it is those closest to me to whom I am most indebted for their ever-present support. I thank my parents for their encouragement and my children for their patience; but I am most indebted to Lisa who became the real driving force. vii

8 TABLE OF CONTENTS ABSTRACT... iv DEDICATION... vi ACKNOWLEDGEMENTS... vii LIST OF ILLUSTRATIONS... xi LIST OF TABLES... xiv LIST OF ACRONYMS, ABBREVIATIONS, AND SYMBOLS... xv I. INTRODUCTION... 1 Use of Composites in Large Structures... 1 Design Challenges of Composite Structures... 2 Composite Failure Analysis and Prediction... 4 Predictive Modeling... 8 II. A REVIEW OF THE ELEMENTS OF PROGRESSIVE DAMAGE MODELING Failure Initiation Criteria viii

9 Modeling Delamination Modeling Matrix Cracking III. A REGULARIZED, EXTENDED FINITE ELEMENT METHOD Finite Element Method The Extended Finite Element Method The Regularized Extended Finite Element Method The Cohesive Formulation Element Size Effects Application of Cohesive Zone to Mesh Independent Cracks Delaminations and their Interaction with Mesh Independent Cracks Implementation IV. VERIFICATION Double Cantilever Beam Problem Description Double Cantilever Beam LEFM Solution Double Cantilever Beam Results Double Cantilever Beam Discussion Multiple Crack DCB ix

10 V. APPLICATION OF THE METHOD TO OPEN HOLE LAMINATES Problem Statement and Boundary Conditions Comparison with Experimental Failure Loads and Damage Patterns SUMMARY CONCLUSIONS AND RECOMMENDATIONS REFERENCES APPENDICES I. AN EXAMPLE OF THE REGULARIZED EXTENDED FINITE ELEMENT METHOD FOR A ONE-DIMENSIONAL PROBLEM II. IMPLEMENTATION OF RX-FEM IN ABAQUS x

11 LIST OF ILLUSTRATIONS Figure 1. Pedigree of the LaRC Failure Criteria... 6 Figure 2. Matrix Cracks Around an Open Hole From X-Ray Images... 7 Figure 3. A General Traction-Separation Relationship Figure 4. Subdivision of the Domain by the Signed Distance Function Figure 5. Adaptation of Integration Scheme for x-fem Figure 6. Regularized Step Function for a One-Dimensional Problem Figure 7. Gradient of the Regularized Step Function for a One-Dimensional Problem Figure 8. Traction-Separation and Damage-Separation Relationships Figure 9. Introduction of a MIC Step 1, Determining Zones Figure 10. Introduction of a MIC Step 2, Determining Cracked Element Figure 11. Introduction of a MIC Step 3, Determining Adjacent Elements Figure 12. DCB Specimen Figure 13. Mesh Near Crack Tip for DCB Model xi

12 Figure 14. DCB Stiffness as a Function of Crack Length for LEFM Model Figure 15. Mesh Size Dependence of Rx-FEM DCB Model Compared to LEFM Solution Figure 16. Effects of Mesh Refinement on Cohesive Zone at a Displacement of 4.6 mm Figure 17. Dependence of DCB Response on the Maximum Cohesive Traction Figure 18. Effects of Interfacial Strength on Cohesive Zone at a Displacement of 4.6 mm Figure 19. Dependence of DCB Response on Penalty Parameter Figure 20. Effects of Penalty Stiffness Coefficient on Cohesive Zone at a Displacement of 4.6 mm Figure 21. The Two-Crack Problem Schematics (a) and the Mesh Around the Crack Tip (b) Figure 22. Two Crack Model Failure Progression Figure 23. Load Displacement Relationship for the Two Crack Problem Figure 24. Scaled Geometry of Hole Size Effect Study Figure 25. Mesh Configuration for Tensile Strength Scaling Studies (a) coarse, (b) fine Figure 26. Effects of Hole Size on Average Traction - Elongation Curves xii

13 Figure 27. Delamination Failure Strength for Specimens with Thick Plies Figure 28. Effects of Ply Thickness on Failure Stress for Rx-FEM compared to Experiment Figure 29. Load Displacement Curve for the Case C3 (m=4 and D=6.35 mm) Figure 30. Damage Progression Sequence for the Case C3 (m=4 and D=6.35) at Load Levels (a) 107MPa, (b) 162Mpa, (c) 223MPa, (d) 253MPa, (e) 300MPa, (f) 304MPa, (g) 306MPa, (h) 304MPa Figure 31. CT Images of Damage Patterns Compared to Predicted Damage at Peak Load Figure 32. Separation of Element Structures by Rx-FEM Figure D Regularized Step Function Figure 34. Implementation of Rx-FEM within ABAQUS via User Subroutines UEXTERNALDB and UEL xiii

14 LIST OF TABLES Table 1. Material Properties for DCB Analysis Table 2. Test Cases for DCB Analysis Table 3. Coefficients for DCB Stiffness as a Function of Crack Length Table 4. Unidirectional Stiffness and Strength Properties for IM7/ Table 5. Properties Used in Cohesive Law Table 6. Failure Stress for Various Laminates Subject to Delamination Failure Table 7. Calculations of Regularized Step Function for Example Case Table 8. Calculation of Regularized Step Function at Integration Points for Example Case Table 9. Division of Properties for Example Case xiv

15 LIST OF ACRONYMS, ABBREVIATIONS, AND SYMBOLS Acronyms CDM CZM CBT DCB DOF FEM FRP LARC LEFM MIC MKL NASA PC Rx-FEM UDRI VCCT WWFE x-fem Continuum Damage Model Cohesive Zone Model Corrected Beam Theory Double Cantilever Beam Degree of Freedom Finite Element Method Fiber Reinforced Plastics Langley Research Center Linear Elastic Fracture Mechanics Mesh Independent Crack Math Kernel Library National Aeronautics and Space Administration Personal Computer Regularized Extended Finite Element Method University of Dayton Research Institute Virtual Crack Closure Technique World-Wide Failure Exercise Extended Finite Element Method xv

16 Element Formulation a B D f J k e l n en n int N a u As a subscript used to designate the local node index for an element Strain-displacement matrix Material stiffness (constitutive) matrix Nodal force vector Element Jacobian Element stiffness matrix Index for the integration point Number of element nodes Number of integration points in the element Shape function associated with node a Displacement vector u, v, w Individual components of the displacement W e W l x Strain energy of an element Weight function associated with integration point l Vector of global coordinates, a point in space x, y, z Individual global coordinates ε ξ Strain vector Vector of parametric coordinate ξ, η, ζ Individual parametric coordinates σ Ω Ω e Stress matrix Material domain Element domain xvi

17 Crack Definition (1), (2) superscripts used to differentiate between zones where (1) designates the zone where the signed distance function is negative and (2) designates the zone where the signed distance function is positive f α H H n x α δ δ Γ α The signed distance function associated with crack α The Heaviside (step) function The regularized Heaviside (step) function The normal to the crack surface A point on the crack surface An index to designate an individual crack The delta function The regularized delta function The crack surface Cohesive Law B d g(λ,b) G c G Ic G IIc G e K l cz l e or L e Mode-mixity parameter Damage variable Energy per unit crack area required to produce a separation of λ for a mode-mixity parameter B. Critical energy release rate (fracture toughness) In Mode I In Mode II Cohesive energy absorbed up to maximum traction Interfacial (penalty) stiffness Length of the cohesive zone Characteristic element length xvii

18 N e S t Y Δu Δu n Number of elements in the cohesive zone Failure strength in shear A characteristic thickness (1/2 ply thickness) Failure strength in tension Displacement jump Component of displacement jump normal to the crack Δ 0 Critical separation at onset of damage (d = 0) Δ f Separation at completion of damage (d = 1) η λ τ τ 0 Mode-mixity exponent Separation (magnitude of the displacement jump) Traction Maximum traction Double Cantilever Beam Equations a d d crit f k k 0, k 1, k 2 U W Crack length Total displacement Critical displacement at which crack growth occurs Reaction force at the point of load application Spring stiffness of the DCB defined as f/d coefficients of the quadratic fit to the k vs a response Internal strain energy Width of the DCB xviii

19 Energy and Volume Terms M α n N N el S v W n v Cohesive energy associated with crack α Index associated with ply number Total number of plies Total number of elements Surface area of a crack Strain energy for ply n Volume z n Location of interface between ply n and n + 1. Φ n Interfacial energy between ply n and ply n + 1 xix

20 CHAPTER I INTRODUCTION The need to understand failure is self-evident. We try to determine why failures occur in order to prevent future failures from happening. Barring the ability to anticipate failures we are left with a strictly empirical approach that requires nearly endless testing of every conceivable load and structural configuration. This is the situation we still find ourselves in today when dealing with composite materials. A great deal of work has been done to shed light on the failure mechanisms to which composites are vulnerable, and much progress has been made. It is our intent to take one more step forward in the predictive modeling of composite failures. In this chapter we introduce the problem in general terms including a brief review of failure prediction methods from the literature. A more detailed review of damage modeling is provided in the next chapter. Use of Composites in Large Structures Composites are now well-established materials for use in a variety of industries. The superior strength to weight ratios and tailorable properties of Fiber Reinforced Plastics (FRP) composites make them desirable for use in aviation, where weight reductions translate into improved efficiency. 1

21 The use of composites in aerospace structures is increasing rapidly now that technical and cost obstacles in the manufacturing of large sections have been addressed and largely overcome. This trend can be seen dramatically within Boeing commercial aviation. The Boeing 747, introduced in 1969, has about 1.3 percent composite material by weight. The Boeing 777, introduced in 1997, was designed with about 10 percent composite by weight. The Boeing 787 Dreamliner, introduced in 2011, contains 50 percent composite by weight [1]. Of course, composites are used in other industries as well. In power generation, for example, wind turbines use composite blades. A single turbine blade may weigh up to 6.6 tons and is made primarily out of glass fiber reinforced plastic. These blades are designed to last 20 years with constant exposure to the environment [2]. In order to provide more powerful wind turbines in the future, blades will be designed to be longer and operate at higher wind speeds. The accompanying strength requirements will result in carbon fiber composite blades replacing glass fiber-reinforced blades [2]. Another application where composite materials are competitive with traditional materials is in pressure vessels. Composites are light weight and corrosion resistant, which are desirable qualities for the transportation industry. As compressed natural gas is turned into a portable fuel source, demand for composite pressure vessels likely will also rise [3] Design Challenges of Composite Structures The transition from traditional engineering materials (i.e., metals) to composites can be expected to present challenges beyond manufacturing issues. The design issues facing structural engineers include uncertainty in the durability and toughness of these new materials in service conditions [4]. Since the infamous case of the de Havilland Comet, where the choice of high strength aluminum with low fracture toughness coupled with other design flaws contributed to catastrophic failures [5], aerospace engineers are understandably nervous about the presence of 2

22 unseen dangers. The Comet disasters demonstrated the need to anticipate new failure modes before they are uncovered accidentally in service. Since composites are relatively new in terms of widespread commercial application, unforeseen failure modes may remain. Therefore, it is understandable that when commercial accidents involving composite structures occur, material safety will come under intense scrutiny. The uncertainty that current design methods are sufficient is a result of the additional complexity of composites. Aircraft structures are designed according to a variety of rules; for example, fracture mechanics feature prominently in design methods based on damage tolerance. But the physical mechanisms that determine the toughness of metallic materials dislocation based plasticity, for example are not present in polymer matrix fiber reinforced composites. When failures occur in composite materials, more detailed failure analysis is often required. The complexity inherent in composites requires damage models that are capable of capturing that complexity. The rupture of the composite tail fin of American Airlines Flight 587 led almost immediately to questions about durability of composite materials [6, 7]. Though eventually the cause of this failure was attributed to the pilot s excessive use of the rudder in another aircraft s wake, subsequent failures of other large composite sections continue to bring into question how well composite failures can be anticipated [8]. An analysis of the composite right rear lug of American Airlines Flight 587 carried out by NASA Langley researchers illustrates how failure analysis is currently executed [9]. In the authors words, micromechanics based approaches to failure prediction remain impractical if not intractable. Continuum-based criteria are insufficient beyond failure initiation once multiple damage mechanisms begin to interact. This leaves strength-based approaches, with their 3

23 comparative simplicity as the best practical method for predicting the onset and development of material failure, especially within the well-developed framework of the finite element method. Composite Failure Analysis and Prediction Structural analysis of composites has been a very active field for years. However, even after rather intense effort dating back to the pioneering work of Rosen [10] in the 1960 s and Tsai and Wu [11] in the 1970 s, strength prediction for laminated composite materials is still challenging. This difficulty is partly due to the nearly endless configurations possible in composites, and the broad range of materials to select from. Multiple damage mechanisms and their interactions must be considered together. Matrix cracks and delamination interact strongly in applications such as skin/stringer fuselage structures [12-14]. The World Wide Failure Exercise (WWFE) challenged the composites community to see how accurately failure of laminated composites could be predicted [15-19]. Once the test cases were defined and the input information for the test cases (materials, geometry, and loading) sent to the participants, experts in the field were invited to employ their models and predict the failure mechanism and load in Part A of the exercise [15]. The exercise involved fourteen benchmark cases, with the data consisting essentially of two types: biaxial failure strength envelope predictions, and stress-strain curve predictions. It should be noted that none of the benchmark cases involved unusual stress-raising geometric features such as discontinuous ply layers or holes, but rather involve only gross section behavior. The studied composite materials were not unusual or exotic. Many of the test cases were for so called black aluminum materials with widespread commercial use or even simple unidirectional composites. These results were published individually and compared on a blind basis [19]. In Part B of the exercise, experimental results of 4

24 the tests [17] were provided to the researchers. The results were compared with these experiments and the researchers were allowed to comment and make adjustments to their models [18]. The comparisons showed some models were clearly better predictors than others [19]. Four failure theories were placed in the highest group with very few fundamental weaknesses, though no theory was without many minor weaknesses. To illustrate the complexity of some of these models, consider the approach taken by Puck and Schurmann [20, 21]. Puck s model was developed from the earlier Hashin failure model, which was a strength-based failure criterion [44]. In the WWFE, Puck s model was found to do a superior job. The model contains features such as non-linear stress-strain relationships using the secant method, fiber failure modes in tension and compression, and inter-fiber failure in three distinct modes: (A) transverse tension, (B) moderate transverse compression, and (C) large transverse compression. Puck s model also includes degradation methods which lower the elastic properties in response to stress. In addition to degradation after initial failure, Puck includes degradation prior to initial failure, which represents individual fiber breaks. Despite its being one of the more successful models in the failure exercise, some questions may be raised about the applicability of Puck s model to real design problems. First, it was developed for monotonic analysis; how well does it model cyclic loads? Second, the model was developed in the context of homogeneous loading; what happens when damage occurs in the presence of stress gradients? Furthermore, it is difficult to see where size effects enter into these models, but size effects are important in cases such as notches [22]. Finally, as Davila emphasizes [23], some of the parameters in Puck s model are not physically based and are difficult to determine experimentally. 5

25 More recently, the LaRC failure criterion family [23] and its integration into a continuum damage framework [24, 25] were proposed (Figure 1). The origins of this model come from Hashin and Puck, with new approaches to the calculation of the fracture plane by using a Mohr-Coulomb effective stress. This framework addresses both the complexity of the failure envelope in unidirectional composites, and the mesh density dependence [26] of failure prediction in high stress gradient areas. Strength Based Approach Continuum Damage (CDM) Hashin (1973) Kachanov (1958) Hashin (1980) Sun (1996) Puck (1998) Chaboche and others LaRC 03 (2005) LaRC (2007) Figure 1. Pedigree of the LaRC Failure Criteria 6

26 Experiments on open-hole composites show that there is frequently significant damage accumulation prior to failure (Figure 2). This damage can take different forms such as delamination and matrix cracking in addition to fiber failure. Furthermore, damage does not always originate at a single highest stress location, but often is distributed over a finite region. In the presence of a notch, cracks may initiate, but may not be harmful. In fact, some degree of damage, in the form of matrix cracks or delamination, may increase the failure strength if the damage remains local and lowers the stress concentration. Figure 2. Matrix Cracks Around an Open Hole From X-Ray Images [24] 7

27 One form of damage may initiate and interact with other forms, accelerating or delaying cracking in adjacent lamina. Strength-based models may be appropriate in determining the location and orientation of crack initiation, or where delamination is likely to occur, but lack the capability of predicting the behavior after the onset of damage. The damage evolution must be captured within an analysis method that can accommodate all the important damage modes and their interactions. Capturing damage evolution is not a trivial matter. For example, in the simple case of a plate with a hole, longitudinal matrix cracking leads to redistribution in the stresses which is not adequately described by property degradation methods [27]. Other techniques that deal with macroscopic damage in greater detail are necessary to properly capture this stress redistribution. Predictive Modeling In order to develop more accurate structural models, the composites community is pushing forward with new analysis approaches. Emerging methodology includes both more accurate prediction of the initiation of failure, and of the progression of failure through multiple mechanisms, without knowing how these interact a priori. Though the prediction of initiation can often be accomplished on a continuum or a strength basis, the prediction of damage interaction seems to require explicit damage modeling. A failure typically creates a discontinuity in the displacement or strain field. As the failure surface grows and evolves, this discontinuity grows. The Finite Element Method is the most mature, general purpose, approach to modeling the mechanical response of real structures. Yet, the discretization of a material into elements is designed to produce continuous displacement fields through the continuity of the shape functions. Smoothness of the field variables has, at least implicitly, been a measure of accuracy of the solution. Only at the free surfaces of elements can discontinuities in displacements occur in a 8

28 typical finite element formulation. To represent cracks, new boundaries must be created within the model. Unfortunately, the creation of new boundaries requires either adaptive meshing, which can be numerically costly, or meshing that anticipates the failure directions, which is only possible under simple condition and may change with each loading condition considered. What is needed is a means of describing discontinuities within the field or, in other words, a means of embedding the discontinuities. In summary, certain desirable characteristics of a methodology to model progressive damage can readily be identified. First, the method should be compatible with arbitrary geometric representation; any reasonable solid object should be able to be modeled. This characteristic is desirable so that realistic structural components can be analyzed. Of course, rather than looking for a new method to describe geometries, it would be more desirable to use an existing approach, such as the finite element method. Second, the model should not require special meshing techniques, but should account for damage wherever it occurs within the structure. Accuracy of the method should not be overly sensitive to the discretization of the volume, and should improve as mesh density increases. Third, the method should be general enough to work in parallel with existing techniques. For example, the method should not prevent contact modeling, non-linear materials or nonlinear geometric effects. Fourth, the method should be able to incorporate strength-based and continuum damagebased failure models. It should work alongside the material formulation and be adaptable to changes in the material formulation. 9

29 In order to meet these criteria, a new method called the Regularized Extended Finite Element Method (Rx-FEM) is proposed. The next chapter reviews in more detail the current approaches to progressive damage, and provides the background for development of this new method. 10

30 CHAPTER II A REVIEW OF THE ELEMENTS OF PROGRESSIVE DAMAGE MODELING In this chapter, the background of the proposed damage modeling method is provided in the context of current approaches to progressive damage modeling. The ingredients of the method are: a failure initiation criteria; a model for delamination growth between lamina; and a method of introducing and growing cracks within plies. The review below shows that all of these aspects of the problem have been addressed individually, but a robust method of combining the ingredients is needed. A solution which is able to incorporate all these ingredients is the objective of the current work. Failure Initiation Criteria As outlined above, a great deal of effort has already been directed towards progressive damage modeling of composite laminates. Strength-based failure criteria have been established for both unidirectional composites and more general systems. These criteria seem to be rapidly approaching a mature form and, if not already sufficient to capture the initiation of damage, will be with further incremental improvements. The strength-based LaRC03 criteria are used to determine failure initiation in this work. LaRC03 contains fiber and matrix failure criteria under combined loading conditions, and has been verified for many of the problems described in the WWFE [23]. 11

31 Modeling Delamination The delamination process in layered composites has been modeled successfully in a variety of ways. The most successful approaches are based on the principles of fracture mechanics, such as the critical energy release rate. These approaches borrow some of their methodology from the work of pioneers such as Griffith, Irwin, and Dugdale. In the Griffith energy balance approach [51], fracture occurs when the energy released by an incremental increase in crack length is greater than the energy required to create the new crack surface area. This critical energy per unit area of new crack surface is called the critical energy release rate. In the Virtual Crack Closure Technique, the crack is extended a finite, but small, amount, and then brought to closure by imposing boundary conditions on the new surface [28]. If the mechanical energy required to close the crack is greater than the critical energy release rate, the crack is opened. The VCCT has been validated for many problems and has been commercially implemented [59, 60]. Normally the method requires that the crack opening proceed along a pre-defined direction, but the method has also been modified for cracks growing in two dimensions and three dimensions in arbitrary directions using ABAQUS X-FEM [60]. Another method, the Cohesive Zone Model (CZM), is related to the Dugdale strip model and is also appropriate for describing delamination. Dugdale considered the effects of plasticity on fracture and required that stresses be limited to the yield strength of the material [29]. He calculated the size of the process zone where the crack forms by assuming the plasticity occurred along a thin strip in front of the crack tip. For ductile materials, of course, the plastic zone is much thicker, but in the case of delamination, the assumption is very reasonable, as the inelastic deformation which accounts for the toughness is confined to a thin layer of matrix 12

32 material between the lamina. The cohesive zone is the approximate equivalent to this process zone. Typically CZM techniques employ interface elements between the lamina [30]. These elements are modeled with no thickness but tie together the degrees of freedom of the adjacent layers through an imposed traction-separation relationship. Separation is the deformation of the interface element as it is pulled apart. Traction are the forces that correspond to the separation. 10 Traction K Separation Figure 3. A General Traction-Separation Relationship Tractions both normal to and in plane with the interface are allowed, to account for multiple failure modes. The artificial stiffness associated with traction-separation curve, K as shown in Figure 3, must be carefully chosen as to have minimal effect on the solution and yet not create numerical instabilities. A summary of cohesive zone models is provided in [58]. Once a critical traction is reached, further separation relaxes the stiffness until the tractions are zero and the crack can be said to have extended. If the energy under the traction-separation law is equal to the critical energy release rate, the CZM approach is consistent with the VCCT approach in its basis on fracture mechanics. 13

33 The CZM interface element approach has been implemented within commercial codes and has been successfully validated for certain benchmarks alongside the VCCT methods [30]. Modeling Matrix Cracking Modeling matrix cracking remains somewhat problematic. Matrix cracks occur between fibers but within the lamina. The failure initiation criteria and the ply orientation determine the direction of the normal to the crack; however, unlike delamination, the crack plane is not known prior to the application of loads. For this reason, methods like VCCT are not generally appropriate. To model damage within lamina, two approaches can be taken. The first is to treat damage as part of the constitutive behavior of the material through a Continuum Damage Model (CDM), which allows the material to soften as damage accumulates. The advantage of the CDM approach is fairly clear. Since the damage is incorporated within the constitutive model, the technique is easily adapted to numerical approaches such as the finite element method. However, CDM approaches that permit softening may be highly dependent on the mesh. Iarve [27] explored the use of property degradation CDM in an open hole laminate and found significant shortcomings. Specifically, the CDM method was unable to properly capture the initiation and growth of longitudinal splits when the loading axis and the fiber direction were aligned. A second approach, discrete damage modeling, also has its advantages and disadvantages. The primary advantage is that discrete damage modeling can readily incorporate fracture mechanics concepts such as critical energy release rate. However, adaptation of discrete damage modeling to FEA is somewhat more complicated, since it requires that the cracks be explicitly modeled, which in turn affects the discretization scheme. For a traditional finite element approach this 14

34 means that the problem must be remeshed every time a crack is introduced or extended, unless the crack path is known a priori. In 1999, Belytschko introduced the extended finite element method (X-FEM) to address the problem of handling discontinuities in a region without resorting to re-meshing [32]. The method enhances the displacement field by adding additional degrees of freedom and partitioning the mesh. Since then, X-FEM has been recast and reinterpreted several times; for example, see [33-35]. Concurrently, Iarve [36, 37] adopted the displacement field partitioning and incorporated it into a spline-based discretization method [45, 50]. As part of his approach he regularized the crack definition in order to maintain the method s integration scheme. Iarve applied this method successfully to model cracking and delamination and their interaction in laminated composites; however, the spline-based method proved unwieldy for arbitrary geometry. The method developed and described in subsequent chapters incorporates these ingredients: the failure initiation criterion; CZM description of the delamination process; and a regularized crack description within the framework of a finite element discretization. The next chapter focuses specifically on the numerical method for adaptive description of a crack. 15

35 CHAPTER III A REGULARIZED, EXTENDED FINITE ELEMENT METHOD In this chapter, the method of modeling cracking and delamination is presented in detail. First, the development of the Regularized Extended Finite Element Method (Rx-FEM) is described. A brief overview of the Finite Element Method (FEM) is provided for context, and to highlight certain aspects of element formulation. The Extended Finite Element Method (X-FEM) is then presented in order to introduce the concept of displacement field partitioning. The modifications to the element formulation that result in the Regularized Extended Finite Element Method (Rx- FEM) are then described. Next the implementation of the method is presented. Both delamination and Mesh Independent Cracking (MIC) formulations employ the Camanho cohesive zone model (CZM) to model crack growth. The mathematical description of CZM as applied to MICs is formulated. Issues of choosing parameters for the CZM and effects of element sizes are addressed. Finally, the method by which a crack is introduced into the finite element data structure is described. Finite Element Method Under the standard finite element method, the displacement field is described by defining its value at discrete points, or nodes. These nodes are connected by elements. Most often, both the 16

36 physical domain of the element and the displacement field within the element are described using the same parametric equations, or shape functions: x n en ( ξηζ,, ) N ( ξηζ,, ) = x a= 1 a a (1) u n en ( ξηζ,, ) N ( ξηζ,, ) = a= 1 a u a (2) Here x a are the nodal values of the coordinates, and u a are the nodal displacements. Thus, continuity of these shape functions, N a, assures both continuity of the domain and continuity of the displacement fields within the element. The strain field is calculated from derivatives of the displacement field with respect to the spatial dimensions. The local strain can be calculated from the derivatives of the displacement field: ε ( ξ) = B ( ξ) u (3) B is the strain-displacement matrix which contains derivatives of the shape functions. The local stresses are determined from the strains through the material modulus matrix, D: σ ( ξ) = Dε ( ξ ) (4) The strain energy within an element can thus be determined from integrating the strain energy density over the element: W 1 = ubdbu dω (5) e T T 2 Ωe 17

37 And, finally, the strain energy for the volume is calculated from the summation of the strain energy for all the elements. The relationship between the nodal forces and displacements is determined by minimizing the strain energy for the entire model, which results in a global stiffness matrix. The contribution to the stiffness from each element is: e T k = B DB dω (6) Ωe The integration is carried out numerically, typically using a Gauss quadrature rule where the volume integration is approximated by the summation: n int e T k ( B DB J ) Wl (7) ξ l = 1 l Here, the integrand is evaluated at n int integration points. The location of the integration points are specified by the parameters ξ l, and the weighting functions W l, depend on the order of the quadrature. J is the Jacobian matrix which maps the derivatives with respect to the parametric variables (ξ) to the spatial derivatives. The great variety of elements available in FEM comes from the choices of nodal configuration (tetrahedral, wedge, hexahedral), shape functions (first order, second order), and integration schemes (full, reduced, selective). Introducing a discontinuity, such as a crack, within the finite element model presents a challenge. The simplest place to introduce a discontinuity is along the boundaries of elements, since continuity within the element is one of the requirements for convergence of the solution. Limiting discontinuities to element boundaries requires that either the crack location is taken into account during the initial meshing of the structure before the start of the analysis, or the structure is remeshed as cracks appear and grow during the analysis. Both of these approaches to representing discontinuities have serious limitations. It is not always possible to know where 18

38 cracks will appear before the analysis begins, and imposing the locations of cracks a priori is contrary to the goal of predictive modeling of damage development. Remeshing the model is certainly possible, and many commercial codes do have adaptive mesh techniques, but their utility in the case of laminated composites is questionable because of the multiple meshing constraints occurring at ply interfaces, where cracks with different orientations in adjoining plies have to be accommodated. The Extended Finite Element Method The Extended Finite Element Method (X-FEM) was proposed to allow the introduction of cracks and discontinuities without the need to remesh [32]. X-FEM enriches the displacement field by introducing additional degrees of freedom to account for discontinuity, and then divides the domain into regions where these additional degrees of freedom act. The domain boundaries, which represent the discontinuity, may cross element boundaries. x f α > 0 nx ( ) Γ α x f α < 0 Figure 4. Subdivision of the Domain by the Signed Distance Function 19

39 The crack surface Γ α, where the displacement function is experiencing a discontinuity, may be described by the signed surface distance function f α, such that f α ( x) = sign( n( x) ( x x))min x x x Γα (8) where n( x) is the normal to the crack surface Γ α at the point x (Figure 4). This function is defined at an arbitrary point x of the domain and is equal to the distance from that point to the surface of the crack and is positive above the crack and negative on the other side of the crack surface. The Heaviside step function, ( ) H x 0 x < 0 = 1 x 0 (9) is used to describe the displacement jump at the crack surface Γ α in the following way: Let u (1) be the displacement field associated with the domain below the crack (f α (x)<0) and u (2) be the displacement field associated with domain above the crack (f α (x)>0). Then the entire displacement field may be described through: ( 1 H ( fα ( ))) H ( fα ( )) (1) (2) u = x u + x u (10) It should be noted that the Heaviside function falls into the class of generalized functions [55]. Generalized functions possess certain additional characteristics which permit relaxation of the rigorous definitions of operations such as integration and differentiation performed on normal functions. The derivative of the Heaviside function is the Dirac delta function, δ. The delta function is also a generalized function [56]. Clearly over most of the domain, its value is zero, but at the origin 20

40 where the Heaviside function is discontinuous, the function is not defined. However the requirement is made that the integral of the delta function across the origin is one. The delta function has the important selection property that it can select the value at a specific point along an integral. If g(x) is an arbitrary function of x, then ( ) ( ) ( ) δ x x0 g x dx= g x 0 (11) Different forms of approximation of the fields u (1) and u (2) have been proposed over the years, including a particularly elegant approach by Hansbo and Hansbo [33] in which the displacements u (1) and u (2) are approximated with the original shape functions and result in duplicate (phantom) nodes in the enriched element. Domain (1) x x Original Domain x x x x x x x x x x x Real Nodes Phantom Nodes Integration Points Domain (2) Figure 5. Adaptation of Integration Scheme for x-fem 21

41 Common to all such approaches is the fact that the integration domains for the functions u (1) and u (2) are different, namely the sub-volumes f α (x)<0 and f α (x)>0 respectively. A custom integration scheme in each area (sub-volume) is needed. Such a construction is possible, but is quite tedious and involves many possible combinations of crack locations with respect to element boundaries. For example, in the case of a quadrilateral element, the division may result in two quadrilaterals, or it may result in a triangle and a five-sided region that will need to be further subdivided. In three dimensions there are even more combinations to consider. Moreover, at the interface between two plies with different fiber orientation, it is necessary to take into consideration a double partition of the element surface in order to describe the delamination evolution from the crack intersections. Thus, just as it is advantageous to preserve the mesh structure, another representation of the crack that does not involve the recalculation of integration point locations is desirable. The Regularized Extended Finite Element Method The term regularization has been applied in various ways. For example, in signal processing, the term may be used to describe methods that reduce noisy information to a smoother form. In solid modeling it may mean replacing a complex surface with a set of simpler surfaces. Loosely, any smoothing or filtering operation could be thought of as a regularization process when the regularized form is used in place of the original. Here the term regularization is used in the sense that a function ill-suited for analytical purposes is replaced by a smoother approximation which captures the essential qualities of the original function [57]. In applying the regularization, the crack boundary, defined to this point using the Heaviside step function, is replaced by a continuous function. This is equivalent to replacing the crack surface by a process zone which, in the three dimensional case, encompasses a narrow volume 22

42 surrounding the crack surface. There is some physical justification for this approach. For example, matrix cracks follow tortuous paths, branch, and form networks of cracks. In many cases, there is indeed a physical dimension to the crack. Moreover, the regularization of the Heaviside function is carried out in a manner compatible with the finite element method itself, and will be described using the element s own shape functions. The approximation to the Heaviside step is n en H ( ξ ) = N ( ξ ) H (12) a l a l a= 1 Where the coefficients H a are calculated as follows: ( x) α ( x) ( x) ( x) 1 Na f dv 1 v H a = + (13) 2 N v a fα dv This definition of the regularized step function involves only continuous functions, and therefore can be calculated throughout the element s domain. The magnitude of the gradient of the regularized step function,, is the regularized equivalent of the Dirac delta function. δ = H (14) For a one-dimensional case the regularized step function and its derivative are shown in Figure 6 and Figure 7 respectively. Please see Appendix I for the specific details of the calculation. 23

43 Regularized Step Function Integration Points Nodes Coordinate, x Figure 6. Regularized Step Function for a One-Dimensional Problem 1 Gradient of Regularized Step Function Coordinate, x Figure 7. Gradient of the Regularized Step Function for a One-Dimensional Problem The regularized step function is a piecewise linear function and continuous (C 0 ); its derivative is discontinuous and undefined at the nodal locations. In order to make use of these regularized functions mathematically, they are also classified with the Heaviside and Delta Functions as generalized functions. The relationship between the step function and its gradient, 24

44 d δ = = dx ( x) dx H ( x) dx H ( x) (15) requires that the gradient be piecewise integrable. That is to say, the discontinuities in do not add to the integral. Mathematically, for all values of x, 0 ( x ) lim δ + s ds = 0 (16) This condition (16) allows the numerical integration to proceed when the integrand contains the gradient of the regularized step function. Furthermore, it completes the analogous behavior between δ and the Delta function. As with the x-fem, two independent displacement fields u (1) and u (2) operate within the volume of the element. Each field has its associated degrees of freedom, and these degrees of freedom are generally independent. It would be tempting at this point to describe the location of the discontinuity as the level set 1 H ( x) = (17) 2 However, such a description of the crack would be imprecise and is unnecessary. The original crack definition as the surface Γ α is retained, and the domains remain separated by the sign of f a (x). As a consequence of the independence of the displacement fields, there are also independent strain and stress fields, independent strain energy functions, and independent stiffness matrices in the subvolumes of an element surrounding the crack. 25

45 ( 1 H ( )) H ( ) (1) (2) u= x u + x u (18) Because the strain fields in each zone are independent, the strain fields are not calculated from derivatives of the combined displacement field. Instead, strains are calculated from the individual displacement fields using the original strain-displacement relationships: ε ε = Bu (1) (1) = Bu (2) (2) (19) The stresses are in turn computed from the strains σ σ = Dε (1) (1) = Dε (2) (2) (20) For visualization purposes, the stresses and strains may be computed in the elements containing the discontinuity through ( 1 H( )) H( ) ε = x ε (1) + x ε (2) (21) and ( 1 H ( )) H ( ) σ = x σ (1) + x σ (2) (22) This measure ensures that even though a steep displacement gradient is displayed on the mesh, if the crack relieves all stresses, the strains and stresses will appear as 0. Perhaps the most important feature of the proposed approach is that as a result of using the original shape functions, the integration domain for the displacements u (1) and u (2) coincides with 26

46 the original element integration domain, and integrations can be performed without subdivision of the element. The stiffness matrices associated with the displacement fields u (1) and u (2) are computed using the Gauss integration points of the original element: n int T ( ) ( 1 ( l) ) (1) K B DB J ξ ξ l= 1 l H W l (23) n int (2) K B DB J ξ ξ l= 1 l T ( ) ( l) H W l (24) It can readily be seen that this amounts to a partitioning of the stiffness matrix, as K (1) + K (2) = K where the partitioning is done spatially based on the value of the regularized step function. With the introduction of the additional degrees of freedom, the load-displacement relationship for the element becomes (1) (1) (1) f K 0 u = f 0 K u (2) (2) (2) (25) The off-diagonal 0-matrices represent stiffness associated with the separation of the two fields. When a cohesive zone model (CZM) is used to simulate tractions at the crack surfaces, non-zero stiffness coefficients will appear in these positions. It is important to emphasize that the geometric description, Γ α, of the discontinuity still remains independent of the mesh; hence the term Mesh Independent Crack (MIC) is used to describe this approach. However, the mathematical representation of the crack is mesh-dependent, as it is in standard X-FEM. Refining the mesh will result in narrower and narrower crack zones, since the transition across the discontinuity occurs over the same length scale as the elements that traverse it. Ultimately, Rx-FEM approaches standard x-fem in the limit as elements are refined. 27

47 The Cohesive Formulation The cohesive zone model as formulated by Turon [38] is used in the present work. The formulation is summarized here for clarity. In this model, the cohesive traction on the crack surface is related to the displacement jump Δu in the following way ( ) τ = Δ u+ Δ n 1 d K dk un (26) In this equation K is a constant defining the initial stiffness of the surface bond, n is the normal vector to the crack surface and d is the damage variable. The component of the displacement jump normal to the crack is Δu n = ( Δu n ) (27) In equation (26) only the first term is responsible for the irreversible fracture energy, whereas the second term is present to prevent interpenetration of the damaged surfaces. The brackets 1 x = ( x+ x) represent the Macaulay operator. 2 In the fracture of composite materials, the fracture energy depends strongly upon the opening mode of the crack or delamination. According to the methodology proposed by Turon et al. [38], the cohesive energy can be written in invariant form as a function of the absolute value of the displacement jump, or separation, λ= Δu, and a mode mixity parameter, B, defined as: 2 Δu B = 1 n (28) 2 λ 28

48 Because Δu n is the component of the displacement jump normal to the crack surface it represents the mode I component of the total separation. B = 0 represents pure mode I opening, and B = 1 represents any combination of the shear opening modes II and III. τ 0 τ d 1 K G c Δ 0 Δ f λ Δ 0 Δ f f λ Figure 8. Traction-Separation and Damage-Separation Relationships For a fixed-mode fracture (B = constant), a bi-linear traction-separation relationship is assumed. Once a maximum traction is reached at a separation of Δ, the cohesive traction is reduced linearly until at the final separation Δ the cohesive traction is zero (see Figure 8). Kλ, λ <Δ0 Δ f λ τ( λ) = τ0, Δ 0 < λ <Δ Δ f Δ 0 0, λ >Δf f (29) where τ is the cohesive strength. For continuity of the traction-separation equation, the critical separation must be a function of the penalty parameter K, and it may not be chosen independently. τ K 0 Δ 0 = (30) 29

49 The damage variable may be determined in terms of the separation by substituting equation (29) into (26). d ( λ) 0, λ <Δ Δ λ Δ f 0 =, Δ 0 < λ <Δ f λ Δ f Δ0 1, λ >Δ 0 f (31) And the damage rate law can be found from the derivative 0, λ <Δ ΔΔ d 0 f =, 2 Δ 0 < λ <Δ f λ λ ( Δ f Δ0 ) 0, λ >Δ 0 f (32) At this point it is worth remarking that, as implemented, the damage variable is only allowed to increase throughout the analysis. Thus, effectively, both the cohesive strength and the stiffness are reduced by prior damage. If the cohesive zone is unloaded, damage does not accumulate again until the new maximum traction associated with the prior damage is reached. The cohesive strength also depends on the mode mixity parameter B as: ( ) ( ) τ 0 = Y + S Y B η (33) where Y and S are the interfacial normal and shear strengths, respectively, and η is a material parameter. All parameters entering the analysis, such as the fracture toughness and strength values, are material properties that can be measured using standard test methods. The energy 30

50 g(λ,b) required to create a displacement jump λ at the given point of the fracture surface is the area under the τ(λ) curve, so that λ g( λ, B) = τ( q, B) dq q= 0 (34) For the crack propagation to exhibit the correct fracture response the energy required to achieve complete separation must be equal to the fracture energy; that is to say the following condition must be satisfied: ( f, ) g Δ B = G (35) c where the critical energy release rate, G c, is assumed to be a function of the mode mixity as follows ( ) G = G + G G B η (36) c Ic IIc Ic G Ic and G IIc are experimentally measured fracture toughness values and η is an influence parameter also determined experimentally. In the case of the bi-linear τ(λ) relationship, the energy to failure is calculated by integrating (29): 1 g( Δ f, B) = τ 0 ( B) Δ f (37) 2 or, simply, G Δ 2 c f = (38) τ 0 31

51 Element Size Effects Turon [38] also provides guidance for element size effects and dependence on the selection of properties for the cohesive law. An upper bound on the size of the cohesive zone can be approximated through [39] for mode II opening 1/4 G c lcz = E t 2 ( τ 0 ) 3/4 (39) where the thickness, t, refers to the half thickness of the laminate. For Mode I opening, l cz EG = (40) τ c 2 0 As reported by Turon, suggestions for the appropriate number of elements in the cohesive zone range from two to five [41] to more than ten [32]. From (39) it can be seen that the cohesive zone length is inversely proportional to the fourth root of the interfacial strength. If the correlation between the Griffith model and CZM holds, the determining factor for crack growth is not the strength but the energy release rate. Adjusting the interfacial strength could provide a means of allowing the mesh to be coarsened without affecting the results. On this basis, Turon [38] derives the following equation for interface strength: 9π EGc τ 0 = (41) 32Nl ee As the authors point out, the effect of reducing the interface strength is to stretch out the cohesive zone, which may lead to differences in the local stress distribution near the cohesive zone. Citing 32

52 Alfano and Crisfield [30], a longer cohesive zone may provide better numerical stability with small sacrifice of accuracy. A simple analysis of the cohesive law also will prove useful. From inspection, the maximum elastic energy in the cohesive law prior to the initiation of damage is G e 1 = τ 0 Δ 0 (42) 2 Substituting (42) into (30), and requiring that G e < G c yields ( τ ) 2 0 2K < (43) G c This relationship (43) places restrictions on the selection of the penalty stiffness, K, in terms of the interfacial strength. It is also indicative of the size of the cohesive zone. A higher strength will result in a larger elastic fraction and a smaller cohesive zone. Conversely, a larger penalty coefficient will result in more of the energy being associated with the softening part of the law and a more extensive cohesive zone. Application of Cohesive Zone to Mesh Independent Cracks The critical aspect of the cohesive zone crack propagation model, which makes it suitable for extension to the regularized MIC formulation, is the fact that the crack tip energy release rate concept of classical fracture mechanics is replaced by a surface fracture energy at a given point. The development of a unit length open crack surface where λ f requires an energy release rate G c, according to (35). This relationship between crack opening and energy release rate is consistent with the Griffith definition of G c. A similar argument is used to calculate the fracture energy of a MIC in the regularized formulation. 33

53 Within each cracked element one can define the displacement jump as Δu=u (1) -u (2). In the original x-fem formulation this jump is confined to the crack surface Γ α, whereas in the regularized formulation, the displacement jump has relevance within the so-called gradient zone (a narrow volume surrounding the crack surface), where the approximate step function gradient is present, i.e. H > 0. The traction-separation relationship, equation (26), is used to calculate fracture energy at each integration point of the gradient zone. The only parameters needed are the displacement jump vector, Δu, and the normal vector to the crack surface, which can either be provided in the crack definition, or be calculated as a normalized gradient of the step function approximation n= H / H. It still remains to determine the relationship between the fracture energies g(λ, B) computed in the volume of the gradient zone and the surface energy of the crack. The basic properties of Dirac s delta function provide the necessary mathematical tools [36]. The surface area of a crack enclosed in a volume v and defined by signed distance function f α can be obtained as: S ( ) v = δ D dv v f α (44) where δ D ( f α ) is the Dirac s delta function of the signed distance function. Equation (44)is the three-dimensional extension of (11). For an arbitrary continuous function, g(x), defined in a volume v, a relationship between the surface integral over the crack surface Γ α (Γ α v) and a volume integral exists so that Γα v g( x) ds = g( x) δ ( f ) dv v D α (45) 34

54 This relationship can be established by applying equation (44) in small adjoining volumes encompassing surface Γ α to develop the integral sums representing the left- and right-hand sides of (45). In the case of the regularized formulation, the approximate value of the right hand side is computed by replacing δ ( ) by the magnitude of the gradient of the approximate step D f α function H. As described previously in equations (14) and (15), the function H may be regarded as the regularized form of δ ( ). D f α The continuous function g(x) defined over the volume is replaced with the pointwise cohesive energy of crack opening, which depends implicitly upon the spatial coordinate as a function of the separation, λ, and the mode mixity, B. Thus the fracture energy required for crack surface opening within the arbitrary subvolume v is equivalent in the regularized and conventional crack surface formulations and can be calculated from: v g( λ, B) δ ( f ) dv g( λ, B) H dv (46) D α v Note that the volume v must be sufficiently large with respect to the mesh size for the crack surface area computed by using the step function gradient to be equal to the exact crack surface area. Thus, equation (46) is valid when all elements with a non-zero gradient of the regularized step function are included in the fracture energy computation. Since the right hand side integral consists exclusively of continuous functions, its integration within an element may be calculated using standard numerical integration methods such as Gauss quadrature. 35

55 To summarize, the fracture energy of a MIC has been evaluated using the regularized formulation. Over the set of all elements within the ply, the fracture energy takes the form Nel α M = g( λ, B) H dω l= 1 Ωl (47) Here the superscript α denotes a MIC defined by the signed distance function of the surface Γ α. Delaminations and their Interaction with Mesh Independent Cracks The delamination between the two plies in a laminate is modeled using the cohesive zone method outlined above. Consider the interface between plies n and n + 1, where the enriched displacement approximation (18) is used. The displacement jump between the two plies is then computed as: Δ u = H u + (1 H ) u H u (1 H n ) u (1) (2) (1) (2) n+ 1 n+ 1 n+ 1 n+ 1 n n n (48) which can be evaluated at an arbitrary point on the surface between the two plies (the subscript denotes the ply number). The fracture energy of a delamination between the two plies is computed by integrating the pointwise fracture energy (34) over the area of the interface between the plies Φ = n z= zn g( λ, B) ds (49) The minimum potential energy principle for deriving the equilibrium equations for an N-layer laminate, which contains cohesive zones between the plies and MIC in each ply, can now be expressed as 36

56 N N 1 α δ Wn + Mn + Φn A = 0 n= 1 α n= 1 (50) The summation includes the strain energy of each ply, W n, the fracture energy of all MICs, and the delamination fracture energy between plies. W n is calculated from the summation over all the elements in the ply. As outlined previously, the regularized formulation allows the computation of all quantities in equation (50) by Gauss integration over the entire element domain. As a result, a system of equations for computing the enriched displacement approximations in each ply can be obtained without introducing any special partitions or integration rules and without regard to the mesh orientation in the plies. Implementation This method is implemented within an existing software framework, BSAM (B-Spline Analysis Method) previously developed by Iarve and others [45] for composite analysis. The element chosen for development is a three-dimensional eight node isoparametric element with full integration. For details see [53]. The phantom degrees of freedom may be introduced into the model in different ways. Here, the following method is employed. First, for a crack defined by the surface Γ α, the signed distance function is calculated at all nodes. If f α is less than zero, the node is assigned to zone 1; otherwise it is assigned to zone 2 as shown in Figure 9. All elements are then interrogated; any element which contains multiple zones is designated as a cracked element since the crack surface passes through the element domain. Subsequently, the nodes of these elements are designated as twinned nodes (Figure 10). When a node is twinned, a new node with a new set of degrees of freedom is created at the same location. The twin node takes on the zone number opposite that of the original node. The elements are interrogated one more time (Figure 11). If 37

57 the element contains any twinned nodes it is labeled as an adjacent element if it has not already been included in the original set of cracked elements. All nodes on any adjacent element are also twinned. This method guarantees that the shape functions of any node which contributes to the crack zone are included in all volume integrals relating to the crack. nodes assigned to zone 1 Γ α nodes assigned to zone 2 n f α > 0 f α < 0 Figure 9. Introduction of a MIC Step 1, Determining Zones Nodes assigned to zone 1 Γ α Nodes assigned to zone 2 n Cracked elements f α > 0 Twinned nodes (2 nd ) f α < 0 Figure 10. Introduction of a MIC Step 2, Determining Cracked Elements 38

58 Nodes assigned to zone 1 Γ α Nodes assigned to zone 2 n Cracked elements f α > 0 Twinned nodes (2 nd ) Adjacent Elements f α < 0 Twinned Nodes (3 rd ) Figure 11. Introduction of a MIC Step 3, Determining Adjacent Elements Each cracked and adjacent element is twinned by replacing it by two sub-elements, one for each of the zones. The stiffness of each element twin is calculated in accordance with (23) and (24). The connectivity of each element twin is established by the zone numbers on the nodes of the original elements. Zone 1 element twins connect the degrees of freedom of zone 1 nodes and zone 1 node twins. Zone 2 element twins connect to zone 2 nodes and zone 2 node twins. This method of creating the additional degrees of freedom helps reduce calculations and conserve memory, since twins always inherit properties from the original nodes and are only created when needed. For details of the implementation within the commercial FEA software ABAQUS see Appendix II. All routines are written and compiled in Intel Visual Fortran and make extensive use of Intel s Math Kernel Library (MKL). MKL provides a parallelized sparse solver PARDISO as well as a set of linear algebra libraries (LAPACK) to handle matrices. The subsequent analyses have been performed on various PC platforms using Microsoft Windows XP, Vista and 7. The software has also been compiled for UNIX. 39

59 For pre-processing the commercial finite element software, ABAQUS/CAE is used to create the finite element meshes for the solid models. The mesh is then imported into the code as text-based input. Routines to perform simple manipulations of the mesh (scaling, rotation, offsets) are also incorporated within BSAM. For output, the displacement field is calculated from (18). Strains and stresses are calculated at the integration points by (21) and (22) respectively and then are extrapolated to the nodes and averaged over elements. The regularized step function (13) is reported at the nodes as va1. The damage variables associated with MIC and delamination are va2 and va4 respectively. All post-processing of results is performed in Tecplot 360, a commercial data visualization software. Failure of the cohesive zone associated with either delamination or cracking is considered to have occurred when the damage variable exceeds In the next two chapters Rx-FEM is applied to problems involving the initiation and growth of cracks. Rx-FEM is shown to be able to reproduce the results of CZM with interface elements for both single-mode crack growth and more complex crack conditions. The effects of mesh size and cohesive zone parameters are explored. Finally, the method is used to simulate failure of openhole laminated composites. 40

60 CHAPTER IV VERIFICATION In this chapter, the Rx-FEM approach is applied to two benchmark problems. The first problem is the failure of a simple single crack Double Cantilever Beam (DCB) specimen. The DCB specimen has been used previously to verify the CZM with interface elements by comparison to experimental results [43]. This problem establishes that Rx-FEM can reproduce the results of conventional interface elements and explores the sensitivity of the results to the parameters of the cohesive zone model. The second problem involves a double cantilever beam specimen with two offset cracks. This specimen has been used as a benchmark for multi-mode fracture [30] and confirms the ability of Rx-FEM to model crack growth in multiple fracture modes. Double Cantilever Beam Problem Description To compare the Rx-FEM approach to interface-based Cohesive Zone Models and investigate the effects of mesh refinement and separation-traction parameters on the solution, a double cantilever beam (DCB) model has been constructed (Figure 12). The DCB configuration has the advantage of exhibiting stable crack growth behavior. 41

61 Figure 12. DCB Specimen The material and dimensions for the DCB model follow that described by Turon [43]. Mechanical properties are shown in Table 1. For all cases the critical energy release rate is N/mm. The other parameters for the cohesive law are listed by case in Table 2. Table 1. Material Properties for DCB Analysis Material Property Value E GPa E 22 = E GPa G 12 = G GPa G GPa ν 12 = ν ν G IC N/mm Test Case Nominal Element Size, L e Table 2 Test Cases for DCB Analysis Penalty Stiffness, K Interfacial Strength, τ 0 G e /G c Peak Load (mm) (N/mm) (N/mm2) (-) (N) TD-Analytic TD E E TD E E TD E E TD E E TD E E TD E E TD E E TD-16-A E E TD-16-B E E TD-16-C E E

62 The penalty stiffness coefficient is investigated over six orders of magnitude, and several values for the maximum traction have been evaluated. The cases include four different unstructured meshes with nominal element sizes of 1 mm to mm in the crack zone as shown in Table 2. The effect of mesh refinement on the regularized step function is shown in Figure 13. Calculation of the cohesive zone size from (40) using the minor modulus, E 22, predict l cz = 1 mm. Thus the nominal element sizes, L e, in this study range from 1/8 l cz to 1 l cz. (a) (b) (c) (d) Figure 13. Mesh Near Crack Tip for DCB Model (a) L e = mm, (b) L e = mm, (c) L e = mm, (d) L e = mm Double Cantilever Beam LEFM Solution A solution based on Linear Elastic Fracture Mechanics (LEFM) techniques can be found following the compliance method [52]. The crack growth for the DCB specimen is calculated using the Griffith criterion for crack extension, 43

63 U a WG c (51) where W is the crack width, and the strain energy U is determined through the work done by the external forces (see Figure 12). Defining the spring constant k as the ratio of the resultant load, f, to the total opening of the DCB, d, at a given crack length, a, the internal energy is simply: U ( ) 2 = 1 k a d (52) 2 Where k is a function of the crack length. Substituting (52) into (51) and solving for the critical displacement at which crack extension occurs: d crit = k a 2WGc (53) What is left is to determine the dependency of the stiffness on the crack length. Solutions exist in the literature for the DCB test [46], which are based on corrected beam theory (CBT). These corrections take into account the stiffening of loading blocks and non-linear geometric effects and are appropriate for experimental work [47]. Here the focus is on comparisons with other numerical techniques. A series of increasing crack length FE models using the nominal mm element size mesh provides the basis for determining the stiffness as a function of crack length. From the models, the stiffness of the DCB is determined. A quadratic function (54) is fit to the stiffness as shown in Figure 14. The coefficients for the equation are listed in Table 3. ( ) 2 k a = k a + k a+ k (54)

64 Table 3. Coefficients for DCB Stiffness as a Function of Crack Length Term Value k E+01 N/mm k E+00 N/mm 2 k E-02 N/mm Stiffness, K (N/mm) y = 1.18E-02x E+00x E+01 R² = 1.00E Crack Length, a Figure 14. DCB Stiffness as a Function of Crack Length for LEFM Model 45

65 Double Cantilever Beam Results Figure 15 shows a comparison between the load-displacement relationship from Rx-FEM and the LEFM solution. Mesh refinement clearly has a profound effect on the size of the cohesive damage zone. As the mesh is refined both the size of the cohesive zone for a given displacement (Figure 15) and the peak loads decrease (Table 2). The response after peak load, however, is not as sensitive to mesh refinement. Figure 15. Mesh Size Dependence of Rx-FEM DCB Model Compared to LEFM Solution 46

66 Nominal Element Size (mm) Cohesive Damage Zone le = le = le = le = Figure 16. Effects of Mesh Refinement on Cohesive Zone at a Displacement of 4.6 mm 47

67 The cohesive traction strength, τ 0, also affects the peak load. Increasing interfacial strength above 50 MPa increases the peak load while reducing the size and intensity of the cohesive zone for a given displacement (Figure 19). In contrast lowering the strength below 60 MPa only slightly softens the load-deflection response. Beyond the peak load, the response is largely independent of the choice of cohesive strength. Figure 17. Dependence of DCB Response on the Maximum Cohesive Traction 48

68 Interfacial Strength MPa Cohesive Damage Zone τ0 = 30 τ0 = 60 τ0 = 90 τ0 = 120 Figure 18. Effects of Interfacial Strength on Cohesive Zone at a Displacement of 4.6 mm 49

69 The penalty stiffness parameter has little impact on the peak load over eight orders of magnitude. A slight reduction in peak load and post peak-load behavior can be seen (Figure 19). However, it has a profound effect on the size of the cohesive zone (Figure 20) and the numerical stability. From 3.36 mm to 8 mm displacement, a total of 524 Newton-Raphson iterations are required when K = 1.00E+04 N/mm. When K = 1.00E+12 N/mm, 2,311 iterations are required over the same displacement range. Figure 19. Dependence of DCB Response on Penalty Parameter 50

70 Penalty Stiffness (N/mm) Cohesive Damage Zone K = 1.00 E+04 K = 1.00 E+06 K = 1.00 E+09 K = 1.00 E+12 Figure 20. Effects of Penalty Stiffness Coefficient on Cohesive Zone at a Displacement of 4.6 mm 51

71 Double Cantilever Beam Discussion Several aspects of the cohesive zone model warrant explanation. First, even with fairly small elements, the model tends to over-predict the initiation of crack growth compared to LEFM results. This may be attributed to the size of the cohesive zone which effectively blunts the crack and reduces the energy release rate. We can deduce, however, that the mm mesh density is close to the converged solution for the simple reason that the post peak behavior for all meshes is very similar. When the post-peak load-displacement is extrapolated backwards, the line intersects the loading curve at nearly the same load for all cases. Second, Wang [40] notes that to achieve an equivalence of CZM with Linear Elastic Fracture Mechanics (LEFM) the cohesive zone must be very small. The most direct way of attaining a smaller cohesive zone is to increase the maximum traction (equation (40)). Unfortunately, as is evident in Figure 17, for a given mesh density, artificially increasing the cohesive traction delays the initiation of failure and leads to higher calculated peak loads. As noted above, without a sufficient mesh density, the cohesive zone is not properly captured and the result is an overprediction of peak load. The fact that the 30 MPa and 60 MPa traction cases yield similar results indicates that the mesh density is sufficient for their relatively larger cohesive zones. Finally, it may appear that the lower penalty stiffness coefficient reduces the size of the cohesive zone around the crack tip. However, the cohesive zone size is not determined by K, but from the other parameters as equation (40) shows. The penalty stiffness parameter does change the damage characteristics within the zone. This is simply because more of the material in front of the crack tip remains in the elastic portion of the traction-separation region up to failure. 52

72 Multiple Crack DCB The intended application of the Rx-FEM is modeling of multiple cracking and therefore a two crack model example is selected for validation. This example has been considered by several authors for verification of cohesive zone [30] and automated virtual crack closure techniques (VCCT) [42]. (a) (b) Figure 21. The Two-Crack Problem Schematics (a) and the Mesh Around the Crack Tip (b) The specimen geometry is shown in Figure 21(a). Two initial cracks are placed in the beam, offset in the ply stack-up direction by two layers. The meshes around the crack tips are shown in Figure 21(b). The meshes in the vicinity of the cracks are unstructured; however, due to high aspect ratio of the sample the mesh is aligned with the crack direction over most of the length. Mesh independent cracks, closed, are placed at the tip of each of the initial cracks. For each 53

73 crack the signed distance function is predefined and the respective elements enriched by node duplication. The load is applied by prescribing vertical displacement of equal magnitude and opposite direction to the left hand side arms of the double cantilever specimen. Figure 22 illustrates the simulated damage progression, with elements containing fully opened cracks shown in red. The load displacement curve predicted by Rx-FEM and the experimental data [30] are shown on Figure 23. Both the cracking sequence and the load displacement response predicted by Rx-FEM match the experimental data well. A double peak load drop associated with the second crack propagation initiation is not observed in the experiment; however, it has been predicted both by conventional cohesive zone models [30] as well as VCCT [42]. It is therefore concluded that the MIC modeling of Rx-FEM simulates multi-crack propagation scenarios at the level of accuracy of the conventional techniques where cracks are meshed explicitly. Figure 22. Two Crack Model Failure Progression 54

74 Figure 23. Load Displacement Relationship for the Two Crack Problem 55

75 CHAPTER V APPLICATION OF THE METHOD TO OPEN HOLE LAMINATES The numerical results presented below address one of the key areas of concern in application of composite materials, namely the strength of laminated composites containing holes. Tensile strength scaling effects in quasi-isotropic laminates [45 m /90 m /-45 m /0 m ] ns have been studied experimentally and analytically [48, 49], for different combinations of the number of blocked plies, m, and ply groups (parameter n-number of sublaminates) along with the in-plane dimension scaling. The plate width, W, to the hole diameter, D, ratio is kept constant W/D = 5 as is the ratio of length, L, (distance between the grips) to the hole diameter, L/D = 20. With increasing numbers of blocked plies, a change of failure mode from fiber failure to delamination failure is observed. In the case of single ply blocks, m = 1, the fiber failure mode determines the laminate s strength and the well-known trend of strength decrease with increasing hole diameter is observed [48]. For high blocked ply numbers (m = 4, 8) specimens delaminate prior to fiber failure, resulting in load drops followed by overall stiffness reduction, visual detection of specimen failure, and loss of integrity. It is observed [48] that the delamination failure in specimens with smaller hole sizes occur at lower applied stress than in ones with larger hole diameter and hence the behavior of the high blocked ply number specimens requires explanation [48]. The proposed Rx-FEM technique is applied to predict the failure in these specimens. 56

76 Figure 24 Scaled Geometry of Hole Size Effect Study Problem Statement and Boundary Conditions The open hole coupon shown in Figure 24 is considered. Tensile loading in the x-direction is applied by incrementing the displacement, u x, at the edges x = 0, L, so that 1 ( 0,, ) ( 0,, ) u y z = u y z Δ (55) i i i x x and i x i 1 (,, ) (,, ) i u Lyz = u Lyz +Δ (56) x where Δ i is a constant and i is the loading step number. This incremental formulation is required to properly account for the thermal curing stresses prior to mechanical loading. The displacement field, u 0 x, appearing in equations (55) and (56) is computed by solving a thermal-mechanical expansion problem under boundary conditions which simulate free expansion and only restrict rigid body motion, i.e. 57

77 u u u u 0 x 0 x 0 y 0 z ( ) ( W ) ( W ) ( x y ) 0,0,0 = 0 0,,0 = 0 0,,0 = 0,,0 = 0 (57) The incremental loading boundary conditions, equations (55) and (56), are supplemented at the lateral edges x = 0 and L, so that i y i y 0 ( 0,, ) = y( 0,, ) 0 (,, ) (,, u y z u y z u L y z = u L y z) y (58) i z i z ( 0,, ) = 0 z ( 0,, ) (,, ) = 0 (,, ) u y z u y z u Lyz u Lyz z (59) Unidirectional ply properties used for analysis are shown in Table 4 and include the stiffness as well as strength properties. Table 4. Unidirectional Stiffness and Strength Properties for IM7/8552 E GPa E GPa G GPa G GPa ν 12 = ν ν Y t 76 MPa S 90 MPa Y c 275 MPa α E-07 1/C α 22 = α E-05 1/C ΔT -150 C The ply level strength properties are used only for MIC initiation. Crack initiation is determined by checking the ply level failure criterion, LARC03, at each integration point. If the criteria is 58

78 met or exceeded, a closed MIC is inserted into the model. The MIC is placed with normal in the plane of the lamina and perpendicular to the fiber direction. Propagation of MICs is governed by the cohesive law (26), where all input parameters are listed in Table 5 and the initial strength values are equal to that of the unidirectional ply. Table 5. Properties Used in Cohesive Law K 2.71E+08 N / mm 3 G 1c N / mm G 2c N / mm The same parameters were used for delamination propagation as well. Comparison with Experimental Failure Loads and Damage Patterns Two unstructured meshes are used in this study: a coarse mesh with 32 elements around the perimeter of the hole and a fine mesh with 80 elements around the hole, as shown in Figure 25. The software allows these meshes to be scaled in any dimension; thus two mesh configurations are employed for each specimen in the study of scaling effects on tensile strength of open-hole laminates. 59

79 (a) (b) Figure 25. Mesh Configuration for Tensile Strength Scaling Studies (a) coarse, (b) fine CASE Table 6. Failure Stress for Various Laminates Subject to Delamination Failure No. Blocked Plies Ply Thickness Overall thickness Hole Diameter m T ply (mm) T (mm) D (mm) Experiment Failure Stress (MPa) Coarse C B C C C C C C D D Fine F 60

80 Because of the large gradients in mesh size within the models, the crack spacing is set to approximately half the hole diameter, although for case C4C (see Table 6) the allowable crack spacing is ¼ of the hole diameter. Furthermore, because the number of degrees of freedom in the model increase rapidly as MICs are introduced, an upper limit of ten cracks per ply has been permitted in the simulation. Table 6 lists the summary of the results of the simulations and the experimental results reported by Wisnom and Hallett [48]. In this study only a subset of data presented in Table 6 is considered. All selected configurations fail in the delamination mode. These cases are one sublaminate, n = 1, and blocked ply cases with m 2. The table is organized in the order of increasing ply block number (ply thickness) and increasing hole diameter. The delamination failure mode does not correspond to a complete loss of load-carrying capacity; rather, it indicates a disintegration of the specimen, where some load can still be carried by the plies with fibers oriented in the loading direction. Average Traction (MPa) mm, Coarse Mesh 6.35 mm, Coarse Mesh 12.7 mm, Coarse Mesh 25.4 mm, Coarse Mesh Strain (mm/mm) Figure 26. Effects of Hole Size on Average Traction - Elongation Curves 61

81 The load-displacement curves predicted with the coarse mesh for four hole size dimensions (3.175mm, 6.35mm, 12.7mm and 25.4mm) for m = 4 are shown in Figure 26. It is evident that the load drop experienced by the large hole-size specimens occurs at higher applied stress levels and is consistent with experimental data. The comparison with experimental data is shown in Figure 27 both for the coarse and fine mesh analysis results. The analysis shows the correct trends for the delamination failure loads for both the coarse and fine mesh analysis. The maximum relative error for the coarse mesh varies from 8 to 12 percent for all hole sizes, whereas the fine mesh produced results within 5 percent for the case 3 and case 4 hole sizes and an 11 percent error for the 12.5mm diameter hole specimen. Similarly, the model-predicted ply thickness effect agrees well with the experimental values as shown in Figure 28. Figure 27. Delamination Failure Strength for Specimens with Thick Plies 62

82 Figure 28. Effects of Ply Thickness on Failure Stress for Rx-FEM Compared to Experiment In addition to failure stress calculations, the failure progression sequence is investigated. The results obtained with the fine mesh configuration for case C3 with m = 4 and D = 6.35mm are discussed below. The load-displacement curve for this case is shown in Figure 29. Eight load levels labeled (a)-(h) are selected, and the respective damage patterns are shown in Though the analysis is three dimensional, all damage events are superimposed as in a radiograph, and drawn on the undeformed specimen. These damage patterns are compared to the experimental radiograph at several load levels. The thin black lines represent matrix cracks in the various plies which retain no more than a small portion of the original strength of the material (damage parameter d > 0.995). The red zones are areas of delamination between the surface (+45 ) and the 90 ply. Delamination between the 90 and -45 plies is shown in green, and delamination between the -45 and 0 plies is shown in blue. The load levels (a)-(d) correspond to gradual damage accumulation, primarily in the form of matrix cracking. The amount of cracking in the 63

83 radiographs is slightly larger in all cases and the length of splitting in the 0 0 ply (vertical direction, coinciding with the loading direction) is longer as well. With increasing load, delaminations begin to form at the edge of the hole, as seen in Figure 30 (d) (e), (f), (g), (h) Average Traction (MPa) (a) (b) (c) (d) Strain Figure 29. Load Displacement Curve for the Case C3 (m=4 and D=6.35 mm) 64

84 (a) (b) (c) (d) (e) (f) (g) (h) Figure 30. Damage Progression Sequence for the Case C3 (m=4 and D=6.35) at Load Levels (a) 107MPa, (b) 162Mpa, (c) 223MPa, (d) 253MPa, (e) 300MPa, (f) 304MPa, (g) 306MPa, (h) 304MPa 65

85 Further increase of the load causes delamination initiation at the free edge of the specimen (Figure 30(e)). Delamination from the free edges then links up with the delamination at the edge of the hole, Figure 30 (f) and (g). At these load levels, the interface between the -45 and 0 ply begins to delaminate and the delamination quickly extends through large portions of the specimen (Figure 30 (h)), causing the load drop and the associated disintegration of the specimen despite some residual load-carrying capacity in the 0 ply. The predicted damage patterns at peak load agree well with CT images of cracking and delamination from experiment as shown in Figure 31. This sequence of events is consistent with experimental observations [48]. Failure occurs when delamination spreading from the edge of the hole links up with delamination from the free edge. Thus, the failure strength is related to the width of the ligament, W D, as opposed to simply the ratio W/D. Figure 31. CT Images of Damage Patterns Compared to Predicted Damage at Peak Load 66