DELAMINATION MODELLING AND TOUGHENING MECHANISMS OF A WOVEN FABRIC COMPOSITE

Transcription

1 DELAMINATION MODELLING AND TOUGHENING MECHANISMS OF A WOVEN FABRIC COMPOSITE Tadayoshi Yamanaka Department of Mechanical Engineering McGill University, Montreal February 2011 A thesis submitted to McGill University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Tadayoshi Yamanaka, 2011

2 Imagination is more important than knowledge. Albert Einstein( ) ii

3 Abstract Efficient and accurate numerical simulation methods for the damage tolerance analysis and fatigue life prediction of fibre reinforced polymers are in high demand in industry. Problems arise in the development of such a simulation method due to the limitations from numerical methods, i.e., delamination modelling, and understanding of damage mechanism of woven fabric composites. In order to provide effective and accurate delamination modelling, a new crack modelling method by using the finite element method is proposed in this study. The proposed method does not require additional degrees-of-freedom in order to model newly created crack/delamination surfaces. The accuracy of delamination growth simulation by the proposed method and that of a commercial FEA package are in good agreement. The damage mechanisms of five harness satin weave fabric composite is studied by creating a multiscale finite element model of a double cantilever beam specimen. The weft and warp yarns, where the gaps are filled with matrix, are individually modeled. Cohesive zone model elements are pre-located within the matrix and interfaces of matrix-yarns and weft-yarns and warp yarns. These meso-scale parts are bonded with homogeneous parts that are used to model regions where no damage is expected. This constitutes a multiscale model of a DCB specimen. The simulation results are in good agreement with the lower bound of experimental results. The toughening mechanism contributed from the weave structure was revealed. This study contributes to knowledge by introducing crack modelling methods and by providing more information in order to understand damage mechanisms of 5HS weave fabric composite laminates during delamination growth. iii

4 Résumé Les méthodes de simulation numériques efficaces et exactes pour l'analyse de l endommagement et la prédiction de vie en fatigue des matériaux composites sont essentielles pour l'industrie. Les problèmes surviennent dans le développement d'une telle méthode de simulation en raison des restrictions des méthodes numériques, c'est-à-dire, modélisation de la délamination et compréhension des mécanismes de rupture de composites à base de fibres tissées. Pour developer un modèle de délamination efficace et précis, une nouvelle méthode est proposée dans cette étude en utilisant la method des éléments finis. La méthode proposée n'exige pas de degrés-de-liberté supplémentaires pour créer de nouvelles sufaces de fissures/ délaminations. Le résultat de simulation de délamination par la méthode proposée est comparé avec un logiciel d'éléments finis commercial, et les résultats se comparent bien. Les mécanismes d endommagement d un composite tissé typique five-harness satin sont le sujet d une étude. Ceci est fait en créant un modèle d'éléments finis méso-échelle en utilisant l exemple d un spécimen d essais Mode 1 (spécimen DCB). Le tissu est modélisé avec les trajectoires exactes des fibres dans les deux directions, et les espaces entre les fibres sont remplis de la matrice. Des éléments cohésifs sont insérés entre la matrice et les interfaces des fibres. Les composants méso-échelles sont joints avec des parties homogènes qui sont utilisées pour modéliser des régions où aucun endommagement n'est prévu. La combinaison des ces parties constitue un modèle multiéchelle d'un spécimen DCB. Les résultats de simulation d un essai sont en accord avec les résultats expérimentaux, du côté conservateur. Le mécanisme renforçant des ultant du type de tissage a été démontré. iv

5 Cette étude contribue à la science en présentant de nouvelles méthodes pour modéliser les fissures et pour comprendre les mécanismes d endommagement des composites tissés pendant la croissance des délaminations. v

6 Acknowledgement Support for CRIAQ Project 1.15: Optimized Design of Composite Parts was provided by Bell Helicopter Textron Canada, the National Research Council of Canada (Aerospace Manufacturing Technology Centre, Institute for Aerospace Research), Delastek Inc., McGill University, École Polytechnique de Montréal, the Natural Sciences and Engineering Research Council of Canada (NSERC), and the Consortium for Research and Innovation in Aerospace in Quebec (CRIAQ). The McGill Composite Material and Structures Laboratory is a member of the Centre for Applied Research on Polymers and Composites (CREPEC). I would like to sincerely thank my supervisor, Prof. Larry Lessard (McGill University), for his kind support. He gave me many invaluable opportunities throughout the Ph.D. program. I would like to acknowledge Victor Feret for his experimental investigation on mode I delamination of the double cantilever beam specimen. His test results inspired me to work on the multiscale analysis. I would like to thank Steven Roy for his testing on a practical application of five harness satin weave carbon fibre fabric composite and discussions on the damage behaviour. His work certainly gave me valuable information to decide the sub topics of my work. I am grateful to Vahid Mirjalili for discussions on the general aspects of fracture mechanics. Finally, I would like to thank all my colleagues in the McGill Composite Materials and Structures Laboratory for their support throughout my studies. vi

7 Table of Contents Abstract..... iii Résumé...iv Acknowledgement..vi 1 Introduction Crack modelling method (ADD-FEM) Introduction Formulation Problem statement Formulation details Elemental level tests Rigid body motions (zero strains) Constant strains over the elements Linear strains Selection of constraint equations Numerical examples Mesh L2-norm error distribution Stress distribution Convergence of Strain Energy Release Rate (SERR) Delamination growth simulation Conclusions Multiscale finite element analysis of a double cantilever beam specimen made of five harness satin weave fabric composite Introduction Failure behaviour of five harness satin weave carbon fibre fabric composite vii

8 3.1.2 Multiscale finite element analyses Hypothesis from experimental results R- curves in delamination growth simulation Summary Meso-scale parts of a multiscale FE model TexGen Creating geometric models and meshing Material properties Element length, DCB specimen size, and contact stiffness Solution procedure Numerical results Comparison with experiments Energy released by CZM elements Toughening mechanisms Discussions Future work Concluding remarks Contributions of this thesis Future work Reference 113 viii

9 Table of Tables Table 2.1. Crack tip displacement field for Mode I and Mode II [38] Table 3.1. DCB specimen dimensions [22] Table 3.2. Ratios of experimental results to FEA results Table 3.3. TexGen input values used to create unit cell of 5HS weave fabric Table 3.4. Dimensions of multiscale 5HS weave fabric composite DCB model Table 3.5. Material properties of cured epoxy resin [63] Table 3.6. Elastic constants of carbon fibre [64] Table 3.7. Fibre volume fractions Table 3.8. Geometries and volumes of 5HS weave fabric s unit cell Table 3.9. Material properties of 5HS weave carbon fibre fabric composite calculated at F.V.F.= Table Material properties of 5HS weave fabric composite at F.V.F.=0.55. [63] Table Number of iterations for various contact stiffnesses Table Numbers of elements for the element types ix

10 Table of Figures Figure 2-1. Finite element model containing a crack Figure 2-2. Local node numbering and the corresponding coordinates (in parentheses) based on the natural coordinate system for the Q4 element Figure 2-3. Elemental level test models and their global node numbering and the corresponding coordinates in the global coordinate system Figure 2-4. Strain and recovered deformation: (a) Translation in the x-direction; (b) Translation in the y-direction; (c) rigid body rotation with angle Figure 2-5. Strain and recovered deformation: (a) Translation in x-direction; (b) Translation in y-direction; (c) rigid body rotation with angle Figure 2-6. Principal strain and deformation: (a) Uniform load applied in x-direction; (b) Uniform load applied in y-direction; (c) pure shear applied Figure 2-7. Principal strain and deformation: (a) Uniform load applied in x-direction; (b) Uniform load applied in y-direction; (c) pure shear applied Figure 2-8. Examples of strain extrapolation Figure 2-9. An example of FE model with distorted mesh Figure Mesh, Master nodes, Slave nodes and constrained DOFs for a bar under uniform tension Figure A beam under pure bending Figure An infinite space containing through crack at the centre in which is applied far from the region of interest Figure Definition of the polar coordinate system ahead of a crack tip Figure Difference of normalized L2-norm error under uniform tensile loading Figure Difference of normalized L2-norm error under pure bending Figure Difference of normalized L2-norm error under pure Mode I Figure (a) structured mesh; (b) distorted mesh; (c) deformation and slave nodes (in red) for structured mesh; (d) deformation and slave nodes (in red) for distorted mesh. 29 Figure L2-norm error in displacement for three load cases Figure Comparison of stress distributions obtained by ADD-FEM and standard FEM Figure Convergence of strain energy release rate obtained by VCCT for structured mesh Figure Convergence of strain energy release rate obtained by VCCT for distorted mesh Figure Convergence of strain energy release rate obtained by VCCT under mixedmode boundary conditions Figure Double cantilever beam specimen Figure The delamination growth simulation algorithm with ADD-FEM Figure Comparison of load-opening displacement of a DCB specimen Figure 3-1. An example of composite structure made of 5HS weave carbon fibre fabric reinforced epoxy [41] x

11 Figure 3-2. Damage on (a) the bracket on the left and Figure 3-3. Correction factor of modified beam theory [60] Figure 3-4. Mode I critical energy release rates of 5HS weave carbon fibre fabric composite Figure 3-5. R-curve applications Figure 3-6. Load-Displacement curves of 2D plane strain models of DCB specimen Figure 3-7. Load Delamination extension curves of 2D plane strain models of DCB specimen Figure 3-8. Screenshot of TexGen GUI to create 5HS weave fabric Figure 3-9. Created 5HS weave fabric by TexGen Figure Surfaces of 5HS weave fabric model imported to ANSYS Figure Matrix and yarn volumes of a 5HS weave fabric unit cell Figure Yarns embedded within matrix domain Figure Finite element model of a multi-scale 5HS weave carbon fibre fabric composite DCB specimen Figure Warp yarns and weft yarns with their numbering Figure Enlarged meso-scale model used in the FE DCB model Figure Side view of enlarged meso-scale model used in the FE DCB model Figure Contact elements used in the FE DCB model Figure Typical contact elements used with a cohesive law Figure Side view of contact elements used with a cohesive law showing empty space indicating no damage within yarns is assumed Figure Boundary conditions for the FE DCB model Figure Unit cell of unidirectional fibre composite Figure E 11 variation of warp yarn 2 due to the slope of yarn Figure Unit cell of 5HS weave carbon fibre fabric composite for elastic constants calculations Figure Bilinear cohesive law Figure D plane strain long DCB model with element length 0.12mm Figure D plane strain short DCB model with element length 0.12mm Figure Homogeneous 3D short DCB model with contact elements bonding 5 parts together Figure Mode I energy release rates obtained by homogeneous DCB models Figure Ratios of Mode I critical energy release rates Figure Initial linear slope of DCB models with various contact stiffnesses Figure Contact elements coloured in red, blue, and green used in homogeneous 3D short DCB model Figure Load-Displacement curves up to initial stage of delamination growth of homogeneous DCB models with various contact stiffnesses Figure Time-step increment history over the entire simulation of Multi Figure R-curves of the 5HS weave carbon fibre fabric composite DCB specimen and the multiscale FE models xi

12 Figure Load-displacement curves of the 5HS weave carbon fibre fabric composite DCB specimen and the multiscale FE models Figure Released energies of the multiscale FE model of 5HS weave carbon fibre fabric composite DCB and 2D plain-strain homogeneous FE model Figure Delaminated areas at the end of each zone of the multiscale FE model of 5HS weave carbon fibre fabric composite DCB Figure Percentage of released energy by tangential debonding within the total released energy by CZM elements Figure Contour plot of on the yarns at Point A p and delaminated elements coloured in pink Figure Contour of contact pressure of the multiscale FE model and delaminated elements (coloured in pink) Figure Contour of contact pressure of the meso-scale FE model under in-plane tensile loading Figure Contact pressure distribution on at Point A p Figure Contour plot of at Point A p clipped at and delaminated elements (coloured in pink) Figure near delamination front at Point A p and reversed contact stress obtained by in-plane loading Figure Delamination front development during the load drop from Point A p to A b Figure The z-coordinates of delaminated elements showing the branching at Point A b Figure Length of positive from the delamination front at Point A b Figure distribution history from Point A b to B b Figure Delaminated CZM elements from Point A b to B b Figure Weft yarn bridging of Multi2 observed at Point B b Figure of CZM element on the delamination front edge with the delamination front z-coordinates at Point B b Figure Delamination area of Multi2 versus delamination length Figure Delamination area of Multi1 versus delamination length Figure Weft yarn bridging of Multi1 at Point B b xii

13 List of Symbols = crack length =the strain-displacement matrix =width of a DCB specimen =material moduli tensor = compliance = artificial damping = derivative operator matrix = correction factor = Load point displacement = damage parameter in mixed mode = damage parameter in normal direction =Young s modulus = error =small strain tensor =strain field within an element = normal strain energy release rate = tangential strain energy release rate = strain energy release rate in Mode I = strain energy release rate in Mode II = Mode I fracture toughness = Mode II fracture toughness =the boundary of = height = thickness =moment of inertia =global stiffness matrix = normal contact stiffness of cohesive zone model = tangential contact stiffness of cohesive zone model =stress intensity factor for Mode I =stress intensity factor for Mode II =Kolosov constant = length from delamination tip to specified magnitude of =length =Moment =direction vector =shear modulus =shape function =unit normal vector =Poisson s ratio = the domain of a body = the domain of an element = percentage of normalized L2-norm error = Load =force vector xiii

14 and and =field of real numbers =polar coordinate system =Cauchy stress tensor = ultimate tensile strength = ultimate shear strength =tensile stress = current time =prescribed tractions = time interval = released energy =displacement vector =prescribed displacements =displacements at node in the system = critical normal separation = critical tangential separation =displacements at node in the system =global coordinate system = natural coordinate system = width =gradient operator direction of the global coordinate direction of the global coordinate xiv

15 1 Introduction Fibre Reinforced Polymers (FRP) are being used in the various industries. For example, Glass Fibre Reinforced Polymers (GFRP) are used for making wind turbine blades [1-4]. Carbon Fibre Reinforced Polymers (CFRP) are used for making more weight critical components, such as suspensions of formula one cars [5-6] and airframes [7-8]. It has been proven that FRPs have better performance than the other materials in certain applications. Computer Aided Engineering (CAE) is an essential step in the structure analysis, especially in the early design phases. Finite Element Analysis (FEA) is the most widely used method for analyzing the solid structures over other numerical methods. The strength of composite structures is then predicted by using failure criteria. For instance, Maximum stress, Maximum strain, Tsai-Wu, Hill, and Hoffman failure criteria are supported by MSC.Nastran [9]. In most cases, the design of composite structures is finalized by using the failure criteria [10] in order to predict initial damage (First Ply Failure). If one wants to analyze beyond First Ply Failure, progressive damage modelling, which reduces the material moduli upon failure of elements, is available in MSC. Nastran. However, delamination, which is one of the most critical types of failure in composite laminates, is not explicitly modeled by the progressive damage modelling technique, which is based on continuum damage mechanics. Accordingly, it is not very easy to predict the onset and propagation of delamination in a composite structure. One example illustrating the difficulty of prediction is the delaminations that occurred in the stringers of wings and centre wing box made of CFRP on a Boeing 787 [8, 11]. Due to this delamination damage and resulting redesign of the structures, the development of Boeing 787 has been further delayed. 1

16 Some aspects of delamination analysis are provided by commercial software. Virtual Crack Closure Technique (VCCT) is available in ABAQUS [12] and MSC.Nastran. VCCT is used for calculating the energy release rate at a delamination tip and requires existing delaminations. Thus, it is used for damage tolerance analysis and propagation analysis of existing delaminations. Cohesive zone models are available in ANSYS [13], ABAQUS and MSC.Nastran. This feature can predict the onset of delamination by using the out-of-plane ultimate strength of composite laminates. Both approaches, however, require advance specification of delamination growth path in order to analyze the propagation. This determination artificially limits the flexibility of the delamination growth path regardless of the criteria that are used. Accordingly, FE model may require a very large number of elements in order to give maximum flexibility for delamination growth or require experience in order to guess the location of the potential regions [14-16]. Neither the VCCT for ABAQUS nor the cohesive zone model is very efficient or useful in optimizing the design of a structure. In order to overcome the difficulties caused by modelling of delaminations, a crack modelling method is proposed in Chapter 2. A new delamination modelling technique is developed in Chapter 2. However, in order to use it for a practical case, it is necessary to understand the failure mechanism and environment in which the actual application is used. For composite laminates, delaminations may occur as a result of cyclic loadings. For example, wind turbine blades will experience more than 10 8 load cycles in the lifetime of 20 years, according to Ref. [1]. Fatigue life predictions based on FEA are conducted by [2-3] for a wind turbine blade and [17] for a tail cone exhaust structure. The S-N curves of the sample coupons and stresses obtained by FEA are used to predict the fatigue life [2]. This method does not consider the stress re-distributions due to accumulated damages by cyclic loadings. Progressive 2

17 fatigue failure analyses based on continuum damage mechanics are applied by [3, 17]. As continuum damage mechanics do not explicitly consider delaminations, this approach may not be very suitable for some structures that are subjected to the loadings causing delamination damage, e.g., the suspension system of a Formula 1 car[6]. To overcome this limitation within continuum damage mechanics, the idea of using a cohesive zone model for delamination onset and propagation due to cyclic loadings is suggested by [18-21]. The cohesive zone model for fatigue failure is developed based on the Double Cantilever Beam (DCB) specimen because it is used for the fatigue testing of composite laminates. In order to use the cohesive zone model validated by DCB specimens and to provide better correlation with experiments, it is necessary to understand the damage mechanisms of DCB specimens. This is very important for some types of composite laminates that have very complex failure mechanism. The complex failure mechanisms are believed to contribute to the increasing Resistance curve (R-curve). For example, it is reported that 5 harness satin weave fabric composite has toughening up to certain crack extension [22-23]. The analyses of toughening mechanisms were conducted by post experimental observations. Analyzing the damage mechanism by only experimental observations may not suffice because it is not very easy to visualize the internal damage development and the stress/strain distributions within the DCB specimen during the test. To reveal the damage and toughening mechanisms of a DCB specimen made using five harness satin weave fabric composite under static loading, which is essential to the development of cohesive zone model for fatigue loadings, the delamination growth is simulated by FEA in Chapter 3. The technology required by industries is an efficient and accurate damage prediction capability under static and fatigue loadings, as clearly stated by the author of Ref. [6]. This study partially contributes to knowledge by introducing crack modelling methods and by providing more information in order to better 3

18 understand the damage mechanisms of five harness satin weave fabric composite laminates during delamination growth under static loadings. 2 Crack modelling method (ADD-FEM) 2.1 Introduction Damage tolerance analyses and fatigue life simulations are an important topic for researchers and engineers. At the same time, Finite Element Methods (FEM) are the most widely used numerical method for solving structural applications for design. Commercial FEA software packages, e.g., ANSYS, ABAQUS, MSC.Nastran, have crack modelling features which require a user to specify the possible crack propagation path by inserting interface/contact elements. This process can take significant amounts of time to create a FE model if the possible crack propagation path is complex and/or the model itself is complex. This is because the entire structure should be divided into two or more components and interface/contact elements must be inserted at the interfaces of components. If one wants to add maximum flexibility for the possible crack growth path by inserting interface/contact elements, there will be the following issues: 1. A very high number of components (unmeshed volumes) is required. 2. The modelling time for inserting interface/contact elements could be high. 3. The Newton-Raphson method for nonlinear analysis with a cohesive law is not guaranteed to converge, depending on pre-defined crack paths. In addition to these issues, a Cohesive Zone Modelling (CZM) element requires a small enough element length depending on the materials used [24]. If the initial model ends up with convergence difficulties during the crack propagation analysis, one will need to revise and re-do the modelling again until one achieves 4

19 a successful result. Due to all the above reasons, crack modelling methods that do not require extensive modification of the geometric model are very attractive for damage tolerance analysis and/or fatigue life simulation. Performing fatigue life simulation at the design phase may reduce the risk of redesigning without full-scale model experiments, which consequently reduces the cost of development. In the past decades, the strong discontinuity approach has been popular for solving crack propagation problems by FEM. This approach is capable of containing a crack, i.e., strong discontinuity, within an element. Consequently, crack propagation analysis by this approach will result in less remeshing during the solution phase and less modifications of the geometric model. One of these approaches is called extended FEM (XFEM), which can model a crack within an element and enriches the singularity field near a crack tip with additional degrees-of-freedom (DOFs), as found in Ref. [25-26]. Embedded FEM (EFEM) is another method that can model a crack within an element by strain softening with a jump parameter, such as in Ref. [27-29]. The additional jump parameter will be condensed before assembling the global stiffness matrix. Therefore, there are no additional DOFs to model a discontinuity for the purposes of modelling a crack. The drawback of EFEM is the lack of ability for modelling the crack tip. The fracture problem considered by linear elastic fracture mechanics does not have crack tip opening displacement. However, it is not possible to prevent the crack tip opening displacement by using EFEM. This drawback limits the crack propagation criterion that can be used with EFEM. Boundary element methods (BEM) can also deal with crack propagation analysis [30]. Although the BEM provides better solution accuracy compared to FEM for same level of discretization, the displacements, stresses, and strains at internal points by BEM require Gaussian integrations over all the boundary elements [31]. When there is no initial crack inserted and stresses are used to find the location of crack nucleation, BEM will certainly require time consuming Gaussian 5

20 integrations over the boundary elements many times. It is stated by the authors [31] that If, however, the solution is required throughout the domain of the body, the FEM program, for a given level of solution accuracy, runs faster than the BEM program. Accordingly, the BEM is not very suitable for crack growth analysis without any initial crack inserted, which consequently requires a criterion based on internal stresses/strains to predict crack nucleation. In addition to the crack modelling methods, remeshing approaches could be an alternative solution for crack propagation problems. The remeshing approach generates a new mesh that follows the crack propagation path. As examples, crack propagation obtained by the remeshing approach for various problems can be found in Ref. [32]. This remeshing approach is more frequently used in simulating crack growth of isotropic materials than for laminated composite materials. This is because it is more difficult to analyze the delamination within CFRP by using a remeshing technique due to the fact that more complex material properties, ply-orientation, and the geometry of laminate need to be considered as variables while remeshing. An alternative remeshing-like technique used for modelling delaminations within CFRP can be found in Ref. [33]. This technique separates the nodes in order to create a delamination. Therefore, matrix cracks and delamination locations are limited to inter-element interfaces. In this chapter, a new approach to model a displacement discontinuity within quadrilateral elements without additional DOFs is developed and presented. There are two steps in the procedure to obtain the stiffness matrix: (1) constructing constraint equations according to the geometries of elements, and (2) applying constraint equations by using a transformation matrix to reduce the size of the stiffness matrix. The first step uses the extrapolation of the displacement gradient of adjacent element to the element containing a crack, i.e., the target element. The extrapolation is obtained by forcing the shared nodes of adjacent and target elements to have the same displacement gradient. This condition enables one to find displacements of a node on the crack face as a 6

21 function of the nodes of an adjacent element. Accordingly, there is no need to add extra nodes by introducing a crack within an element. The constraint equations are obtained for a bilinear quadrilateral element and are used for modelling delamination in beam/shell structures. One of the disadvantages of the proposed method is relatively high error in the region where the displacement gradient is high, i.e., near a crack tip. This error is due to the extrapolation of the displacement gradient. In other words, this extrapolation gives better performance where the displacement gradient is low, i.e., far from a crack tip. The other disadvantage is that the slight stiffening effect is observed. This effect results from the error caused by extrapolation as well. It is observed that the stresses in the elements containing crack are higher than those of standard FEM. In this research, only the formulation for a bilinear quadrilateral element is provided. The most suitable application for this type of element is delamination growth simulations because of the number of elements used and moderately accurate energy release rate can be obtained by VCCT. This proposed method could be generalized to obtain the constraint equations for other types of elements that are more suitable for other applications. However, the generalization is not the focus of this research. This research rather focuses on the practical application for which the proposed method can be immediately applied. It should be noted that a delamination is a type of crack that commonly occurs in a laminate. In this paper, delamination refers to a particular type of crack, whereas the term crack is used to express the more general case of a crack. 7

22 2.2 Formulation Problem statement Delaminations within CFRP, especially with a brittle matrix, i.e., epoxy, can be successfully predicted by linear finite element methods with linear fracture mechanics, i.e., VCCT [34]. While delamination problems sometimes require a large deformation formulation, for the sake of development, the focus here is on the linearized strain-displacement relationship defined as (2.1) where is the small strain tensor, is the displacement, and is the gradient operator. The body force term in the equilibrium equations is neglected, i.e., in (2.2) where is the Cauchy stress tensor, and is the domain of the body. Stress-strain relationships are given by (2.3) where is material moduli tensor. The essential and natural boundary conditions are on, on (2.4) where is the boundary of with unit normal vector, prescribed displacements, and prescribed tractions. The displacement discontinuity considered in this study is shown in Figure 2-1. On the crack faces, traction-free conditions are applied. The bilinear quadrilateral element (Q4) is chosen for the development since quadrilateral elements are one of the suitable elements for delamination growth simulation of composite laminates. 8

23 (a) Element with through crack line (b) Sub-element A with attached element and sub-element B Figure 2-1. Finite element model containing a crack Formulation details Figure 2-1b shows the sub-elements divided by a crack line and the attached element above sub-element A. These sub-elements have slave nodes degrees of freedom that will be eliminated later by applying proper constraint equations that is expressing the DOFs of slave nodes as functions of master nodes DOFs. In order to condense the slave nodes DOFs, the solution of the attached element is extrapolated to sub-element A. The extrapolation is managed by assuming the derivative of the displacement field at nodes shared by the attached element and sub-element A are the same. For the case considered in Figure 2-1b, the assumption can be written as (2.5) 9

24 where is the displacement vector of the attached element, is the displacement vector of the sub-element, and are the natural coordinate system as shown in Figure 2-2, and is the direction which satisfies the condition derived in Appendix A. The condition obtained is that the direction of cannot be parallel to the edge from Node 3 to Node 4 in Figure 2-1 (b) in order to obtain the constraint equations. It is noted that the derivative of the displacement field is not identical at the element boundary in the displacement based FEM while Eq. (2.1) assumes them to be the same. 1 Figure 2-2. Local node numbering and the corresponding coordinates (in parentheses) based on the natural coordinate system for the Q4 element. The basic notations used in the Q4 element are reviewed before deriving the constraint equations. The shape functions of the Q4 element are given by (2.6) where and are coordinates of and at local node number given in Figure 2-2, respectively. The displacements at node in and of the global coordinate system are given by and, respectively. Another local node 10

25 numbering is introduced for the attached element and sub-element A as shown in Figure 2-1b in order to derive the constraint equations. The displacement gradient along a direction of can be expressed as a function of nodal displacements, i.e., at a point within an element where (2.7) By using Eq. (2.7), the displacement gradients along attached element are respectively given by of sub-element A and the (2.8) and 11

26 (2.9) where superscripts,,, and indicate sub-element, attached element, slave node and master node, respectively. Now, the displacement gradients along of sub-element and attached element are expressed by a linear function of the DOFs. By using the assumption Eq. (2.5), Eq. (2.8), and (2.9) can be equated, and isolating gives (2.10) where is the component of the derivative operator matrix obtained for the location of node 1 defined by Figure 2-3. As shown in Eq. (2.10) above, the DOF of a slave node is expressed by a linear function of master nodes. The component of derivative operator matrix can be computed once the coupling of attached element and sub-element are modeled. Therefore, the constraint equation can be explicitly obtained prior to the solution procedure of the FEM. 12

27 The analogous procedure is applied to obtain the constraint equations for the rest of the slave nodes for which the derivations are given in Appendix B. (a) Structured mesh (b) Distorted mesh Figure 2-3. Elemental level test models and their global node numbering and the corresponding coordinates in the global coordinate system. Once the set of constraint equations for each set of the attached elements and sub-elements are obtained, they can be rewritten in matrix form. The displacement vector of the entire system including slave nodes is expressed by (2.11) where is a transformation matrix obtained by the constraint equations and is the displacement vector of all master nodes. The system before condensation is given by (2.12) where is the global stiffness matrix and is the force vector. By using the matrix, the condensed system is given by where (2.13) 13

28 . This method is a general method for applying linear constraint equations without re-ordering the stiffness matrix. It should be noted that it is not necessary to assemble the global stiffness matrix before applying the constraint equations. To take advantage of this method, it is preferable to apply the constraint equations while assembling the condensed stiffness matrix. Any type of methods that utilizes the idea of transformation, e.g., Ref. [35-36], can be used to apply the constraint equations. The condensed global stiffness keeps the same number of DOFs while introducing new slave nodes for modelling cracks. Therefore, by using the proposed method, the crack growth simulation does not increase the number of DOFs regardless of the increase in the number of slave nodes. This additional DOF elimination procedure to model the cracked faces is named Assumed Displacement Discontinuity Finite Element Method (ADD-FEM). 2.3 Elemental level tests The constraint equations should not inappropriately lock the element behaviour as has been observed in EFEM [29]. As a minimum requirement, the subelements should be capable of undergoing rigid body motions and have adequate extrapolation of the derivatives of displacement fields from the attached element. The strain field extrapolated to the sub-element is investigated in this section to understand the behaviour of the constraint equations under prescribed strain on the attached element. The two models described in Figure 2-3 are used for numerical verifications. Figure 2-3a shows a model (Test case A) having identical element shape for the attached-element and sub-element. Figure 2-3b shows another model (Test case B) having different and distorted element shapes for the attached-element and sub-element. 14

29 The strain field within an element is given by (2.14) where is the strain-displacement matrix. The strain field within the subelement for the test cases is given by where (2.15). Accordingly, the strain fields of the attached element and the sub-element are functions of. This relationship suggests that the transformation matrix controls the strain field of the sub-element. The following sub-sections describe the behaviour by using various numerical examples Rigid body motions (zero strains) The constraint equations derived in the previous section should not induce extra-constraints preventing rigid body motions of the system. When displacements causing rigid body rotations or translations are applied to the master nodes of an attached element, the sub-element has to be able to undergo the rigid body motion as well. All master nodes displacements have to be prescribed in order to have rigid body motion. Accordingly, the solving process is not required for this test since the slave nodes displacements are directly recovered by using Eq. (2.11). Also, material properties are not required for this test as constraint equations and strains are independent of them. Figure 2-4a and b show the maximum principal strain and the recovered deformation of sub-element and attached element. The magnitude of maximum principal strain is zero in the attached element and sub-element for both cases. Figure 2-4c shows the shear strain, i.e.,, whose value is also nearly zero. It should be noted that strain components and are not zero under rigid body rotation due to the infinitesimal strain assumption. These examples show that the constraint equations do not prevent the required rigid body motions. In 15

30 other words, the zero-strain field of the attached element is extrapolated to the sub-element successfully. (a) (b) (c) Figure 2-4. Strain and recovered deformation: (a) Translation in the x-direction; (b) Translation in the y-direction; (c) rigid body rotation with angle. The results of test case B are shown in Figure 2-5. Analogous to Figure 2-4, Figure 2-5a and b show the maximum principal strain and the recovered deformation of sub-element and attached element. Figure 2-5c shows the shear strain under rigid body rotation. The distorted geometry of elements does not influence the zero-strain extrapolation property of the constraint equations. 16

31 (a) (b) (c) Figure 2-5. Strain and recovered deformation: (a) Translation in x-direction; (b) Translation in y-direction; (c) rigid body rotation with angle Constant strains over the elements The next test is to check the strain extrapolation when the attached element has constant strains applied. When the displacement gradients at shared nodes have the same value, the constant strains should be exactly extrapolated to the subelement. To verify this, the displacements that cause constant strains of 0.2 on the attached element were applied to the master nodes. In these cases, the subelement is expected to have exactly the same strain field. A Young s modulus of 1.0 Pa and a Poisson s ratio of 0.3 were used. The strains in the sub-element and the attached element are identical as shown in Figure 2-6 for all cases. Since the superposition principle holds within linear finite element methods, the sub-element and the attached element have identical strains under any combination of constant strains. It is also verified that there is no effect of mesh distortion as shown in Figure

32 (a) (b) (c) Figure 2-6. Principal strain and deformation: (a) Uniform load applied in x- direction; (b) Uniform load applied in y-direction; (c) pure shear applied. (a) (b) (c) Figure 2-7. Principal strain and deformation: (a) Uniform load applied in x- direction; (b) Uniform load applied in y-direction; (c) pure shear applied. 18

33 2.3.3 Linear strains Besides constant strains, the Q4 element is capable of handling bi-linearly distributed strains within the element. As shown in Eqs. (2.14) and (2.15), the strains of the attached element and sub-element are functions of displacements at the nodes of the attached element. Displacements are the unknown variables used to obtain the strains in the attached element and sub-element. By introducing the new strain-displacement matrix for the sub-element, Eq. (2.15) can be rewritten as where (2.16) Strains of the attached element and sub-element are explicitly defined by Eq. (2.14) and (2.16), respectively. The difference between the strains is governed by the difference between and. If the attached element and sub-element are rectangular, the derivative of and with respect to the local coordinate system shown, in Figure 2-3a, yields (2.17) The components of matrix and are constants in this case. This fact suggests that the gradient of strain is constant for both attached and sub-element meaning that the gradient of strain in the attached element is extended to the sub-element. An example of linear strain computed at the nodal position is visually shown in Figure 2-8. Displacements are prescribed at the attached element s nodes as in the previous section. A Young s modulus of 1.0 Pa and a Poisson s ratio of 0.3 were used. Under this boundary condition,, and show the linear extrapolation of strain. The characteristics of linear strain extrapolation are only observed when both the elements have a rectangular shape. It should be noted that the components of matrix and are not constant when either of the elements is not rectangle. However, it is not practical to have the 19

34 strain extrapolation characteristics for all possible shapes of the element set. The example shown in Figure 2-8 provides easier understanding of the solution behaviour due to the assumption made by Eq. (2.5). Figure 2-8. Examples of strain extrapolation Selection of constraint equations When distorted elements are used as shown in Figure 2-9, there are two possible constraint equations for slave node 8; Case 1: Node 8 s constraint equation is obtained by using the set of attached element 1 and sub-element 1 and Case 2: Node 8 s constraint equation is obtained by using the set of attached element 2 and sub-element 2. Since only one constraint equation is allowed to be assigned for a slave node, only one constraint equation can be chosen among them. In order to assess the difference caused by the selection of constraint equations, two possible constraint equations, i.e., Case 1 and Case 2, are compared. As the strain extrapolation depends on the boundary conditions, three types of boundary conditions are tested. 20

35 Figure 2-9. An example of FE model with distorted mesh. The first problem is a bar under uniform tensile loading as an example of uniform strain cases as shown in Figure The exact displacement solution is given by [37] (16) where is the Poisson s ratio, and are the displacement in and, respectively. It should be noted that notation is hereafter used to prevent the reader from misreading the notation for the Poisson s ratio. The exact displacements are applied at master nodes of elements extracted from the bar under tensile loading as shown in Figure The blue nodes are the master nodes with prescribed displacements indicated by green triangles. The red nodes are the slave nodes. 21

36 Figure Mesh, Master nodes, Slave nodes and constrained DOFs for a bar under uniform tension. The second problem is a beam under pure bending as an example of a linearly distributed strain case as shown in Figure The exact displacement solution is given by [37] (17) where is Young s modulus and. The exact displacements are applied at master nodes of elements extracted from the beam under pure bending as shown in Figure Figure A beam under pure bending. 22

37 The third problem is that of an infinite space containing a through crack at the centre. The schematic of this pure mode I case is shown in Figure The exact displacement solutions of Mode I and Mode II at near crack tip are listed in Table 2.1. The exact displacements are applied at master nodes of elements extracted from just above the crack face as shown in Figure Figure An infinite space containing through crack at the centre in which is applied far from the region of interest. Table 2.1. Crack tip displacement field for Mode I and Mode II [38]. Mode I Mode II 23

38 and are the stress intensity factors for Mode I and Mode II, respectively. is the shear modulus. and define the polar coordinate system as defined in Figure is the Kolosov constant defined for plane strain:, plane stress:. Crack Figure Definition of the polar coordinate system ahead of a crack tip Numerical results In order to assess the differences due to the constraint equation selection, the normalized L2-norm is defined in following way. First, L2-norm error is given by (2.18) where and is the domain of an element. The L2-norm error is normalized by the L2-norm of the exact displacement, which is expressed as (2.19) where. The difference of normalized L2-norm error due to the constraint equation selection is defined by where the sub-script indicates the case of constraint equation selection. (2.20) 24

39 Figure 2-14Figure 2-16 show the differences of normalized L2-norm error under uniform tensile loading, pure bending and pure Mode I loading, respectively. Each figure contains the 4 types of mesh. Under uniform tensile loading as shown in Figure 2-14, for which a constant strain field is expected, there is no significant influence of mesh type selected. Under pure bending and pure Mode I loading as shown in Figure 2-15, some differences caused by the selection are observed. The distinguishing difference is the error in alternating behaviour observed for the pure Mode I loading case as shown in Figure The difference is positive as shown in Figure 2-16b and goes to negative as shown in Figure 2-16c. There is no consistent way to choose the best constraint equation from Cases 1 and 2 for all types of mesh and BCs. Moreover, the difference is relatively small, i.e., within 1%, for the tested mesh and BCs. It is, therefore, concluded that the selection of constraint equations does not lead to significant difference in the solutions, thus any of them can be arbitrarily picked if there are more than one constraint equation that can be obtained for a slave node. 25

40 (a) (b) (c) (d) Figure Difference of normalized L2-norm error under uniform tensile loading. 26

41 (a) (b) (c) (d) Figure Difference of normalized L2-norm error under pure bending. 27

42 (a) (b) (c) (d) Figure Difference of normalized L2-norm error under pure Mode I. 2.4 Numerical examples Crack problems taken from linear fracture mechanics are chosen in order to demonstrate the capabilities of ADD-FEM. The exact boundary conditions listed in Table 2.1 are applied to the outer boundary of the FE model as depicted in Figure Mixed mode boundary conditions are obtained by superimposing the displacements of pure Mode I and Mode II. By this superposition, any mixed mode ratio can be achieved. 28

43 (a) (b) (c) (d) Figure (a) structured mesh; (b) distorted mesh; (c) deformation and slave nodes (in red) for structured mesh; (d) deformation and slave nodes (in red) for distorted mesh Mesh Two types of FE model used for comparative studies are introduced here. The first type of FE model (10 10 uniform mesh) is shown in Figure 2-17a. The dimensions for the model are: and. The second type of FE model (10 10 distorted mesh) is shown in Figure 2-17b. The domain size and the crack length are the same as those of the first model, but new 29

44 parameters are introduced to describe distorted mesh: and. For convergence tests, is fixed, but the divisions of,, and are changed. The applied exact boundary conditions at the boundary nodes are indicated by green triangles, as shown in Figure 2-17c and d. The nodes at the edge of crack have active DOFs in order to properly apply the displacement boundary conditions. Each set of elements that are formulated as ADD-FEM has the same color. The red nodes indicate the slave nodes that are eliminated during the stiffness matrix assembly and recovered in the postprocessing. For the comparison, the same mesh was used in a standard FEM model by simply changing the slave nodes to master nodes. The exact displacement fields near crack tip are shown in Table 2.1. An isotropic material, which has Young s modulus of 206.9GPa and Poisson s ratio of 0.29, under plane stress conditions, is used throughout the comparative studies L2-norm error distribution The drawback that results from modelling without adding DOFs is the accuracy of solutions and so its error pattern should be clearly understood. The proposed extrapolation method is similar to the Euler method in the sense of using the first order derivative of primary solutions. However, differences in error propagation exist. The Euler method sequentially solves an ODE with a time increment, so the approximation error propagates forward. The FEM solution is obtained by solving the matrix simultaneously, so the error due to the constraint equations will propagate spatially. In order to capture the error propagation caused by eliminating the additional DOFs, the L2-norm error distribution of standard FEM and that of ADD-FEM are compared. The L2-norm error of standard FEM is defined by (2.21) where. The L2-norm error of ADD-FEM is defined by 30

45 (2.22) where. The difference of error index is then given by The L2-norm error distribution under pure Mode I with shown in Figure 2-18a, pure Mode II with is (2.23) is shown in Figure 2-18b, and mixed Mode is shown in Figure 2-18c. There is no consistent pattern for error propagation observed by comparing the figures. The figures also clearly show that the error propagation pattern depends on the boundary condition applied. The common behaviour throughout these three examples is that the error seems to be large at the bottom region just ahead of the crack tip and at the set of attached and sub-elements. Even though it does not seem to be possible to generalize the error propagation behaviour, the error difference between standard FEM and ADD-FEM is not unacceptably large and it ranges from -1.5 to 1.5%. 31

46 (a) Pure Mode I (b) Pure Mode II (c) Mixed Mode Figure L2-norm error in displacement for three load cases. 32

47 2.4.3 Stress distribution The post-processed solution is also an important result that can be obtained by FEM simulations. Stresses and strains are obtained from post-processed solutions, by using displacement based FEM. These stresses and strains are used in various failure criteria and also used to extract the strain energy release rate by the energy domain integral method [39]. The L2-norm error distribution shows that the error ranges for the three examples are not significantly different from each other. Therefore, only the pure Mode I boundary condition was chosen to compare the stress distribution obtained by standard FEM and by ADD-FEM. Figure 2-19a shows the contour lines of stress obtained by standard FEM with dashed lines and that by ADD-FEM with solid lines. The largest difference appears above the crack face where the extrapolation is used. Except in this region, there is a good agreement. The contour lines of stress are shown in Figure 2-19b. The contour lines of standard FEM and ADD-FEM agree very well up to 200MPa. There are some differences below 100MPa. These lower values are observed at the sub-elements in the ADD-FEM formulation. Figure 2-19c shows the contour lines of stress. The closer agreement is observed below the crack face. Overall, the stresses obtained by ADD-FEM capture a similar stress distribution to that obtained with standard FEM. 33

48 (a) ; solid line: ADD-FEM, dashed line: FEM (b) ; solid line: ADD-FEM, dashed line: FEM 34

49 (c) ; solid line: ADD-FEM, dashed line: FEM Figure Comparison of stress distributions obtained by ADD-FEM and standard FEM Convergence of Strain Energy Release Rate (SERR) Extracted SERR values from the FE model are compared with critical values to check whether the crack will advance or not. Accordingly, the error in SERR obtained by ADD-FEM has to be comparable to that by standard FEM in order to use it for adequately accurate crack growth simulation. Also, the convergence rate of the SERR is an important factor to give an idea of the similarities and differences between standard FEM and ADD-FEM. The convergence is studied for the relative error in the SERR is defined by (2.24) Figure 2-20 shows the convergence of SERR by standard FEM and ADD-FEM with a structured mesh as shown in Figure 2-17a. Figure 2-20a shows the convergence under pure Mode I deformation. The difference in relative errors of standard 35

50 FEM and ADD-FEM is very small. Also, the convergence rate is almost identical. On the other hand, there is a difference in convergence rate under pure Mode II as shown in Figure 2-20b. The convergence rate by ADD-FEM is not constant; rather it rapidly approaches to the exact value. Figure 2-21 shows the convergence by using the distorted mesh shown in Figure 2-17b. As observed by the convergence of the structured mesh, the difference in relative errors of standard FEM and ADD-FEM is very small under pure Mode I. The convergence rate is almost identical as well. However, the convergence under pure Mode II by ADD-FEM approached below-zero values while that of standard FEM stays as positive error. By means of this convergence tests, it can be concluded that ADD- FEM attains good agreement under pure Mode I. However, the convergence is more rapid and converged value goes to negative under pure Mode II. The difference in convergence behaviour is not significantly affected by the distortion of elements. (a) Convergence for pure (b) Convergence for pure Mode I Mode II Figure Convergence of strain energy release rate obtained by VCCT for structured mesh. 36

51 (a) Convergence for pure (b) Convergence for pure Mode I Mode II Figure Convergence of strain energy release rate obtained by VCCT for distorted mesh. Convergence tests of mixed Mode I and II are shown Figure 2-22 with the structured mesh. The mixed mode ratio is defined as follows: (2.25) This mixed mode boundary condition is applied by superimposing the exact displacement listed in Table 2.1 in order to make an arbitrary combination of mixed mode ratio. Figure 2-22a shows the convergence of the SERR with. The relative error in Mode II SERR by ADD-FEM is relatively larger and the relative error in Mode I SERR by ADD-FEM is slightly lower than that by standard FEM. Figure 2-22b shows the convergence of SERR with. The difference of relative error between Mode II SERR by ADD-FEM and that by standard FEM is reduced. On the other hand, it is increased for Mode I SERR. Considering the results for pure mode boundary conditions, it can be concluded that the SERR obtained by ADD-FEM has a 37

52 smaller difference in the dominant mode. In other words, when Mode I is dominant as in the pure Mode I case, the difference in the relative error becomes very small and vice versa. The influence of this error on determination of crack growth or crack growth direction is not analyzed in this paper as this paper focuses on introducing the novel technique itself. a) Convergence for b) Convergence for Figure Convergence of strain energy release rate obtained by VCCT under mixed-mode boundary conditions Delamination growth simulation The ADD-FEM method is developed in order to simulate delamination growth in laminated composite materials. To demonstrate its capability, we used the double cantilever beam (DCB) Mode I fracture toughness test shown in Figure 2-23 where the following dimensions and material properties have been assumed for the beam material [40]: 38

53 where, and are the length, width and height of the beam, is the initial delamination length,,,, and are the elastic constants and Poisson s ratio of the unidirectional composite. The Mode I fracture toughness of the composite,, is used coupling with the following delamination propagation criterion:. (2.26) In order to evaluate from the solution of ADD-FEM, VCCT was used. It is assumed that delamination propagation direction does not change. Also, the DCB specimen was modeled with a 2D plane strain assumption. Accordingly, a 2D rectangle with dimensions is meshed. The initial delamination is modeled by using ADD-FEM. Figure Double cantilever beam specimen. The algorithm used in the demonstration is shown in Figure There are constant inputs, i.e., material properties, geometry and initial crack length, and a variable input, i.e., opening displacement increasing step-by-step. The opening displacement is updated according to the step. When, the global stiffness matrix should be assembled from the scratch. In this particular case, constraint equations are also computed in order to consider the initial crack. Next, boundary conditions are applied to the system of equations. Then the equations are solved. evaluated by VCCT is then compared with the mode I fracture toughness. If is greater than or equal to, the crack length is 39

54 extended by an increment of, which is the element length ahead of the crack tip, and this loop continues until drops below. During this loop, only the components of global stiffness matrix influenced by ADD-FEM formulation are changed to model the delamination. After that, the boundary condition at next step will be applied. The load-opening displacement of a DCB specimen simulation by ADD-FEM is compared to that of VCCT for ABAQUS which is available for ABAQUS version 6.8 [12]. The element type used for ABAQUS is CPE4, which is the same as the Q4 element used for ADD-FEM. The mesh has 600 divisions in length and 8 divisions in height, i.e., a mesh for both FE models. The opening displacement is constantly increased by 0.025mm up to 5mm for ADD-FEM. VCCT for ABAQUS has the feature to adapt the increment to minimize the unnecessary solving procedure in the linear part. Therefore, the increment is not constant throughout the analysis. Figure 2-25 shows the simulation results of ADD-FEM and VCCT for ABAQUS. There are two differences observed when examining the two cases. The first one is the difference in the initial slope. The ADD-FEM result has 97.5 N/mm while ABAQUS result has 93.7 N/mm. Accordingly, ADD-FEM shows 4% higher stiffness than ABAQUS. This difference makes the opening displacement required for the initial delamination to propagate 4% faster as well. Since the same element formulation is used, the cause of this increase is mainly due to the ADD-FEM formulation. The second difference is observed in the delamination propagation part of the simulations. 40

55 Start Material Properties, Geometry, BC(t) Yes No Modify global stiffness matrix Assemble global stiffness matrix Solve Yes No Output (Optional) Stop Figure The delamination growth simulation algorithm with ADD-FEM. 41

56 This difference is caused by the tolerance in fracture criterion used in VCCT for ABAQUS. The value is set to 0.01 which is the smallest possible value to use [12].Therefore, VCCT for ABAQUS considers the delamination to propagate when the following criterion is met: (2.27) Even though the two differences are observed, the overall load-opening displacement has a good agreement. Therefore, the ADD-FEM can be used for simulating delamination growth as accurately as when standard FEM is used. Also, ADD-FEM provides easier modelling as it does not require for the user to change the geometric property of the FE model in order to consider a crack. This is a very powerful feature when the initial delamination size and location is not known in advance. Figure Comparison of load-opening displacement of a DCB specimen. 42

57 2.5 Conclusions A new crack modelling method is developed by extrapolating the solutions of master nodes near crack faces to slave nodes at the crack faces. The derivatives of displacement at shared nodes by the attached elements and sub-elements are assumed to be same in order to extrapolate. The extrapolation gives the transformation matrix to eliminate the slave nodes DOF from the global system. The new method is developed in order to eventually provide the mesh independency for crack modelling. However, the concept is only shown by using the inter-element cracks due to the rack of ADD-FEM s modelling capability for arbitrary located crack tip within an element. Elemental tests are carried out to understand the characteristics of ADD-FEM. The choice of two possible constraint equations for one slave node does not make significant difference. It gives easier computer implementation. The extrapolation behaviour is also checked for constant strain and linear strain. Constant strain over the attached element is extrapolated to the sub-element. Linear-strain is also extrapolated when both attached and sub-elements are rectangular. Two linear fracture mechanics problems are used to show the convergence behaviour of ADD-FEM and that of standard FEM. Both results show similar behaviour especially if the case is under pure mode I or II. Slight differences are observed in the mixed mode case. Finally, delamination propagation simulation under pure mode I is conducted by ADD-FEM and VCCT for ABAQUS. The loadopening displacement curves are in good agreement for a practical case. Accordingly, the ADD-FEM gives adequately accurate results by modelling a crack within elements without adding any DOFs. 43

58 For future work, the computational cost should be compared to see whether the fact that the use of no extra DOFs will contribute to cost. Also, the influence of numerical error due to the assumption should be studied considering the determination of crack growth direction by using stresses. 3 Multiscale finite element analysis of a double cantilever beam specimen made of five harness satin weave fabric composite 3.1 Introduction Failure behaviour of five harness satin weave carbon fibre fabric composite Five Harness Satin (5HS) weave carbon fibre fabric is frequently used as a reinforcement in composite structures due to its better damage behaviour and handling in manufacturing processes compared to unidirectional fibre layers. For such composite laminates, Mode I delamination is one of the weakest modes of damage. An example of damage development in a composite structure is shown in Figure 3-1. The composite structure is made of 5HS weave carbon fibre fabric reinforced epoxy manufactured by resin transfer moulding. The airfoil is loaded in bending and the failed brackets show delaminations. The delaminations at the early stage of damage development are shown in Figure

59 Figure 3-1. An example of composite structure made of 5HS weave carbon fibre fabric reinforced epoxy [41]. Figure 3-2. Damage on (a) the bracket on the left and (b) the bracket on the right [41]. Since delaminations are the most critical type of damage, the comprehensive investigation of Mode I, Mode II and Mixed-Mode I-II delamination tests of 5HS weave fabric composites are reported by [22] and [23], independently. Research in [23] reported X-rays images of delamination surfaces showing the sub-surface damages. The sub-surface damage is the damage that occurred between warp (longitudinal) yarns and weft (transverse) yarns, but not within the interlaminar 45

60 region where dominant delaminations grow. The sub-surface damages and the measured interlaminar fracture toughness seem to have a good correlation. The possible effect of the transverse yarn debonding mechanism of 5HS weave fabric composite on the toughness is mentioned in [42]. The delaminated specimens are analyzed to understand delamination mechanisms of composite laminates. However, the experimental results did not provide enough information to give deterministic conclusions on the source of toughening because the fractography data does not provide the physical states during delamination growth Multiscale finite element analyses Finite element analysis (FEA) can potentially provide more information than experiments especially during delamination growth if the phenomenon is modeled properly. The accuracy of analysis depends upon the FE model because many simplifications are generally applied in order to create a FE model. The key point in modelling is not to consider all physical phenomena, but wisely simplify the problem so that it can be solved. One of the examples is the meso-scale FE models of woven fabrics, i.e., a unit cell made of the warp and weft yarns, used to predict the elastic constants [43-46]. The elastic constants are calculated by averaging stress over the unit cell obtained by FEA and applied averaged strain by periodic boundary conditions [47-50]. The elastic constants obtained are usually in good agreement with experimentally obtained values. The major assumptions typically used for unit cell analyses are; perfect bonding between the yarns and matrix, the yarns are homogenous materials, the unit cells are repeated infinitely in all directions, and no voids nor damages are considered. Misalignment of fabric plies occur in real composite laminates. Woo and Whitcomb [51] considered the effect of misalignment on elastic constant prediction. Misalignment gives 10% difference in the in-plane modulus and 20 to 46

61 50% difference in the Poisson s ratios. It seems that it is worthwhile to consider the misalignment if each problem can be solved in a practical time range. Beyond the elastic properties of woven fabric composites, the unit cell approach is also used to predict damage behaviour. Karkkainen and Sankar [52] obtained the failure envelope of plain weave carbon fibre fabric composites. Daggumati et al. [53] explored the local damage in 5HS satin weave carbon fibre fabric composites with focus on the accuracy of unit cell analysis. Further damage development within woven fabric composites are studied by utilizing continuum damage mechanics in which the stiffness matrix is degraded according to the failure criterion. Plain weave composites under in-plane tensile loading were investigated and the stiffness degradation was compared with that of corresponding experiments [54]. Various types of weave, i.e., plain, 4-, 5-, 8- harness satin and twill, were tested under in-plane tensile and compressive loading conditions [55]. A comprehensive road map to multiscale FE modelling from generating textile models to progressive damage analysis is provided by [56]. Key et al. [57] showed the multiscale progressive failure of woven fabric. This multiscale approach decomposes the failure into its constituents, i.e., matrix and glass fibre. Gorbatikh et al. [58], on the other hand, showed results that expose the inadequate use of continuum damage mechanics by providing an example of using continuum damage mechanics with an embedded crack. The multiscale simulation of a notched beam specimen made of braided composite considering damage using a cohesive zone model is reported by [59]. Although the majority of multiscale FE analyses so far are based on continuum damage mechanics, it is probably better to model cracks/delaminations explicitly in order to understand the delamination growth behaviour of 5HS weave fabric composites. 47

62 3.1.3 Hypothesis from experimental results Modelling of a phenomenon usually starts from the observation of experiments. Within the framework of this project, Feret [22] conducted pure mode I fracture toughness tests by following ASTM standard D5528 [60]. The schematic of a typical DCB specimen is shown in Figure 2-23 and the corresponding dimensions are listed in Table 3.1. It is worth noting that the initial delamination front is straight and perpendicular to the specimen edge. Table 3.1. DCB specimen dimensions [22]. Length Length of Teflon Initial delamination Width Thickness [mm] insert [mm] length [mm] [mm] [mm] Seven samples were tested and their Mode I critical energy release rates were calculated by using the Modified Beam Theory (MBT) method. According to the MBT method, the mode I critical energy release rate is expressed by (3.1) where load, load point displacement, specimen width, delamination length, and correction factor. The correction factor is introduced to overcome the overestimation of due to the rotation that occurs at the delamination front. The correction factor is graphically expressed by plotting a least squares plot of the cube root of the compliance, function of delamination length as in Figure 3-3., as a Critical energy release rates of seven samples of 5HS weave composite are plotted in Figure 3-4. Although there is a huge scatter within the critical energy release rate, i.e.,, the critical energy release rate curves, also called R-curves, of all samples reach a plateau when the delamination length increment reaches 7mm, which is approximately the same as the width of three tows 48

63 Figure 3-3. Correction factor of modified beam theory [60]. Figure 3-4. Mode I critical energy release rates of 5HS weave carbon fibre fabric composite. A hypothesis has been put forward that the initial delamination, which has straight front, needs several tows to develop the delamination pattern of 5HS weave fabric composite. In order to validate this hypothesis, a multiscale FE model, which is shown in detail in later sections, is generated and critical energy 49

64 release rates obtained by FEA is compared with those of experiments. The FEA results provide more internal damage information during the delamination growth, which is not easily observed during the experiments nor by fractography R- curves in delamination growth simulation Though the focus of this study is to understand the toughening mechanism observed at early stages of delamination growth, understanding the delamination front development phase also gives very important information to extend the use of R-curves to more practical applications where there is no initial delamination modeled or a very small initial delaminations exists. The basic concept of using R-curves from 5HS weave fabric composite is shown in Figure 3-5. R-curve is unable to be used directly. R-curve Delamination analysis without initial delamination Delamination analysis with initial delamination Practical and useful for more applications Very limited applications Figure 3-5. R-curve applications. One of the applications for which the R-curve can be used directly is the FEA of a DCB specimen. This successful use of R-curves is shown in Figure 3-6 and Figure 3-7. The FE model of the DCB specimen is identical to that used in Chapter 2 except the dimensions. Accordingly, there is no inhomogeneity consideration 50

65 except the R-curve which inherently has the inhomogeneous property. Two extra cases, i.e., the minimum N/m and the value at the plateau N/m, are also tested. As expected, the case using R-curves is closest to the experimental result, Ex1, for both Displacement-Load and Delamination Extension Length-Load relationships. On the other hand, the case with minimum underestimates the peak load and overestimates delamination extension length. The case with at plateau slightly overestimates the peak load and delamination extension length. The ratios of experimental results to FEA results are listed in Table 3.2. Figure 3-6. Load-Displacement curves of 2D plane strain models of DCB specimen. 51

66 Figure 3-7. Load Delamination extension curves of 2D plane strain models of DCB specimen. Table 3.2. Ratios of experimental results to FEA results. R-curve Minimum Maximum Maximum delamination extension length ratio Maximum load ratio Summary Delamination failure could possibly be the initial failure that occurs for 5HS weave fabric composite structures. The prediction of failure behaviour of structures requires a good understanding of delamination growth mechanisms. 52

67 The Mode I delamination test using a DCB specimen shows an increasing R- curve. The implementation of R-curves into a delamination growth model is crucial to improve the accuracy. However, the experimentally obtained R-curve is not directly used for any type of applications. In order to extend the use of the experimentally obtained R-curves to structural applications, the toughening mechanism must be understood. For this purpose, the multiscale FE analysis of 5HS weave carbon fibre fabric composite DCB specimen with CZM elements to model delaminations and matrix cracks is conducted in this study. 3.2 Meso-scale parts of a multiscale FE model TexGen The 5HS weave fabric used as reinforcement of the composite material investigated in this project is categorized as a 2D woven fabric. In order to create a geometrical model of a 2D woven fabric that could be transferred to CAE software, TexGen [61] and WiseTex [62] were considered for the modelling. WiseTex is a commercial software developed by the composite materials group at Katholieke Universiteit Leuven in Belgium. WiseTex is able to create models that can be generated in ANSYS Mechanical APDL. This capability is very attractive and reduces the time required to create an FE model for many types of analysis. However, the fabric model must be modified in order to insert cohesive elements for delamination growth analysis. Thus, the capability is not a big advantage for this particular case. On the other hand, TexGen is a free software developed by Textile Composites Research at the University of Nottingham in the United Kingdom. TexGen can create an IGES format file that can be read by CAD software as well as FEA packages including ANSYS. It also generates meshed models with tetrahedral elements. Although both of them are capable of creating 5HS weave fabric models that can be imported to ANSYS, TexGen was selected over WiseTex. 53

68 The TexGen website has extensive information on its software on the Documentation page. The User Guide is well written and Graphical User Interface (GUI) of TexGen is well designed, so it is not very difficult to create a fabric model by using TexGen. The following data are required for creating a 2D fabric model; number of warp yarns, number of weft yarns, yarn spacing, yarn width, fabric thickness and gap size. The values are listed in Table 3.3 and the corresponding screenshot of TexGen GUI is shown in Figure 3-8. While the yarn spacing and the fabric thickness were measured values of the actual 5HS weave fabric composite, the yarn width was changed to have large enough gap between yarns. This gap is determined by trial and error, i.e., the processes from model generation by TexGen to meshing by ANSYS, were repeated several times. Although it is possible to make large area or many plies of 5HS weave fabric in TexGen, only one ply and the minimum number of yarns to make a unit cell were created before exporting the model to an IGES file. This helps to reduce the processing time to modify and simplify the geometry for FEA. Even though the minimum number of warp and weft yarns required for a unit cell is five, seven warp yarns and seven weft yarns are needed as input to TexGen to make a unit cell that has identical surfaces at each facing edge. Figure 3-9 shows the created 5HS weave fabric in which a yarn at each edge is deleted for exportation. An IGES format file is created by TexGen and imported in ANSYS as shown in Figure The imported data is only the surface information to create the outline of warp yarns along the x-direction and weft yarns along the y-direction. 54

69 Table 3.3. TexGen input values used to create unit cell of 5HS weave fabric. Number of warp yarns Number of weft yarns Yarn spacing Yarn width Fabric thickness Gap size [mm] [mm] [mm] [mm] * * 0 *measured value Figure 3-8. Screenshot of TexGen GUI to create 5HS weave fabric. 55

70 Figure 3-9. Created 5HS weave fabric by TexGen. Warp yarns Weft yarns Figure Surfaces of 5HS weave fabric model imported to ANSYS Creating geometric models and meshing No mesh dependency on delamination growth and its direction is desired. However, there is no practically useful modelling tool causing no mesh dependency available in either ANSYS version 12.0 or ABAQUS 6.8. CZM seems 56

71 be more suitable for this application over VCCT available in ABAQUS 6.8 because there will be multiple cracks, which do not have self-similar crack growth, within the meso-scale region. ANSYS version 12.0 offers two types of element to use CZM, i.e., interface elements and contact elements. Since the delaminated surfaces may be in contact again during the delamination growth simulation, contact elements are used in order to model the cohesive zone. These contact elements with cohesive laws can only be inserted between the volumes that are components consisting of elements. Accordingly, the delamination growth direction is constrained by the fineness of volumes used to model the mesoscale region. The possible delamination growth is restricted to occur within the matrix, at the interfaces of matrix and yarns, and at the interfaces of weft yarns and warp yarns. Therefore, no damage is assumed within weft and warp yarns. In other words, no transverse cracks or fibre peeling is modeled. Even though the larger number of generated volumes for the model tends to give more freedom for a delamination to grow in favoured directions, it will also significantly increase the database size and meshing time for contact elements. The grid size in the x- and y-directions of 0.24mm seems to be fine enough to model the yarns and limits the database size, i.e., 400MB for a single unit cell. The created volumes before meshing are shown Figure The embedded volumes of yarns are plotted with transparent volumes of matrix as shown in Figure Also, very small gaps exist at the overlaps of weft yarns and warp yarns. The gaps, smaller than 0.025mm, were removed by modifying yarns geometries. The warp and weft yarns are completely embedded in the matrix domain with length 12mm, width 12mm, and thickness 0.36mm. Similar small gaps exit between the outer boundary of the matrix domain and yarns. These gaps were removed in the same manner as previously mentioned gaps. 57

72 It is noted that the gap where the warp yarn gap and the weft yarn gap meet together is left empty because it is difficult to create the volumes that contact elements can be inserted with cohesive laws by the program code written by the author. Figure Matrix and yarn volumes of a 5HS weave fabric unit cell. 58

73 Figure Yarns embedded within matrix domain. In order to minimize the problem size and save computational time, the DCB specimen with meso-scale 5HS weave fabric composite is modeled and meshed as shown in Figure 3-13 and the dimensions are listed in Table 3.4. Homogeneous parts are bonded to meso-scale parts by contact elements. This simplification significantly reduces the problem size. The delamination extension length required to reach the plateau is around 3 weft yarns in length. Therefore, the length of DCB specimen model is also shortened. The effect of this simplification on the FE result is studied in the section The width is around the half of a DCB specimen which contains four warp yarns as shown in Figure The enlarged meso-scale mesh is shown in Figure The mesh length in the x- direction is no longer than 0.12mm. In section 3.2.4, it is verified that this length is short enough to give accurate results by using a cohesive law, using the material properties. 59

74 The initial delamination tip is shown in Figure The weft and warp yarns have nine divisions in the yarn width direction. In order to have a better transition from meso-scale parts to homogeneous parts, an initial delamination was inserted up to the forth division of the weft yarns, i.e., 1.08mm of the mesoscale parts has an initial delamination. Figure Finite element model of a multi-scale 5HS weave carbon fibre fabric composite DCB specimen. Warp yarns Weft yarns Figure Warp yarns and weft yarns with their numbering. 60

75 Table 3.4. Dimensions of multiscale 5HS weave fabric composite DCB model. Length (mm) Initial delamination length (mm) Width (mm) Thickness (mm) Figure Enlarged meso-scale model used in the FE DCB model. 61

76 Figure Side view of enlarged meso-scale model used in the FE DCB model. Figure 3-17 shows the contact elements inserted within the simplified DCB model. Two types of contact elements are used for the interface, which are coloured in blue and green, and also for CZM elements, which are coloured in red. The contact elements used for cohesive zone modelling are enlarged and shown in Figure 3-18 and Figure All delaminations occur only along the contact elements. As verified in Figure 3-19, no contact elements are inserted within yarn spaces and these spaces are left empty so that yarns cannot have any damage. 62

77 Figure Contact elements used in the FE DCB model. 63

78 Figure Typical contact elements used with a cohesive law. Figure Side view of contact elements used with a cohesive law showing empty space indicating no damage within yarns is assumed. 64

79 The boundary conditions applied to the multiscale DCB model are shown in Figure The displacements in the z-direction are applied at the end of each beam to open up the DCB model, while the displacements in the x- and the y- directions are applied to prevent rigid body motions. The red arrow shown on the FE model indicates the acceleration due to the gravity, 9.8m/sec 2. Although there is no consideration of inertia term in this FEA, the force of gravity, which is calculated by the mass of elements, is applied to the nodes. This force, however, does not seem to have a significant influence on the result, but it slightly improves the convergence of Newton-Raphson solutions. Acceleration Displacements Figure Boundary conditions for the FE DCB model Material properties In creating the multiscale DCB model, there are three regions with different material properties. The matrix region within the meso-scale parts has properties of epoxy matrix, which are listed in Table 3.5. It is noted that the Mode II critical energy release rate and ultimate shear strength of cured epoxy resin are assumed by the Mode I critical energy release rate and ultimate tensile strength because the values were not available. The weft and warp yarns are 65

80 assumed to have the same properties as unidirectional fibre composite. Its mechanical properties are obtained by a unit cell analysis of unidirectional fibre reinforced composite with hexagonal packing as shown in Figure The material properties of carbon fibre are listed in Table 3.6. The Fibre Volume Fraction (F.V.F) of unidirectional fibre composite listed in Table 3.7 is calculated from the volumes listed in Table 3.8. The volumes are obtained by the function that calculates volumes of elements available in ANSYS. The periodic boundary conditions proposed by [49] were used to obtain elastic constants. The calculated mechanical properties of unidirectional fibre composite are listed in Table 3.9. The density, however, is calculated by using the rule of mixtures. Table 3.5. Material properties of cured epoxy resin [63]. E ν Density (GPa) (g/cm 3 ) G IC_matrix G IIC_matrix * * (N/m) (MPa) (N/m) (MPa) *Note: and. Table 3.6. Elastic constants of carbon fibre [64]. E 11 E 22 =E 33 G 23 G 12 =G 13 ν 23 ν 12 = ν 13 Density (GPa) (GPa) (GPa) (GPa) (g/cm 3 ) Table 3.7. Fibre volume fractions

81 Table 3.8. Geometries and volumes of 5HS weave fabric s unit cell Unit Cell Thickness (mm) Unit Cell Length (mm) Unit Cell Volume (mm 3 ) Matrix Volume (mm 3 ) Tow Volume (mm 3 ) Figure Unit cell of unidirectional fibre composite. 67

82 Table 3.9. Material properties of 5HS weave carbon fibre fabric composite calculated at F.V.F.= E 11 E 22 =E 33 G 23 G 12 =G 13 ν 12 = ν 13 ν 23 Density (GPa) (GPa) (GPa) (GPa) (g/cm 3 ) Figure E 11 variation of warp yarn 2 due to the slope of yarn. The warp and weft yarns have a wave pattern for which the elastic constants are not uniform over the yarn length. One of the warp yarns, i.e., warp yarn 2, was taken to investigate the effect of waviness on the value of E 11. The slope angle and E 11 along a yarn are shown in Figure The maximum difference of E 11 is -3% at. This difference is small and does not likely to cause significant 68

83 effect on the simulation result if it is not considered. Thus, material properties are considered to be uniform over the weft and warp yarns in the meso-scale model. The averaged mechanical properties over a unit cell of 5HS weave composite were also obtained by unit cell analysis. The slightly different meso-scale model of 5HS weave fabric composite unit cell is used as shown in Figure Unlike the meso-scale model for DCB specimen, there is no need to insert contact elements and it has more flexibility to mesh the model with smaller yarn gaps. Elastic constants obtained by the unit cell analysis are compared with those experimentally obtained, in Table It seems that the calculated elastic constants are not very far from the experiments. Accordingly, the similar magnitude of agreement is assumed to exist for the other values that are not available by experiments. Figure Unit cell of 5HS weave carbon fibre fabric composite for elastic constants calculations. 69

84 Table Material properties of 5HS weave fabric composite at F.V.F.=0.55. [63] E 11 E 22 E 33 G 12 G 13 =G 23 ν 12 ν 13 = ν 23 Density [GPa] [GPa] [GPa] [GPa] [GPa] [g/cm 3 ] Unit Cell Exp n.a. 4.8 n.a. n.a. n.a. n.a. The material properties required for a bilinear cohesive law for Mode I debonding are critical strain energy release rate, which is the area of shaded triangle, and strength, which is the peak normal stress, as shown in Figure The critical normal separation is given by (3.2) The softening is described by the degradation of stiffness by up to after reaching the peak. Once reaches one, the contact elements no longer have a cohesive law constitutive equation, but acts as standard contact elements with no friction. The analogous relationship is applied to Mode II debonding. The power law based energy criterion is used to define the completion of debonding in ANSYS : (3.3) where and are the normal and tangential strain energy release rate, respectively. It should be noted that ANSYS does not explicitly differentiate Mode II and Mode III, but tangential strain energy release rate is used instead. 70

85 Figure Bilinear cohesive law. Since the multiscale model has many pairs of contacts, the solution behaviour is strongly affected by the magnitude of. In order to evaluate the effect, the DCB model shown in Figure 3-13 was loaded up to mm, which is still in the elastic range, with the increment, mm for three values. The maximum number of Preconditioned Conjugate Gradients (PCG) iterations and the total number of iterations for Newton-Raphson solution for the test are listed in Table It is clearly shown that the higher the value, the larger is the number of iteration needed. The maximum number of PCG iterations may be reduced by decreasing the load increment. However, it inevitably increases the time required for the solution. Accordingly, the initial slope stiffness,, is determined as N/m. Table Number of iterations for various contact stiffnesses. ( N/m) Maximum number of PCG iterations Total number of iterations for Newton-Raphson solution

86 Noted that ANSYS enforces the following relationship for the mixed mode: (3.4) where the subscripts indicates tangential variables. This expression is needed as the damage parameter for mixed mode,, is used for calculating the normal and the tangential contact stresses written as (3.5) and (3.6) Therefore, normal and tangential debondings are controlled only by using Eqs. (3.2), (3.5) and (3.6) with, Eq. (3.4) is rewritten as By (3.7) This expression is used to give tangential contact stiffness, material properties., for the given Artificial damping is used in ANSYS to overcome the convergence difficulties in the Newton-Raphson solution. The normal contact stress at CZM is expressed as: (3.8) where time interval, and is the artificial damping. It is stated in ANSYS Theory Reference, 4.13 Cohesive Zone Material Model[13], that the damping coefficient has units of time, and it should be smaller than the minimum time step size. sec. is selected from the preliminary tests. Large artificial damping may stiffen the cohesive zone overestimation and prevent load drop off during delamination growth. Time used in the cohesive zone material model for static analysis is not the actual time as in timedependent analysis, but it is artificial time that monotonically increases with load steps and used to apply the displacements with a function of the time, i.e., 72

87 (3.9) where is the current time and the constant has a unit of (mm/sec) Element length, DCB specimen size, and contact stiffness Three FE models are prepared in order to verify the material property selection. Figure 3-25 shows the 2D plane strain model using the dimensions listed in Table 3.1. The model has a very fine mesh, i.e., element length 0.12mm with aspect ratio 1. This result is considered a reference to ascertain the effects of a shortened DCB specimen model and the contact stiffness of elements bonding the meso-scale parts and the homogeneous parts. Figure 3-26 shows the 2D plane strain version of the shortened DCB specimen model. As it does not have bonding contact elements, this analysis shows only the end edge effect caused by the shortened DCB specimen model. Figure 3-27 shows the 3D DCB specimen model without meso-scale parts, but has contact elements to bond homogeneous parts. This analysis shows the effect of contact stiffness as well as the DCB specimen length. First of all, it should be verified that the maximum element length in the x- direction is 0.12mm, N/m and sec. are appropriate selections of values for delamination growth simulation. If the 2D plane strain long model gives energy release rate that is close to given as a material property, the model and selected material properties are considered to be within an appropriate range. Calculated is shown in Figure The correction factor introduced in Eq.(3.1) is used for this case only. The averaged value is 204N/m which is just 2% larger than the given. Accordingly, the selected element length and material properties are used for all the other tests in this study. The rest of the models are also tested and their energy release rates are shown in Figure 3-28, with the 2D long DCB model as a reference. 2D and 3D DCB models 73

88 show the monotonically decreasing caused by edge effect of the shortened specimen model. This tendency is similarly obtained for the case with N/m, i.e., MPa and mm. Due to the strong edge effect, the correction factor did not work properly and was not applied for these cases resulting a slight overestimation of at the very beginning of delamination growth. The decreasing trend is more clearly shown with the ratio of to as shown in Figure The trend is almost identical for 2D and 3D models regardless of critical energy release rate. This result could possibly be used to correct the energy release rate obtained by multiscale DCB model as the tendency is not significantly affected by the magnitude of energy release rate. The correction factor is the function of delamination length written as: (3.10) where is the coefficient determined by polynomial curve fitting of the ratio of to for 3D model. Dividing by gives the corrected, i.e.,, expressed by (3.11) Figure D plane strain long DCB model with element length 0.12mm. 74

89 Figure D plane strain short DCB model with element length 0.12mm. Figure Homogeneous 3D short DCB model with contact elements bonding 5 parts together. 75

90 Figure Mode I energy release rates obtained by homogeneous DCB models. 76

91 Figure Ratios of Mode I critical energy release rates. The contact stiffnesses at the bonding areas used for 3D models are investigated by varying their values. Figure 3-30 shows the load displacement curves within elastic range by varying the values of, and defined in Figure 3-31 with fixed tangential contact stiffnesses, i.e., N/m and. It shows that N/m gives fairly close response to the 2D reference. From this test, three candidates, i.e., K3, K4 and K1, are chosen for another test and the result is shown in Figure It seems that the difference between K3 and K4 are small. Also, it is already shown in Table 3.11 that the smaller contact stiffness tends to give smaller number of iterations. Thus, K1 and K4 are chosen for the delamination growth simulations 77

92 of multiscale DCB model and the results are presented in the next section as Multi1 and Multi2, respectively. Figure Initial linear slope of DCB models with various contact stiffnesses. 78

93 Figure Contact elements coloured in red, blue, and green used in homogeneous 3D short DCB model. 79

94 Figure Load-Displacement curves up to initial stage of delamination growth of homogeneous DCB models with various contact stiffnesses. 3.3 Solution procedure In addition to the FE mesh and material properties, the parameters that are chosen for the solution procedure affect the results. The most influential factors are the convergence criterion for the Newton-Raphson solution and time-step increment that controls the load-step increment and the amount of contact stress update described as Eq. (3.8). Optimized default values for the convergence criterion, i.e., L2 norm of force tolerance equal to 0.5%, recommended by ANSYS has been used. A homogeneous DCB model uses a large time-step increment until the initiation of delamination and then uses a 80

95 smaller one during delamination growth. In principle, the multiscale DCB model follows the same rule as a homogeneous DCB model. However, crack-arrest occurs during the solution which is in contrast to a homogeneous DCB model. Accordingly, a very aggressive time-step increment change has been made as shown in Figure Figure Time-step increment history over the entire simulation of Multi2. The minimum time-step increment is set to [sec] which is slightly larger than that is [sec]. Therefore, the minimum time-step increment remains larger than for the entire solution as suggested by ANSYS. To reduce the number of iterations required for each step of Newton-Raphson procedure, the predictor and the line search were activated. The predictor is used to predict the displacements of current time by using last time-step. This option reduce the number of iterations in general, but will predict too much displacements and/or distortion resulting in termination of solution when one or 81

96 more elements are about to have rigid body motions. The predictor was deactivated when the solution stopped due to occurrence of excessive prediction. The line search is recommended to use with contact elements. The line search updates the displacements of the next iteration by using those of the current iteration. The factor used for update is automatically determined by minimizing the energy of the system. Unlike the predictor, the line search must be activated during the entire solution in order for the solution to converge. Large rotation of elements is expected to occur during the delamination growth. Accordingly, the option to activate the updated Lagrangian method is set to ON. More information on the convergence criterion, time-step increment, the predictor, the line search, and the large rotation option can be found in ANSYS Theory Reference [13]. The PCG solver is chosen over the most robust sparse direct solver due to the advantage of computational time. Also, the multiscale DCB model has a very large number of contact and target elements as listed in Table Such a model does not have good scalability of Distributed ANSYS and has better performance using shared-memory parallel processing. As the scalability of shared-memory parallel is limited by memory access issues, this type of analysis requires a huge computational time. Table Numbers of elements for the element types. Type of element Number of element SOLID185 (8-node brick element) CONTA173 (4-node contact element) TARGE170 (4-node target element) 249, , ,810 82

97 Finally, element death options are activated to the element that causes convergence difficulties in the Newton-Raphson solutions. Local instability due to losing constraints by cracks growth is one of the cases. 3.4 Numerical results Comparison with experiments It is essential to compare FEA results with experimental results in order to correctly understand the toughening mechanism of the 5HS weave carbon fibre fabric composite DCB specimen by the model. calculated by Eq.(3.1) for the multiscale FE model and the corrected by using Eq. (3.11) are shown in Figure 3-34 with two experimental results that are the representatives of lower and upper bounds. The noticeable difference in the results is the zigzag pattern in the multiscale FE result. In the experiments, there is a limitation on measuring the delamination growth increment. The actual delamination growth of the 5HS weave fabric composite DCB specimen is not very smooth, but rather repeats rapid growth and arrest. The delamination growth length is measured while delamination arrests and the load tracked right before the next delamination growth. This procedure gives such a smooth R-curve connecting the local maxima of. On the other hand, the multiscale FE result provides not only local maxima of, but the entire history of. It should be noted that there is no consideration of an inertia term in the multiscale FE model that may affect the results. Both models, called Multi1 and Multi2, have very good correlation with the lower bound of experimental results, called Ex2. When the correction is applied, of Multi1 has slightly higher value at point C p while Multi2 gives good correlation with Ex2. 83

98 Further comparisons are made by using load-displacement curves as shown in Figure As the of multiscale FE models gives better correlation with Ex2, the load-displacement curve also has closer agreement with that of Ex2 than Ex6. Comparing Multi1 and Multi2, the steeper initial slope of Multi2 suggest that Multi2 is stiffer than Multi1 due to the higher contact stiffness in the bonding region. The initial slope of the load-displacement curve of Multi2 has better agreement with Ex2 than Multi1. The lower contact stiffness of Multi1 requires more displacement to reach the critical load at which the delamination starts growing. Figure R-curves of the 5HS weave carbon fibre fabric composite DCB specimen and the multiscale FE models. 84

99 Although the raw value of Multi2 seems to be lower at point C p, the corrected is able to achieve slightly higher value than that of Ex2. Considering the better correlation of Multi2 with Ex2, the result obtained by Multi2 seems to be better representing the 5HS weave carbon fibre fabric composite DCB specimen and the results of Multi2 is mainly analyzed. Figure Load-displacement curves of the 5HS weave carbon fibre fabric composite DCB specimen and the multiscale FE models Energy released by CZM elements Understanding the toughening mechanisms of 5HS weave carbon fibre fabric composite DCB specimen is the main objective of this study. The energy release rate obtained by experiments is based on the change of energy in global scale. FE results, on the other hand, are able to provide alternative methods to quantify the released energy. First, the released energy at delamination length is defined by 85

100 (3.12) where is the initial delamination length and is the critical energy release rate. The released energy due to normal and tangential delamination can be obtained by replacing by and, respectively. The energy release rate is averaged over each element and multiplied by its area to obtain the released energy. Figure 3-36 shows the released energies of Multi2 and 2D reference calculated by using the results of CZM elements and calculated by MBT. The released energy of 2D reference calculated by MBT gives almost identical value to that of calculated by CZM elements. This result is very straight forward as only CZM elements release the energy due to delamination growth. On the other hand, the total released energy by CZM elements of Multi2 is smaller than that obtained by MBT, i.e., global approach. This result indicates that the multiscale DCB model were able to store more strain energy than 2D homogeneous model for the same delamination length. The total released energy by CZM elements consists of normal and tangential components. seems to be very similar to that of 2D reference except slight increase in the slope around delamination length 53mm. The increase seems to be caused by the sub-surfaces that are created by lifting up weft yarn 2. For the sake of analysis, the released energy is divided into three zones, i.e., Zone A covering from the initial point to Point A b, Zone B covering from Point A b to B b, and Zone C covering from Point B b to the ends. The delaminated elements at the end of each zone are shown in Figure The darker portions of the delaminated area show multiple layers, i.e., sub-surfaces. The black portions show the killed elements due to convergence difficulties. It is shown that subsurfaces appear in significant amounts from the end of Zone B and keep increasing within Zone C. This result confirms the increase in the slope of. It 86

101 should be noted that there is sharp drop in the released energy due to the killed elements during the solution. This drop should not be considered as any physical phenomena. The released energy by tangential debonding, i.e.,, has as much as 14% of total released energy as shown in Figure As CZM elements are placed around weft and warp yarns, there is always a portion of tangential debonding. Figure Released energies of the multiscale FE model of 5HS weave carbon fibre fabric composite DCB and 2D plain-strain homogeneous FE model. 87

102 Figure Delaminated areas at the end of each zone of the multiscale FE model of 5HS weave carbon fibre fabric composite DCB. 88

103 Figure Percentage of released energy by tangential debonding within the total released energy by CZM elements Toughening mechanisms Although sub-surfaces and tangential debonding are indeed acting as toughening compared to homogeneous model, that amount is not enough to explain the difference in released energy calculated by MBT and CZM elements. The toughening mechanism is discussed starting from Zone A to Zone C, which is the same order as delamination growth. The toughening mechanism observed in the multiscale FE model can be broken down to two mechanisms. The first observed mechanism is the relaxation of stress concentration ahead of delamination front caused by inter-yarns locking with the help of compressive stresses ahead of delamination in DCB specimen. The deformation and stress distribution of at Point A p are shown in Figure Unless otherwise noted, the scaling factor for displaying the deformation is four. The compressive stress ahead of delamination front, i.e., on weft yarn 2, is observed. 89

104 Delamination Figure Contour plot of on the yarns at Point A p and delaminated elements coloured in pink. The yarn interaction is rather easier to see with the contact pressure distribution on the overlapping areas of yarns as shown in Figure It indicates compressive contact pressure, i.e.,, against the yarns as positive value. As compressive stresses exist in the warp and weft yarns at the ahead of delamination front as shown in Figure 3-39, the overlapping areas experience the compressive contact pressure. To understand the contributions from the weave structure, the inter-yarn locking compressive contact pressure under inplane tensile loading at the same points are shown in Figure

105 Delamination Figure Contour of contact pressure delaminated elements (coloured in pink). of the multiscale FE model and Figure Contour of contact pressure in-plane tensile loading. of the meso-scale FE model under The meso-scale parts were taken from the multiscale DCB model with the initial delaminations removed. Constant displacement is applied to the edge normal to the x-direction. The magnitude of applied displacements is controlled to have 91

106 the same range of contact pressure as that of multiscale DCB model. Figure 3-41 shows the upper warp and weft yarns for 2 plies only. The compressive contact pressure indicates that inter-yarn locking exists on the overlapping areas where the warp yarns change their paths to under and over the weft yarns. The negative contact pressure, which is showing the stress to separate the yarns, is also observed where the weft yarns are on the warp yarns. Figure 3-42 shows the contact pressure at overlapping areas on line defined in Figure Contact pressures under two different magnitudes of in-plane tensile loading are plotted with that of the multiscale DCB model. The contact pressure tendency of in-plane loading is not very sensitive to the magnitude of applied load. Accordingly, this tendency itself can be considered remain the same for the multiscale DCB model at which some amount of exists. With this assumption, the contact pressure of the multiscale DCB model at overlapping areas seems to consist of relatively flat compressive pressure inherently caused by the DCB specimen and pressure variations caused by yarn interactions. This implies that there is inter-yarn locking effect in the multiscale DCB model ahead of delamination front. The effect of yarn interactions on the stress distribution near delamination front at Point A p is also investigated. Figure 3-43 shows the view of the upper warp yarns, weft yarns, and matrix having positive clipped at from bottom side. It is noted that the matrix near the overlap of warp yarn 3 and weft yarn 2 has positive indicating stress concentration due to delamination front is slightly relaxed forward. The stress distributions on lines defined in Figure 3-43 and contact pressures of contact elements at under in-plane loading are plotted in Figure The minimum value of is observed on warp yarn 3 and its maximum value is observed on warp yarn 1. The contact pressure, whose sign is reversed for sake of better understanding, shows inter-yarn locking that may reduce the. The 92

107 overlap of weft yarn 1 and warp yarn 3 seems to have the highest reduction on. It is noted, however, that the exact amount of reduction on by yarn interaction cannot be obtained. Figure Contact pressure distribution on at Point A p. 93

108 Delamination Warp yarn 3 Weft yarn 2 Figure Contour plot of at Point A p clipped at and delaminated elements (coloured in pink). By analyzing the results, it seems that inter-yarn locking caused by in-plane deformation in the x-direction at the overlap of weft yarn 1 and warp yarn 3 and the overlap of weft yarn 1 and warp yarn 4 has a toughening effect up to Point A p in addition to the toughening effect by inter-yarn locking of weft yarn 2 and warp yarn 3. The contributions from the two toughening mechanisms cannot be separated. However, the former toughening effect is smaller than latter because the former toughening effect vanishes with delamination growth towards Point B p where the load reaches a higher value than that at Point A p. The load drop toward Point A b seems be triggered by the delamination front under warp yarn 1 and weft yarn 1 because it has the highest stress compared to the others. However, the delamination front seems to grow uniformly as shown Figure

109 Delamination Figure near delamination front at Point A p and reversed contact stress obtained by in-plane loading. 95

110 A p A b Figure Delamination front development during the load drop from Point A p to A b. The toughening effect can be clearly explained by the stress distribution near delamination tip at Point A p. This approach, however, does not work well with the case at Point A b, which is the starting point of toughening toward Point B p, 96

111 because the delamination has already been branched from the initial delamination front. This makes the analysis difficult as shown in Figure Figure The z-coordinates of delaminated elements showing the branching at Point A b. On the other hand, the positive ahead of delamination front seems be an alternative measurement of the stress relaxation at the delamination front. The distance from the maximum delamination front at a given location in the y- coordinate to the specified magnitude of ahead of the delamination front is defined as as shown in Figure is based on stress, which has been divided into 9 color increments as shown in Figure 3-47, and represents the difference in distance between zero stress (color red) and stress of 8MPa (color blue). Figure 3-47 is at Point A b where the delamination growth has just arrested. The longest is obtained on warp yarn 3 up to 7MPa. This warp yarn 3 seems to prevent delamination by inter-yarn locking with weft yarn 2. for = 1MPa shows the representative trend of over the width of the multiscale 97

112 DCB model as shown in Figure 3-48 for various points between Point A b and B b. From Point A b to B p during which the load is increasing, has its highest values on warp yarn 3. on warp yarn 3 is shortened during the transition from Point B p to B b. This suggests that the inter-yarn locking resisted during the transition from Point A b to B p, and is released as it goes toward Point B b. This result seems to support the toughening caused by the inter-yarn locking of warp yarn 3 and weft yarn 2. Figure Length of positive from the delamination front at Point A b. 98

113 The delaminated CZM elements from Point A b to B b are shown in Figure 3-49 with the warp and weft yarns of the upper ply as viewed from the bottom side. Small amounts of delamination growth mainly on warp yarn 1 are observed while the load increasing, i.e., Point A b -B p. This result suggests that warp yarn 1 is not acting as the main source of toughening toward Point B p because there is no load drop due to the growth. By observing Figure 3-49, the delamination growth on warp yarn 1 seems be a trigger for the entire delamination growth because the delamination grows on warp yarn 1 first followed by the rest of yarns. Also, on warp yarn 1 at Point B p is relatively smaller than the rest of them as shown in Figure This implies the correlation between the likelihood of delamination growth and. At Point B p - B b, it is shown that delamination on warp yarn 1 and 2 goes intraply, which is indicated by slightly darker areas because of transparent blue of weft yarn 2, while the delamination on warp yarn 3 keeps going inter-ply. This delamination growth pattern can be clearly seen at Point B b. The exaggeratedly scaled deformation of yarns at Point B b is shown in Figure The upper and lower weft yarn 2, coloured in green and yellow, respectively, are bridging the upper and lower plies. This bridging creates extra delamination surfaces which are observed by X-rays [23] and are called sub-surfaces. The inter-yarn locking ahead of delamination front played a significant role on the toughening up to Point B p. However, it does not seem to continue toward Point C p as weft yarn 3 has no crimping warp yarn. Accordingly, there is no significant amount of inter-yarn locking effect expected ahead of delamination, but the effect of weft yarn bridging gradually appeared from Point B p. 99

114 Figure distribution history from Point A b to B b. The bridging effect reduces the stress concentration ahead of delamination front if the weft yarns carry enough loads to resist. of the CZM element on the delamination front edge at Point B b is shown in Figure 3-51 with the z-coordinate at the delamination front edge. The corresponding delaminated area is shown in Figure 3-49, B b. The values of on the warp yarns are in the range of 60 to 70MPa. This distribution does not seem to show significant amount of stress relaxation at the delamination front unlike that of Point A p as shown in Figure This confliction indicates that there is no evidence of stress relaxing due to the weft yarn bridging. 100

115 Numbering A b A b - B p B p B p - B b B b Figure Delaminated CZM elements from Point A b to B b. 101

116 Figure Weft yarn bridging of Multi2 observed at Point B b. Figure of CZM element on the delamination front edge with the delamination front z-coordinates at Point B b. The other possible source of toughening is simply due to the creation of subsurfaces that increases the delaminated area. To investigate this effect, the delamination area calculated by the mean delamination length multiplied by the 102

117 width, i.e., A1, and the total delaminated area of CZM elements, i.e., A2, are obtained and shown in Figure 3-52 together with their linear curve fittings. The figure shows that the actual delamination area is around twice that of the dominant delamination area, i.e., A1. The global energy release rate contributed by normal debonding of CZM elements can be expressed by (3.13) where and is the width of the multiscale DCB model. The equation states that the global energy release rate due to normal debonding can be obtained by multiplying the critical energy release rate of epoxy matrix by the ratio of delamination area increase rate, i.e., the slope, with respect to the width of the multiscale DCB model. Figure Delamination area of Multi2 versus delamination length. Accordingly, the energy release rate by normal debonding of all CZM elements could be 430N/m. The additional energy release rate due to tangential 103

118 debonding should be added. It has been already shown that the contribution of tangential debonding to the released energy has up to 14% as discussed in subsection If so, the tangential debonding should contribute 70N/m. The total energy release rate by all CZM elements should be 500N/m. On the other hand, value of Multi2 calculated by MBT at Point C p is 486N/m with the correction. The total energy release rate by CZM elements is only 2.88% higher than that of Multi2 obtained by MBT. So far, Multi2 results have been analyzed, but Multi1 results are also available. value of Multi1 has a similar trend to that of Multi2 up to Point B b as shown in Figure However, Multi1 seems to have superior toughening towards Point C p. The analogous analysis approach is used to obtain delamination area increase rate of Multi1. The corrected value of energy release rate at Point C p of Multi1 obtained by MBT is 528N/m. The ratio of delamination area by CZM elements, which is shown in Figure 3-53, with respect to the width of the multiscale DCB model is Assuming 14% of total energy release rate is contributed by local tangential debonding, the increased amount of delamination area would raise the energy release rate to 556N/m, which is 5.30% higher than that obtained by MBT. Although there is a speculative correction to the end-edge effects, this good correlation suggests that the main source of toughening seems be the creation of sub-surfaces. 104

119 Figure Delamination area of Multi1 versus delamination length. The main difference between Multi1 and Multi2 is the contact stiffness that connects the meso-scale parts to the homogenized parts. The weft yarn bridging of Multi1 is shown in Figure If the contact stiffness is the cause of this difference, it implies that the transition from homogeneous parts to meso-scale parts needs to be improved. Using a finer mesh in homogeneous model and longer initial delamination within meso-scale parts would eliminate the effect of the transition on the delamination growth. 105

120 Figure Weft yarn bridging of Multi1 at Point B b. In summary, the toughening of 5HS weave fabric composite DCB model consists of two different mechanisms; 1. Inter-yarn locking ahead of delamination front causing stress relaxation, 2. Sub-surfaces created via the weft yarn bridging providing additional resistance. One of the literatures [42] on the experimental investigations on 5HS weave carbon fabric composite suggested conflicting concluding remarks, i.e., the cause of toughening is the weft yarn, which is acting as periodic obstacles for delamination growth. By analyzing the current results, there is no evidence of weft yarns acting as obstacles. Moreover, the FE model would likely cause delamination arrest by the weft yarn if they actually acted as obstacles because delamination growth direction is restricted to the CZM elements that are inserted. The toughening explanation in [42] is obtained by the experiments results, which are intermittent data on the side edge of specimen and delamination surfaces. This limited amount of information may blind other source of toughening mechanisms. On the other hand, the experiments indeed showed more variable information, such as fibre breakage that suggests fibre bridging [22-23] and transverse cracks within the weft yarn [23]. These types of damage are omitted in the FE model in 106