FEM modeling of fiber reinforced composites

Transcription

1 FEM modeling of fiber reinforced composites MSc Thesis Civil Engineering Elco Jongejans Delft, the Netherlands August

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3 MSc THESIS CIVIL ENGINEERING FEM modeling of fiber reinforced composites The use of GFEM, the embedded reinforcement approach including bond slip and the Interface GFEM approach. Elco Jongejans Delft University of Technology Daily supervisor Dr. A. Simone Responsible professor Prof.dr.ir. L.J. Sluys Other thesis committee members Dr.ir. C van der Veen

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5 Acknowledgements Before entering into math derivations and Matlab code I would like to thank some people. First of all, Angelo Simone who supported me through the week process in weekly meetings, hallway encounters and countless replies often after midnight. Bert Sluys and Cor van der Veen who were always enthusiastic during committee meetings with good feedback and steering. Adriaan Sillem for the many discussions about linear dependency of GFEM, but also on global economy, startup companies and for his general patience with my less mathematical reasoning. Jure and Amin; thanks for being great roommates at room 6., I ve had a great time there with you. Frank Everdij, thanks for the occasional technical support during the project. Lara, thanks for checking my thesis on my correct use of the English language. Last but not least, I would also like to thank Armando Duarte for the reflection on the topic last february over the .

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7 Abstract Recently several studies have been performed at the TUDelft on the use of the generalized finite element method (GFEM) to model fiber reinforced composite materials in a two dimensional space. The GFEM model allows a large number of arbitrarily placed fibers to be taken into account. The fibers are placed on top of the ordinary mesh and therefore do not require aligned meshing to be done. Matrix material, fiber material and the interface between them each have their own material parameters. The discontinuous displacement field on the fiber, also known as the fiber slip is taken into account by the use of extra degrees of freedom. These extra degrees of freedom are placed on the original nodes of the elements crossed by fibers. Fibers are in essence one dimensional objects. The main degree of freedom they should have is the slip in direction of the fiber. It is therefore computationally expensive to use extra degrees of freedom on top of regular element nodes to describe the displacement field of the fiber. Why not inserting the extra degrees of freedom in direction of the fiber on top of the fiber itself? In this thesis a search is done to an efficient element with extra degrees of freedom on the fiber. A first unsuccessful try is the use of a so called Interface-enriched GFEM element (IGFEM). The second successful approach is the slightly different embedded reinforcement approach including bond slip (ERS). This is an element that has been used before in calculating reinforced concrete in several publications. This element is used here for calculating elastic material reinforced with many fibers. In its mathematical derivations, a small extension is made to allow for an arbitrary enrichment function to be inserted on the fiber displacement field analogous to the GFEM derivations. This enrichment function inserts a priori knowledge into the solution allowing it to converge faster. However, enrichments in this research are kept simple. Next to a two dimensional implementation of the GFEM and ERS approach, a three dimensional implementation is presented. Encountered problems and numerical examples for both models are discussed. Both approaches produce very similar numerical results. The ERS approach is more efficient as the current GFEM approach.

8 Contents Introduction 9. The finite element method Fiber reinforced composites Reading guide Fundamental concepts and thesis goals. Fundamental concepts Generalized finite element method (GFEM) Interface-Enriched Generalized FEM (IGFEM) Embedded Reinforcement including bond slip (ERS).. Thesis Goals Primary objectives Secondary objectives Literature study 5. Distributed models Bridged crack models Lattice models Conformal meshing and homogenization Embedded Reinforcement including bond slip (ERS) Mathematical derivations. Continuum without a fiber Problem statement Kinematics Constitutives Equilibrium Weak statement Discretization Continuum with one fiber Problem statement Kinematics Constitutives

9 .. Equilibrium Weak statement Fiber thickness Discretization GFEM and IGFEM Discretization ERS Dimensional check of K Continuum with multiple fibers The enrichment function χ Simple heavyside enrichment Exponential enrichment Elements 6 5. Element overview Standard quad GFEM GFEM D fiber integration IGFEM ERS Element quality 5 6. Sanity checks Introduction Element pulled vertically Element vertically pulled apart Element horizontally pulled Conclusion sanity checks Eigenvector analysis Introduction Overview of eigenvectors Conclusion eigenvector analysis Extension to D 6 7. Elements Standard brick GFEM ERS Linear dependency problem GFEM Problem investigation Eigenvector analysis Possible solutions

10 8 Computer implementation Parsing the fibers D implementation D implementation Integration schemes Iso-parametric fiber implementation Numerical results 7 9. Parameters Results GFEM D Visual comparison Length of fiber (D) Rotation of fiber (D) Convergence (D and D) Different distributions of 9 fibers (D) Different distributions of fibers (D) Different distributions of fibers (D) Different amounts of fibers (D) Different length combinations of fibers (D) Conclusions and recommendations 95. Conclusions Main research goal GFEM IGFEM ERS Reflection on the primary and secondary goals Recommendations A Mathematical derivations 98 A. Derivation of the engineering B matrix in D A. Derivation of the enriched engineering B matrix in D B Code flow C GFEM D eigenvector analysis

11 List of Tables 5. Overview of the different elements with their characteristics Eigenvectors - rigid body modes Eigenvectors - bending and shear modes Eigenvectors - pinching and expansion modes Eigenvectors - rigid fiber modes Eigenvectors - various fiber modes () Eigenvectors - various fiber modes () GFEM D rank deficiency with multiple elements and one fiber 6 7. GFEM D rank deficiency with one element and multiple fibers Different integration possibilities for GFEM in D

12 List of Figures. FEM calculation of an airplane (a) and a car rim (b) Example of fiber reinforcement plastic with low (a) and high (b) amounts of fibers Distributed modeling, converting reinforcement to a homogeneous layer of extra stiffness Bridged crack model. Traction t is applied at the crack surfaces crossed by the fibers Lattice model using springs at the boundaries (a) or fibers as complete lattices including nodes and extra lattices for bond relation (b) Conformal meshing, meshing the fiber together with the matrix 7.5 Element with embedded reinforcement with extra degrees of freedom to model reinforcement bond slip Continuum without a fiber Continuum with a fiber Close up of the fiber inside the matrix Simple discontinuous enrichment (a) and exponential enrichment (b) over multiple quad elements crossed by a fiber Nodes and dof s of a standard quad (a) and shape function belong to node (b) GFEM standard element (a) and shape functions belonging to node and 5 (b) Nodes of a IGFEM D quad element with fiber (a) and shape functions belonging to node and 6 (b) Nodes and dof s of an embedded reinforcement element (a) and shape to model the fiber (b) Sanity check - parameters used Sanity check - influence of the normal bond () Sanity check - influence of the fiber Young s modulus Sanity check - influence of the normal bond () Sanity check - influence of the tangential bond

13 6.6 Sanity check - influence of the poisson ratio of the fiber Eigenvector analysis - parameters used D elements, GFEM (a) and Embedded Reinforcement (b) with in black the standard dofs and in green the enriched dofs 6 7. GFEM D linear dependency - different bending modes on each fiber end GFEM D linear dependency - deformation in the separate planes perpendicular to the fiber GFEM D linear dependency - parallel bending GFEM D linear dependency - rotation and torsion around the fiber axis Parsing a fiber in D Parsing a fiber in D Enriched nodes (blue GFEM, red ERS) and fiber Gauss points (black) ERS (red) and GFEM (blue) deformed mesh and fiber slip Meshes with different fiber lengths Slip curves with different fiber lengths Reaction forces due to different fiber lengths Meshes with different fiber angles Reaction forces due to different fiber rotations Convergence D - all degrees of freedom Convergence D - only enriched degrees of freedom Convergence D - all degrees of freedom Convergence D - only enriched degrees of freedom Meshes with different distributions of 9 fibers Different reaction forces with distributions of 9 fibers Meshes,,,,5 and 6 with different distributions of fibers Different reaction forces with distributions of fibers Different reaction forces with distributions of fibers, x smaller fiber cross section Deformed mesh fibers (red = ERS, blue = GFEM) Deformed mesh fibers (red = ERS, blue = GFEM), x smaller fiber cross section Meshes,,,,5,6 and 7 with different distributions of fibers in D Different reaction forces with distributions of fibers in D Meshes with, 5 and fibers in random and radial distribution Meshes with, 5 and fibers in random and radial distribution

14 9. Reaction force with different numbers of fibers Number of dof s needed with the different amounts of fibers Reaction forces with different fiber length combinations Meshes with short and long fiber combinations (,5/,5), (,5/,), (,5/,) with fibers (left) and 5 fibers (right). 9 C. GFEM D linear dependency - different bending modes on each fiber end C. GFEM D linear dependency - deformation in the separate planes perpendicular to the fiber C. GFEM D linear dependency - parallel bending C. GFEM D linear dependency - rotation and torsion around the fiber axis C.5 Rigid body translation in x and z direction C.6 Rotation around the x-axis and translation in y direction.. 7 C.7 Rotation around the y-axis and z-axis C.8 Rigid rotation of the fiber around y-axis and z-axis C.9 Shear mode of the element and bending mode of the fiber.. 8 C. Bending mode of the fiber and one of the element C. Shear modes of the element C. Shear modes of the element C. Shear mode and bending mode of the element C. Pure shear mode and extension mode both of the element.. C.5 Extension mode and shear mode of the element C.6 Shear mode and bending mode of the element C.7 Bending mode and rigid translation of the fiber C.8 Rigid translations of the fiber C.9 Bending modes of the element C. Bending modes of the element C. Expansion mode and bending mode C. Shear mode and extension mode of the element C. Shear modes of the element C. Extension mode and expansion mode

15 Chapter Introduction This document is the final report for the master thesis project of Elco Jongejans. This master thesis is research based. The topic is described in one sentence as: FEM Modeling of fiber reinforced composites. In this chapter an explanation will be given to the terms in this sentence.. The finite element method Finite element methods are nowadays a must for an engineer designing buildings, cars, airplanes, consumer products etc. Stated in a simple way, the method in application to the mechanics of those products, allows you to solve unknown displacements and movements when applying loads or displacements to the product under study. This usually results in color plots of displacements and tensions in those materials. Two examples of this are given in figure.. Stated in a mathematical way, the finite element method Figure.: FEM calculation of an airplane (a) and a car rim (b) 9

16 solves partial differential equations in the primary unknowns. In application to mechanical engineering, it does this by dividing the object under study in a number of finite elements connected by stiffness relations. Each element has a certain number of degrees of freedom (dofs), the displacements in different directions. Combining all stiffness relations between the different elements yields the stiffness matrix K for the whole object. This multiplied with the unknown displacement vector u yields the force vector f: Ku = f (.) Solving this system will result in the displacements in different points of the whole object due to the forces applied. Finite element problems are roughly tackled in steps. First, a mathematical model is derived of the problem under consideration. Second, the model is discretized into vectors and matrices. Third, the model is programmed into a computer and solved numerically.. Fiber reinforced composites The second term mentioned in the sentence above: what exactly are fiber reinforced composites? Or first of all, what is a composite? A composite is a material consisting of at least sub materials mixed together. There are three distinct parts in fiber reinforced composites when considering it from a finite element point of view: the matrix, the base material connecting everything; the fibers, strengthening the original material; the interface zone between matrix and fiber, relating the previous two through a certain bond stiffness. An example of a fiber reinforced composite can be fiber reinforced concrete. This is concrete in which small steel fibers of around to cm length are mixed. The fibers increase the strength and ductility properties of ordinary concrete and thereby facilitate use of concrete in a sometimes more economical and thinner way. Another example is the material plastic which is often reinforced with fiber like materials. In figure. two illustrations of this are shown. The first figure shows a plastic with a low amount of fibers. The second figure shows a plastic with a high amount of fibers. The matrix material has been burned away here. Since the 96 s the use of composite materials has grown with an average of 5% per year leading to a multi billion dollar market nowadays. Hence, it becomes increasingly more important to be able to model these materials accurately, inexpensively and quickly with computer models such as the finite element method: This is the topic of this thesis.

17 Figure.: Example of fiber reinforcement plastic with low (a) and high (b) amounts of fibers. Reading guide This document is the final report of this master thesis. This chapter served as the introduction to the topic and to point out some practicalities. Chapter provides a general overview of the three calculation methods that are used (GFEM, IGFEM and ERS) and defines the goals of this research. Chapter provides concise background information on other methods available to model fiber reinforced composites in the form of a small literature study. In chapter the mathematical derivations are made for the standard finite element solution and the GFEM, IGFEM and ERS solution including fibers. Several potential elements are derived in chapter 5 and their quality is tested through sanity checks and eigenvector analysis in chapter 6. Chapter 7 describes the extension of the models to D. Chapter 8 describes key implementation issues and chapter 9 deals with numerical tests and applications of the code. ERS and GFEM are being compared and additional tests with different fiber distributions are being made. Chapter contains the conclusions and recommendations for further research. At the end of the thesis appendices with mathematical derivations, code flow, D eigenvectors and bibliography are included.

18 Chapter Fundamental concepts and thesis goals This chapter describes the main content of the research. First it explains the three fundamental finite element concepts behind this thesis.. Fundamental concepts.. Generalized finite element method (GFEM) The generalized finite element method, or sometimes called the Partition of unity approach (GFEM), is a finite element method that can be used to model discontinuities in a solution field in a mathematical way. This is done by super-positioning extra degrees of freedom (b) on top of the existing degrees (a) of freedom in the elements. In combination with the existing degrees of freedom they form the displacement field. The method can be used to model fibers embedded in materials as is done in []. The results obtained in [] will be the main comparison for results from in thesis. With the GFEM approach, the displacement field in discretized form can be formulated as follows: u = N p a + χn p b (.) The displacement field is formed out of the two terms together. The function χ represents the enrichment function which adds a-priori knowledge to the solution such as a displacement jump over the fiber domain. For both parts of the displacement field the same shape functions of the parent element are used... Interface-Enriched Generalized FEM (IGFEM) The IGFEM method builds on the same idea as the GFEM approach. However, the extra degrees of freedom are not positioned on top of the existing

19 nodes, but directly on the discontinuity one wants to model. The discretized displacement field with IGFEM is build up as follows: u = N p a + χn c b (.) In which the first term represents the standard degrees of freedom with the standard shape functions of the parent element. The second term represents the extra added degrees of freedom, now on the discontinuity itself, with additional shape functions. The basic principle is that you split the element over the discontinuity and discretize the extra nodes and dofs by new shape functions of the separated parts, the child elements. The advantage of IGFEM is the use of less degrees of freedom making it more numerically efficient. For a simple quadrilateral element, with a fiber crossing the domain this is immediately clear. In the IGFEM only dofs ( nodes) are needed as compared to 8 dofs on the corner nodes in the GFEM approach. A disadvantage of the IGFEM method is that the shape functions used to discretize the solution have to be determined programmatically to span the child elements. This requires more computational steps considering the different ways an element can be crossed by a fiber. The IGFEM method has been used before in [] to model problems with discontinuous gradient fields... Embedded Reinforcement including bond slip (ERS) Next to GFEM and the idea to use IGFEM for modeling fibers there have been many publications [,, 5] about modeling reinforced concrete using rebars that contain slip degrees of freedom. This method is called embedded reinforcement. An advantage of this method is, that you only add dofs in direction of the fiber. A simple quad element would thus only need extra degrees of freedom to model the fiber crossing it. However, since the slip degrees of freedom are in the local coordinate system on the fiber, this introduces the need for extra transformation matrices to convert between different coordinate systems. This can already be seen in the description in the displacement field in which the slip degrees have to be projected on the global coordinate system: cos(θ) u = N p a + χn c sin(θ) cos(θ) b (.) sin(θ) To the authors knowledge, none of the publications on ERS are using fiber reinforced composites with lots of arbitrarily placed fibers as in []. Also, none of these publications take into account the mathematical freedom to use an arbitrary enrichment function χ to obtain better convergence of the solution. A possible disadvantage of this method is that the fiber is modeled as a one dimensional bar. In GFEM and IGFEM the fiber can be modeled as a two or three dimensional object with material properties in all dimensions.

20 . Thesis Goals Goal of this thesis is to find a more efficient finite element to model fibers in fiber reinforced composites; an element that uses less degrees of freedom for being crossed by a fiber than the current GFEM model, but qualitatively yields the same results. The model should have the mathematical possibility to be enriched with arbitrary functions to obtain even better convergence. Starting point is the IGFEM model published as in [] that models discontinuous gradient fields. This should allow for taking into account a fiber with full two dimensional material properties such as in the GFEM approach by []. The objectives can be summarized as follows:.. Primary objectives Derive the necessary mathematical equations to model fibers with an IGFEM approach Start with a simple FEM model in Matlab with fiber and element and extend it to multiple elements Perform element sanity checks with Maple and make an eigenvector analysis to verify the element quality If IGFEM does not work, try and find another approach that works Compare the solution to the GFEM solution.. Secondary objectives Implement multiple fibers arbitrarily placed Extend the two dimensional GFEM and IGFEM models to three dimensions Try out different enrichment functions χ that lead to a faster convergence Produce a non-linear version of the model

21 Chapter Literature study This chapter summarizes the different approaches to model fiber reinforced composites. It tries to highlight the differences with illustrations and short explanations. It is by no means a complete overview of all methods ever tried or all literature available.. Distributed models A distributed model is the simplest method of modeling fiber reinforced composites. It works best for reinforcement with rectangular patterns such as reinforced concrete. The reinforcement layer is described as one mechanically equivalent homogenous layer. This means adding up the average stiffness introduced by the reinforcement to the material properties already contained in the D matrix. This comes down to increasing the modulus of elasticity E. The process is illustrated in figure.. Downside of this method is that it can not model any local effects such as fiber slip.. Bridged crack models Bridged crack models are non-linear finite element models to describe the cracking of fiber reinforced composites. They do this by modeling the fibers Figure.: Distributed modeling, converting reinforcement to a homogeneous layer of extra stiffness 5

22 . t Figure.: Bridged crack model. Traction t is applied at the crack surfaces crossed by the fibers that are trying to close the crack. They are modeled as discrete reaction forces or traction stresses along the crack surface. Studies of this model can be found in [6] or [7]. An illustration of the cracked matrix still being supported by fibers is given in figure.. The bridged crack models are able to accurately capture local displacement and stress behavior around the crack and fibers. The disadvantage of this method is that it is difficult to use in case of many arbitrarily placed fibers. In that case, it might be very difficult and computationally expensive to determine how the crack propagates and which fibers are still active in stiffening the matrix material.. Lattice models Lattice models offer another way of modeling composite materials. The material is divided in different areas (D) or volumes (D). The centers of these volumes are then given degrees of freedom (translation and or rotation). These centers are then connected by the lattices which are truss or beam elements. In order to simulate the cracking of a material the lattice element with the most critical stress is removed in each calculation loop. Due to this very explicit removal of lattices this model is suitable for fracture modeling. More on the solution algorithms used can be found in [8]. Fiber reinforced composites can be modeled with this technique in different ways. One way is to use an overlay of fibers over the different areas and model the point where the fiber crosses the boundaries as a spring with zero length (see fig.a). This spring stiffness is then transformed to global coordinates and added to the original degrees of freedom of the lattice structure as discussed in [9]. A second method is to model the fibers as separate lattices. This allows to simulate the bond relation matrix-fiber by adding extra nodes on the fiber and connecting them with bond lattices. More information can be found in []. This method is illustrated in figure.b. A disadvantage of the lattice model is that you can not simply implement a Poisson ratio into the model. A second disadvantage of the lattice approach is that it does not represent an elastic homogeneous material due to the arbitrary distributed 6

23 . regular lattice spring stiffness bond lattice fiber lattice (a) (b) Figure.: Lattice model using springs at the boundaries (a) or fibers as complete lattices including nodes and extra lattices for bond relation (b) Figure.: Conformal meshing, meshing the fiber together with the matrix lattices. This can be partially solved by using special Voronoi techniques to distribute the lattices as proposed in [9].. Conformal meshing and homogenization Conformal meshing is a method in which the fibers are made part of the mesh by meshing them along. An example for D is presented in figure.. The obvious downside of this method is the computational expensiveness. You can only calculate small pieces of material. However, if used in combination with the homogenization method as described in [], you are able to calculate general material behavior with it. The procedure calculates a general D matrix and is able to use that at a macro scale for the material as a whole. 7

24 η ζ Figure.5: Element with embedded reinforcement with extra degrees of freedom to model reinforcement bond slip.5 Embedded Reinforcement including bond slip (ERS) An old method to model reinforced concrete accurately is the embedded reinforcement method (ER). The model is based on the virtual work principle. In the simplest case, the reinforcement rebar is aligned with one of the element axes. The virtual work done by the rebar is calculated by integrating over the rebar and adding it up to the standard stiffness matrix of the underlying element. It is thereby assumed that the rebar undergoes the same strains as the element itself, hence, the same B matrix. This leads to the following definition for the stiffness matrix: K = B T DB dω + B T E r B dl (.) Ω Here the subscript r stands for the rebar and the integral over the rebar domain is a line integral. This method was proposed in []. Later extensions [,, 5] implemented the bond slip model (ERS) with possible arbitrary placement of the reinforcement. The bond slip model meant inclusion of additional degrees of freedom in direction of the fiber as displayed for a quad element in figure.5. This led to the following structure of the stiffness matrix for one element: [Kaa K ba K ab K bb r ] [ ] a = b [ fa ] (.) In this equation b represents the vector with additional degrees of freedom that model the bond slip. The complete stiffness matrix following from equation. then results in: Ω BT DB dω + r HT BT r E r B r H dl r HT BT r E r B r H dl r HT BT r E r B r H dl r HT BT r E r B r H dl+ (.) r N r T D b N r dl In this stiffness matrix N r and B r belong to one dimensional shape functions local to the rebar. The matrices H and H are transformation matrices 8

25 that connect the local coordinate system of the fiber to the global coordinate system. The embedded reinforcement method has many mathematical similarities with the GFEM method to model fiber reinforced composites. Compare for example equation. with equation 5 from []. Even so, it has never been used to model material with high amounts of arbitrary placed fibers. The downside of the embedded reinforcement method is that only a one dimensional representation of the fiber is possible in this method. That implies only bond slip in direction of the fiber and the material of the fiber would be modeled with just the Young s modulus and without the Poisson s ratio. 9

26 Chapter Mathematical derivations. Continuum without a fiber In this section a short derivation will be given of a finite element solution in a continuum without a fiber. Primary purpose is to give a recap and introduce the variable names and symbols as used in the rest of this document. The derivation is loosely based on [] and []... Problem statement For the continuum without a fiber we consider a continuous body Ω R n for which the displacement field is described by u. Furthermore, it has the following boundaries on Γ: Γ g, on which a Dirichlet boundary condition (displacement) is applied: u = g Γ h, on which a Neumann boundary condition (traction) is applied: t = h The problem is depicted in figure. Γ g h Ω n Γ h Figure.: Continuum without a fiber

27 .. Kinematics The displacement field is described by u. Following from this, the second order strain tensor is defined as the symmetric gradient of the displacement field: ɛ = s u (.).. Constitutives The constitutive law for linear elastic material connects the second order strain tensor to the second order stress tensor by a fourth order tensor with the material properties: σ = D m : ɛ (.).. Equilibrium The body under consideration (neglecting body forces) should be in static equilibrium. This implies balance of momentum: t dγ = σn dγ = (.) Γ Γ When applying the divergence theorem of Gauss this can also be written as σ dω = (.) Ω To obtain the virtual work equation we pre-multiply by a virtual displacement δu δu σ dω = (.5)..5 Weak statement Ω Next, we integrate by parts and rearrange the two terms to get the weak form: s δu : σ dω = δu t dγ (.6) Ω Γ h We insert the constitutive law defined above for σ: s δu : D m : ɛ dω = δu t dγ (.7) Ω Γ h And following to that we insert the kinematic law for ɛ s δu : D m : s u dω = Ω δu t dγ Γ h (.8) This is the weak equation that describes the solution to the problem. In the next section we discretize this formulation to be able to solve it numerically.

28 ..6 Discretization To solve the problem we discretize the displacement field in i unknowns u i that represent displacements on element nodes. The displacement field between the nodes is the sum of the values u i times their shape functions N i (x). The shape functions are dependent on spatial coordinates x. The displacement values (amplitudes) u i are scalars. u = n N i (x)u i (.9) i= Next, we need to discretize the symmetric gradient of the displacement field. Since we have isotropic material, we can simplify this second order tensor into a vector. This derivation is shown in appendix A. This also implies that the fourth order elasticity tensor D m is simplified into a second order tensor called the D matrix. Using vector a for the amplitude values u i, the displacement field is discretized in the well known FEM matrix notation as follows: u = Na (.) ɛ = s u => Ba (.) Here N is the matrix containing the shape functions and B is the matrix containing the derivatives of the shape functions. In the second equation a => is used on purpose to indicate the transformation from second order tensor to vector. Using the Galerkin approach the virtual displacements use the same basis functions as the ordinary displacements. Hence: δu = Na v (.) s δu => Ba v (.) Next we enter all quantities in equation (.8): (Ba v ) T D(Ba) dω = Ω (Na v ) T t dγ Γ h (.) The nodal unknowns a and the virtual displacement a T v can be moved out of the integral which leads to: a T v B T DB dωa = a T v Ω N T t dγ Γ h (.5) We can cancel the virtual displacement to obtain the finite element definition: B T DB dωa = N T t dγ (.6) Ω Γ h In which the first integral represents the stiffness matrix K and the second integral represents the external forces f

29 Γ g matrix fiber h Ω + Γ + n Γ h Ω Γ Figure.: Continuum with a fiber. Continuum with one fiber.. Problem statement For the continuum with a fiber we consider a continuous body Ω which consists of the fiber domain Ω + and the matrix domain Ω. The problem is depicted in figure. Ω = Ω + Ω (.7) The displacement field is described by: u = û + ũ (.8) where û represents the continuous displacement field of the matrix material. At the location of the fiber the field contains displacement jumps that result in a discontinuous field. This allows the fiber to slip relatively to the matrix. This slip part of the displacement is represented by ũ. For the moment it is left free which degrees of freedom are contained in ũ, since this will differ in the GFEM, IGFEM and ERS implementation. Furthermore the following boundaries are defined: Γ g, on which a Dirichlet boundary condition (displacement) is applied: u = g Γ h, on which a Neumann boundary condition (traction) is applied: t = h Γ +, the fiber matrix boundary seen from the fiber domain. Γ, the fiber matrix boundary seen from the matrix domain... Kinematics As mentioned above the displacement field is described by: u = û + ũ (.9)

30 Following from this, the second order strain tensor is defined as the symmetric gradient of the displacement field: ɛ = s u = s û + s ũ (.) Another quantity we need to describe, in order to describe the bond interface between matrix and fiber, is the displacement jump between matrix and fiber. We define this vector [[u]] as the difference in the displacement field considered from the different fiber boundary sides. [[u]] = u Γ+ u Γ+ (.) If we fill in the displacement field here, taking into account that the fiber slip ũ on the matrix side of the domain is zero and the continuous displacement û field does not change over the boundary, we end up with: [[u]] = (û Γ+ + ũ Γ+ ) (û Γ + ũ Γ ) = ũ Γ+ (.) As expected the displacement jump equals the fiber slip on the fiber domain... Constitutives For the constitutive laws we need to describe three different laws: the matrix, the fiber and the bond between the two. All three are considered to be linear elastic and they read: σ = D m : ɛ (.) σ = D f : ɛ (.) τ = D b [[u]] (.5) In which D is the fourth order material tensor for matrix and fiber respectively. It maps in both cases the second order tensor for the strain ɛ to the second order tensor for the stresses σ. Furthermore we have the second order tensor for the bond between matrix and fiber which is described by D b. It maps the vector for the displacement jump from matrix to fiber to the tractions τ. To make the bond more understandable it can be seen as the fiber being connected to the matrix material with an infinite amount of little springs. For example, in two dimensions this would be springs in vertical and horizontal direction with their spring constants defined on the diagonals in D b... Equilibrium The body under consideration (neglecting body forces) should be in static equilibrium. This implies as in the previous section balance of momentum of the whole body: t dγ = σn dγ = (.6) Γ Γ

31 When applying the divergence theorem of Gauss this can also be written as σ dω = (.7) Ω To obtain the virtual work equation we pre-multiply by a virtual displacement δu δu σ dω = (.8) Ω Now we can make the first splitting step in the derivation by inserting the virtual displacement field δu = δû + δũ. δû σ dω + δũ σ dω = (.9) Ω We can further split up this integral by considering that we can consider the different domains Ω and Ω + δû σ dω + Ω δũ σ dω + Ω + δũ σ dω = Ω (.) If we consider, as before, that the fiber slip is zero on the matrix domain, we are left with only one integral over the fiber and one over the whole domain: δû σ dω + Ω δũ σ dω = Ω + (.) Next we integrate by parts and rearrange the four terms to get the weak form. Ω s δû : σ dω + δũ σn + dγ Γ + s δũ : σ dω = Ω + δû σn dγ Γ h Ω (.) In figure. a close up is given of how the fiber resides inside the matrix. Coming from the figure the following relations exist for the tractions along the outside boundary and the fiber boundary, respectively: t = σn on Γ (.) τ = σn = σn + on Γ + (.) Since the traction around the fiber is defined as positive from the matrix to the fiber, we switched the sign using the normal vector in the above equation. Inserting the traction law and shifting the terms around results in: Ω s δû : σ dω + δũ τ dγ + Γ + s δũ : σ dω = Ω + δû t dγ Γ h (.5) 5

32 matrix fiber τ +τ slip ũ Figure.: Close up of the fiber inside the matrix..5 Weak statement Since the virtual displacement in the above equation is arbitrary, we can split the equation to obtain two separate weak equations. s δû : σ dω = δû t dγ (.6) Γ h Ω δũ τ dγ + Γ + s δũ : σ dω = Ω + (.7) Next, we insert the constitutive laws defined above for the stress tensor σ and thereby splitting the first integral into the fiber and matrix domain: s δû : D m : ɛ dω + Ω s δû : D f : ɛ dω = Ω + δû t dγ Γ h (.8) s δũ : D f : ɛ dω + Ω + δũ D b [[u]] dγ = Γ + (.9) Now we can insert the kinematic relations for ɛ. Thereby we have to keep in mind that the slip part of the displacement field ũ is still zero at the matrix domain Ω. Also for the displacement jump it holds that [[u]] = ũ Γ+ = ũ on Γ + s δû : D m : s û dω + s δû : D f : s û dω Ω Ω + + s δû : D f : s ũ dω = δû t dγ (.) Ω + Γ h 6

33 s δũ : D f : s û dω + s δũ : D f : s ũ dω Ω + Ω + + δũ D b ũ dγ = (.) Γ + Now the first term in equation (.) contains an integral over Ω which is a difficult term to calculate. We can simplify the problem by converting it to an integral over the whole domain Ω if we subtract the double integrated volume part over Ω + from the second term. We end up with the following final weak form for the problem: Ω s δû : D m : s û dω + s δû : (D f D m ) : s û dω Ω + + s δû : D f : s ũ dω = δû t dγ (.) Ω + Γ h s δũ : D f : s û dω + s δũ : D f : s ũ dω Ω + Ω + + δũ D b ũ dγ = (.) Γ +..6 Fiber thickness To discretize the problem we need to determine the dimensions of the problem. If we have a D problem all integrals over a domain Ω are volume integrals. The integrals over a boundary Γ are then surface integrals. Since fibers are in one direction bigger then in the other two, we can simplify the fiber to a line element. Then the integrals over the fiber domain and surface change to line integrals:... dω = A f... dl (.) Ω + l... dγ = C f... dl (.5) Γ + l In this simplification A f and C f are the cross-sectional area and the circumference of the fiber. We assume that they are constant over the fiber length. If one makes a further simplification to a dimensional problem, as done in the rest of this report, one usually takes a unit thickness of to describe the problem sample. This can be done as well with fibers. In this case we get: A f = t f (.6) C f = or (.7) 7

34 In which t f is the thickness of the fiber in the dimensional space. As a parameter of choice the circumference can be or. This simply means you consider the fiber as a dimensional line that is only bonded at side or a more dimensional fiber that has a bond on both sides. In previous publications on GFEM ([]) the fibers are bonded at one side only in dimensional problems. With the above assumptions and simplifications the integral terms in the weak form become better understandable: Ω s δû : D m : s û dω + A f s δû : (D f D m ) : s û dl l + A f s δû : D f : s ũ dl = δû t dγ (.8) Γ h l A f s δũ : D f : s û dl + A f s δũ : D f : s ũ dl l l + C f δũ D b ũ dl = (.9)..7 Discretization GFEM and IGFEM In this section we will discretize the weak equations previously found for the GFEM and IGFEM approach. The next section describes the differences in the ERS approach. The displacement field was equal to: l u = û + ũ (.5) In this we defined ũ as the continuous displacement field responsible for modeling the slip of the fiber. It is still not defined which degrees of freedom are modeled with û. We define the separate parts of the displacement field as: n û = Ni a (x)a i (.5) i= ũ = χ(x) m Nj b (x)b j (.5) j= So to solve the above described problem we discretize the displacement field in i + j unknown amplitudes a i and b j. Next we describe the displacement field as an interpolation of the sum of these amplitudes times their shape functions N i (x) and N j (x). The shape functions are dependent on spatial coordinates x. The amplitudes are then simply scalars. In the above we also introduced a special enrichment function χ which is a scalar that depends on the coordinates. This function is responsible for capturing the discontinuous behavior of the displacement field around the fiber. To illustrate 8

35 this function one can consider the simple case in which χ = on the fiber and χ = elsewhere in the domain. If we write the above in matrix vector notation it results in: û = N a a (.5) ũ = χn b b (.5) Next we need to discretize the symmetric gradient of the displacement field; the strain tensor. Since we have isotropic material we can simplify this second order tensor into a vector. Before doing that we first write down the strain tensor from this enriched solution field. ɛ = s u = s û + s ũ = s (N a a) + s (χn b b) (.55) = ( s N a ) a + ( s χn b ) b (.56) = ( s N a ) a + ( ( s χ) N b + χ( s N b ) ) b (.57) As in the case without a fiber this strain tensor can be simplified to engineering strain which will then become a vector. The only complication is now the extra term with the gradient of the enrichment function. The full derivation from strain tensor to strain vector is shown in Appendix A.. What we end up with for the two parts of the strain vector is: ) s û => (L N a a (.58) s ũ => ( (Lχ) N b + χ(l N b ) ) b (.59) Here L is the matrix containing the directional derivatives. If we insert the standard B matrix: s û => B a a (.6) s ũ => ( (Lχ)N b + χb b ) b (.6) As a side note we can mention that in the simplest case of an enrichment function with χ = on the fiber and χ = elsewhere this strain vector turns into a much easier form since the derivatives of the enrichment function are zero. This form is also used in []. The formulation changes into: s û => B a a (.6) s ũ => χb b b (.6) With (.6) and (.6) as a tool we can also write down definitions for the virtual displacements that use the same shape functions on basis of the Galerkin approach: δû = N a a v (.6) δũ = χn b b v (.65) 9

36 s δû => B a a v (.66) s δũ => ( (Lχ)N b + χb b ) bv (.67) A last term that has to be discussed is the bond term. The bond D b is defined in a local coordinate system with tangential and normal slip coefficients in two dimensions: [ ] kbt D b = (.68) k bn This bond term is oriented in a coordinate system local to the fiber. The slip displacement field in GFEM and IGFEM is global and thus the bond term should be rotated by means of a standard rotation matrix: D b = R T D b R (.69) Now we can insert the above quantities for the displacements and strains in the earlier derived weak form. Note again that not only the second order strain tensors are converted to vectors, but also the fourth order elasticity tensors convert to second order tensors; matrices. This results in the following equations: (B a a v ) T D m B a a dω + A f (B a a v ) T (D f D m ) B a a dl Ω l + A f (B a a v ) T D f ((Lχ)N ) b + χb b b dl = (N a a v ) Γ T t dγ (.7) h l ( ) ((Lχ)Nb ) T A f + χb b bv D f B a a dl l ( ) ((Lχ)Nb ) T + A f + χb b bv D f ((Lχ)N ) b + χb b b dl l ( + C f χn b b v ) T R T D b R χn b b dl = (.7) l We proceed as in the non-fiber case with cancelling the virtual displacements a v and b v and we take amplitudes a and b outside the integral terms since they are not coordinate dependent. Ω Ba T D m B a dωa + A f Ba T (D f D m ) B a dla l + A f Ba T D f ((Lχ)N ) b + χb b dlb = Na Γ T t dγ (.7) h l

37 ( ) T A f (Lχ)Nb + χb b Df B a dla l ( ) T + A f (Lχ)Nb + χb b Df ((Lχ)N ) b + χb b dlb l + C f χnb T RT D b R χn b dlb = (.7) With the above two equations the system is discretized into a form in which it is practical to implement in code. We can write these equations in the matrix form: [ ] ] Kaa K ab K ba K bb l ] [ a b = [ fa (.7) It has the following components: K aa = Ba T D m B a dω + A f Ba T (D f D m ) B a dl (.75) Ω l K ab = A f Ba T D f ((Lχ)N ) b + χb b dl (.76) l ( ) T K ba = A f (Lχ)Nb + χb b Df B a dl (.77) l ( ) T K bb = A f (Lχ)Nb + χb b Df ((Lχ)N ) b + χb b dl+ (.78) l C f χ Nb T RT D b R N b dl (.79) l f a = Na T t dγ (.8) Γ h..8 Discretization ERS The discretization for the embedded reinforcement approach is different since the slip degrees of freedom have their own coordinate system (x ) local to the fiber. The displacement field of the fiber is built up of a regular part similar to the matrix displacement and a slip part. The key to understanding the difference between the approaches is that local degrees of freedom on the fiber require the whole displacement field of the fiber to be local to the fiber. Otherwise a correct assembly in the globally oriented stiffness matrix is impossible. This leads to both parts of the fiber displacement field having their own one dimensional shape functions local to the fiber (N b ). This is fundamentally different with GFEM and IGFEM where the underlying field of shape functions of the matrix is used to describe the displacement of the fiber. In order to assemble this field correctly into the global stiffness matrix we need to transform both parts to global degrees of freedom. We introduce transformation matrices H and H. These transform the local coordinate system to a global system. We use two of them to be able to split

38 the displacement field into the matrix displacement and the fiber slip. This also allows to have an enrichment function χ. In the element derivations in the next chapter the content of these matrices role will become clear. The regular displacement field of the matrix stays similar but the regular displacement field of the fiber domain (Ω + ) is also subject to a transformation matrix. This leads to the following discretization: û = û Ω + = n Ni a (x)a i (.8) i= n Ni b (x )H a i (.8) i= ũ Ω + = χ(x) In matrix notation this becomes: m Nj b (x )H b j (.8) j= û = N a a (.8) û Ω + = N b H a (.85) ũ Ω + = χn b H b (.86) Using the same principles as before the strain field for the different domains can be derived. This results in: s û => B a a (.87) s û Ω + => B b H a (.88) s ũ Ω + => ( (Lχ)N b + χb b ) H b (.89) Working out the equations leads to a system of equations with the same structure as GFEM and IGFEM: [ ] [ ] [ ] Kaa K ab a fa = (.9) b K ba K bb However the terms are slightly changed. First transformation matrices H and H have been introduced. Second the regular displacement field over the fiber in K aa is now described by shape functions belonging to the fiber itself (B b ). Third the bond term inside K bb now does not require a rotation matrix since the slip dof s and the bond term are defined in the same

39 coordinate system K aa = Ω Ba T D m B a dω + A f H T Bb T (D f D m ) B b H dl l (.9) K ab = A f H T Bb T D f ((Lχ)N b + χb b ) H dl l (.9) K ba = A f H T ((Lχ)N ) T b + χb b Df B b H dl l (.9) K bb = A f H T ((Lχ)N ) T b + χb b Df ((Lχ)N b + χb b ) H dl+ l (.9) C f χ Nb T D b N b dl l (.95) f a = Na T t dγ Γ h (.96)..9 Dimensional check of K A short sanity check is made of the dimensions of the derived stiffness matrices. Terms inside the stiffness matrix should always represent a stiffness; [Nm ]. For all methods (IGFEM, GFEM and ERS) the equations are similar in dimensions since the transformation matrices inside the ERS approaches have no dimensions. K aa = [m ][Nm ][m ][m ] + [m ][m ][Nm ][m ][m ] (.97) K ab = [m ][m ][Nm ]([m ][ ] + [m ])[m ] (.98) K ba = [m ]([m ][ ] + [m ])[Nm ][m ][m ] (.99) K bb = [m ]([m ][ ] + [m ])[Nm ]([m ][ ] + [m ])[m ]+ (.) [m][ ][Nm ][ ][m] (.) f a = [ ][Nm ][m ] (.) As can be seen each stiffness term represents a force per length. The right hand side of the equations represents a force.