An Adaptive Hybrid Method for 2D Crack Growth Simulation

Transcription

1 Institut für numerische und angewandte Mathematik An Adaptive Hybrid Method for D Crack Growth Simulation Diplomarbeit eingereicht von Jan Hegemann betreut von Prof. Dr. Martin Burger Prof. Dr. Joseph Teran Dr. Casey Richardson Münster August 009

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3 Abstract We present a new algorithm for the simulation of two dimensional quasi-static crack growth. It combines the extended finite element method (XFEM) with a general cutting algorithm that provides the degrees of freedom for the crack to open as well as determines material connectivity. Further, we introduce a general and easy quadrature scheme. Our hybrid method uses an adaptive integration mesh embedded into a coarse simulation mesh that represents the actual degrees of freedom. Thereby, we gain some of the accuracy of refining locally around the crack as well as a smoother crack path without adding any new degrees of freedom. Also, our approach is easy to implement and does not need a triangulization that incorporates the crack faces. The direction of crack propagation is determined through the stress intensity factors which are computed on the fine integration mesh.

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5 Contents 1 Introduction 1 Physical and Mathematical Background 3.1 Setting Governing Equations Cauchy Stress Tensor Equilibrium Equation Symmetry of the Stress Tensor Linear Elasticity Weak Formulation Fracture Modes The Energy Balance Approach and the J-Integral Crack Propagation Implementation Discretization The Cutting Algorithm Comparison to Heaviside Representation Enrichment Hybrid Mesh Refinement Bound Particles Integration Solving the FEM System CG Algorithm Condition of the Stiffness Matrix Remarks Computation of the Interaction Integral Computation of the Strain i

6 Contents Kinked Cracks Numerical Results Opening Example Shear Example Complicated Geometries Example of Crack Propagation Conclusions and Outlook Conclusions Outlook ii

7 List of Figures.1 Fracture modes Crack tip with integration paths Cutting algorithm: duplicating nodes Cutting algorithm: geometry example Difference between Heaviside representation and cutting algorithm Red-green-refinement Refined geometry processes by the cutting algorithm Binding a virtual node Low resolution projection of F High resolution projection of F Condition: setting of the triangle example Eigenvalues of the stiffness matrix for the triangle example Condition of the stiffness matrix for the triangle example Condition: settings of the square example Condition for the square example: different crack lengths Condition for the square example: shifted crack Condition for the square example: rotated crack Mapping for kinked cracks Setting opening example Results opening example Illustration of the results for the opening example Setting shear example Results shear example Illustration of the results for the shear example Complicated crack geometry: simulation mesh Complicated crack geometry: quadrature mesh Complicated crack geometry: opening iii

8 List of Figures 4.10 Crack propagation: initial configuration Crack propagation: 1 iteration Crack propagation: 5 iterations Crack propagation: 10 iterations Crack propagation: 15 iterations Crack propagation: 16 iterations iv

9 Acknowledgements First of all I would like to thank Prof. Dr. Martin Burger for not only letting me write my diploma thesis with him but also sending me to the University of California, Los Angeles (UCLA) and therefore giving me the unique experience I had there. Additionally, I thank Prof. Dr. Joseph Teran, CAM Assistant Professor at UCLA, who included me in his project and work group and introduced me to the topic of fracture mechanics. During my time at UCLA Dr. Casey Richardson, CAM Assistant Adjunct Professor at UCLA, was a good and productive colleague with whom I enjoyed working. Dr. Eftychios Sifakis, post-doctoral scholar at UCLA, provided a lot of PhysBAM and programming knowledge. Ralf Engbers, Michael Möller, Paul Bunn and Hem Wadhar helped me a lot by proofreading this thesis and making useful remarks. For the same reasons and also her support I thank my girlfriend Rachel Danson. Furthermore, I would like to thank all my friends at WWU, UCLA and elsewhere for a great, productive study environment and their support. Last but not least I am very grateful for my family and all their support as well as providing me with the great opportunities I have had in my life. v

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11 1 Introduction Fracture and crack growth is an important field within the simulation of solid mechanics. Applications range from engineering of buildings and vehicles with a natural interest in a high accuracy to graphical applications where a fast computation along with realistic appearance is more important than actual physical realism. In this thesis we limit ourselves to two dimensional quasi-static crack growth. Quasistatic means that kinetic effects are assumed negligible throughout the whole process and at each time step the system is presumably in a state of equilibrium and therefore the solution is no longer time-dependent. For our approach we will use the finite element method (FEM). This incorporates some difficulties since in a classical setting the topology changes due to the crack propagation make remeshing a necessity. In [BB99] and [MDB99] Belytschko, Moes et al. introduced the extended finite element method (XFEM). The idea is to enrich certain mesh nodes around the crack with additional degrees of freedom and extend the usual FEM function space by basis functions that include the asymptotic near tip solutions and are partly discontinuous across the crack. Therefore, remeshing of the domain can be avoided. These enrichment functions have gradients that are singular at the tip. Since assembling the stiffness matrix for the FEM system requires integration of these singular gradients, some problems arise. The usual approach is to retriangulate the elements that contain the crack and then perform e.g. a Gaussian quadrature. This retriangulization is for integration purposes only and does not introduce new degrees of freedom. However, a complicated crack geometry that includes branching or multiple cracks can lead to a very complicated and expensive implementation. Also, it is not generalizable to three dimensions without introducing new vertices. Further, the computational determination of the material connectivity, which changes as the crack propagates, might cause problems. Since the degrees of freedom that allow the crack to open are created by using a Heaviside-type function that takes the value 1 on one side of the crack and 1 on the other, the orientation for every node with respect to the crack must be determined. This is easy for one single, straight crack. In [DND + 00], [BZMB04] and [ZSB + 04] it has been expanded to the case of branched, multiple and even 1

12 1 Introduction intersecting cracks. However, a general method to handle arbitrary cracks and resolve the material connectivity systematically is missing. Another basis of our work is the cutting algorithm we use. Introduced in [SDF07], it not only allows the creation of the degrees of freedom needed for the crack opening with the use of so-called virtual nodes, but it also resolves material connectivity for arbitrary crack geometries and the orientation of all nodes with respect to the crack. We use this to replace the Heaviside-type enrichment of XFEM. Further, we address the problematic integration near the crack tip with a hybrid approach based on [SSIF07] by Sifakis et al. We solve the equilibrium equation on a simulation mesh with relatively few degrees of freedom but perform all necessary integrations on a quadrature mesh that is refined locally around the crack. This fine mesh is embedded into the coarse one without introducing any new degrees of freedom. Then for quadrature purposes the enriched functions are interpolated by the piecewise linear basis functions of the finer mesh. We therefore gain some of the increased accuracy of refining around the crack but without additional degrees of freedom. In Chapter we will begin with giving some background knowledge about fracture mechanics and the governing equations. We will derive a weak formulation for the later implementation and describe our method to propagate the crack. In Chapter 3 we are going to go into the details of our implementation. We will explain the use of the enrichment as well as our cutting algorithm and hybrid integration. Various numerical examples and results, computed with the use of PhysBAM c 1, will be presented in Chapter 4. These examples will highlight varying considerations, including accuracy, crack propagation, and the ability of our method to handle complicated crack geometries. We will close in Chapter 5 with some conclusive remarks and further sketch the work of [LRS], which might be a good addition to the method we present in this thesis. A summary of our method and results can also be found in [RHS + ]. 1

13 Physical and Mathematical Background In this chapter we give an introduction into the mathematical description of fracture mechanics. We begin with the variables and governing equations used to describe fracture, and we point out common assumptions. We then derive an equilibrium condition, which we also express through its weak formulation suitable for a finite element computation. Lastly, we present some results on crack propagation and the criterion we use for our simulations..1 Setting We consider a body with a material region represented by the domain Ω R with boundary Ω. Ω is subject to forces which consist of a traction t, applied on Γ t Ω, and, in general, body forces f (e.g. gravity). Further, we assume an initial crack Γ C. The goal is to find a deformation φ : Ω R that maps the undeformed configuration to the deformed configuration Ω d. Therefore, we will solve for the displacement defined as u = φ id. (.1) We make the common assumption of a quasi-static evolution, i.e. inertial effects are neglected and the material is assumed to be in an elastic equilibrium at every time step and thus the solution is no longer time-dependent.. Governing Equations In order to stick to common conventions this section discusses the concepts of fracture mechanics in a three dimensional setting. For our two dimensional case we then consider 3

14 Physical and Mathematical Background the x-y-plane with forces acting within the plane...1 Cauchy Stress Tensor Consider a deformable body subject to a force f. For a point P within this body choose a plane passing through P and dividing the body into two parts. Let a be an area on the cutting plane of one of the segments and around P with normal vector n. This area of material is subject to a force p caused by the action of the material of the other part of the body. The traction vector is defined as ([BW08]) p t(n) = lim a 0 a. (.) To obey Newton s third law of motion the traction on one side of the surface must be equal in magnitude and of opposite direction to the traction on the other side, i.e. t(n) = t( n). (.3) The state of stress at point P is now given by the traction vectors for all three Cartesian directions, which can further be decomposed into components in the directions of e 1, e and e 3 yielding t(e 1 ) = σ 11 e 1 + σ 1 e + σ 13 e 3 (.4) t(e ) = σ 1 e 1 + σ e + σ 3 e 3 (.5) t(e 3 ) = σ 31 e 1 + σ 3 e + σ 33 e 3 (.6) or in short form 3 t(e i ) = σ ij e j. (.7) j=1 Equation (.7) allows the following definition. Definition..1. Cauchy Stress Tensor For an arbitrary point P the nine components of the three traction vectors associated with the three Cartesian directions form a second-order tensor which completely describes the stress at P : σ = (σ ij ) ij (.8) 4

15 Physical and Mathematical Background for i, j {1,, 3}. σ is called the Cauchy stress tensor. Among other properties we will see that σ is in fact a second-order tensor... Equilibrium Equation In this section we will derive an equation to describe a general equilibrium state. At first we need an expression for t(n) in terms of σ. Therefore, we consider a tetrahedron that is formed by the three coordinate planes and a small arbitrarily oriented area da with normal vector n. In equilibrium all forces sum up to zero: t(n)da + 3 t( e j )da j + fdv = 0, (.9) j=1 where da j = (n e j )da is the projection of da onto the the coordinate plane orthogonal to e j and dv is the volume of the tetrahedron. Hence, applying (.3) on the second term and rearranging yields t(n) = 3 j=1 t(e j ) da j da f dv da. dv By noticing that lim da 0 da = 0 we obtain (.7) leads to t(n) = 3 t(e j )(n e j ), (.10) j=1 and thus 3 t(n) = σ ij (n e j )e i, (.11) i,j=1 t(n) = σn. (.1) 5

16 Physical and Mathematical Background With the definition of the tensor product we also see from (.11) that σ actually is a second-order tensor: σ = σ ij (e i e j ). (.13) To derive a general equilibrium equation for a deformed body we consider a body with material in the domain Ω subject to body forces f and a traction t on the surface Ω. In an equilibrium state the sum of all acting forces has to vanish: With (.1) this is equivalent to Ω Ω tds + fdω = 0. (.14) Ω σnds + fdω = 0. Ω Gauss theorem leads to Ω (divσ + f)dω = 0, (.15) where we use divσ = i,j σ ij x j e i. (.16) Since this holds for any arbitrary volume of the body the integrand itself must vanish, which leads to the equilibrium equation divσ + f = 0 in Ω. (.17) We will often use the case of an equilibrium with no body forces, i.e. f = 0. The equilibrium equation then simplifies to divσ = 0 in Ω. (.18) 6

17 Physical and Mathematical Background The boundary conditions in both cases are given by u = ū on Γ u (.19) σ n = t on Γ t (.0) σ n = 0 on Γ c, (.1) where Ω = Γ u Γ t Γ c and Γ c denotes the crack surface. Equation (.1) expresses the assumption that crack faces are traction free...3 Symmetry of the Stress Tensor Another important property of the stress tensor is its symmetry. It can be observed by considering a rotational equilibrium of a body which is subject to body forces f and a traction t on the surface. The equilibrium requires that the total momentum about any point vanishes, i.e. for the origin this implies Ω With (.1) we can rewrite (.) as Ω x tds + x fdω = 0. (.) Ω x (σn)ds + x fdω = 0, Ω and by using the third-order alternating tensor ɛ ijk this identity is further equivalent to Ω j,k,l ɛ ijk x j σ kl n l ds + Ω ɛ ijk x j F k dω = 0 for i = 1,..., 3. j,k Finally Gauss theorem and x j x l = δ jl lead to Ω( j,k ɛ ijk σ kj + j,k,l ɛ ijk x j σ kl x l + j,k ɛ ijk x j F k )dω = 0 for i = 1,..., 3. 7

18 Physical and Mathematical Background Since the second and third term cancel out due to the equilibrium equation (.17), we obtain ɛ ijk σ kj dω = 0 for i = 1,..., 3. Ω j,k Again, this identity holds for an arbitrary volume Ω, thus the integrand itself must vanish ɛ ijk σ kj = 0 for i = 1,..., 3, (.3) j,k which results by definition of ɛ ijk in the symmetry of the stress tensor σ ij = σ ji for i, j = 1,..., 3. (.4)..4 Linear Elasticity In our work we consider the case of small deformations, strains and displacements as well as isotropic homogeneity ([UF03]). This allows us some simplifications of the above equations. For small deformations we can make the common assumption of a linear relationship between σ and the Cauchy strain tensor ɛ given by the generalized Hooke s law where C is the fourth-order elasticity tensor. σ = C : ɛ, (.5) For an isotropic homogeneous material this relation can be expressed by ([BW08]) with the Lamé coefficients depending on the material constants σ = µɛ + λ tr(ɛ) I (.6) νe λ = (1 + ν)(1 ν), (.7) E µ = (1 + ν) (.8) 8

19 Physical and Mathematical Background E: Young s modulus, ν: Poisson s ratio. Further, we assume small strains ɛ and small displacements u. Therefore, ɛ is given by the symmetric part of the gradient of u, ɛ = ɛ(u) = u + ut = S u, (.9) or in component notation ɛ ij = 1 ( ui + u ) j. (.30) x j x i Hence, equation (.6) becomes σ = µɛ + λ divu I. (.31) The combination of equations (.18), (.9) and (.31) leads to a second order partial differential equation for the displacement u in an equilibrium state given by µ divɛ(u) + λ grad divu + f = 0 on Γ u (.3) and hence µ div( S u) + λ grad divu + f = 0 on Γ u, (.33) with the same boundary conditions as above u = ū on Γ u (.34) σ n = t on Γ t (.35) σ n = 0 on Γ c. (.36) 9

20 Physical and Mathematical Background.3 Weak Formulation We consider the equilibrium equation (.17) again and derive a weak formulation suitable for the finite element method (cf. [Bra07]). We define the function space H 1 Γ u (Ω) = {v : Ω R v H 1 (Ω), v = 0 on Γ u }, (.37) multiply the equilibrium equation (.17) with test function v [H 1 Γ u (Ω)] and integrate over Ω to obtain 0 = Applying Gauss theorem yields 0 = + Ω Ω σ vdω + f vdω. (.38) Ω σ : vdω + σn vds + Γ t + σn vds σn vds Γ } u {{} =0, (.37) Γ c + } {{ } =0, (.1) f vdω Ω + σn vds, (.39) Γ c } {{ } =0, (.1) where we used the traction free condition (.1) and the definition of the test function space H 1 Γ u (Ω). n denotes the corresponding normal vector. Further, we split v in its symmetric part (see equation (.9)) and its anti-symmetric part S v = ɛ(v) (.40) AS v = v vt. (.41) We then use the symmetry of σ to observe σ : AS v = 0. (.4) Further, with traction boundary condition t = σn, the weak formulation for the equilibrium problem (.17) is given by Ω σ : ɛ(v)dω t vds f vdω = 0 v [HΓ 1 u (Ω)], (.43) Γ t Ω 10

21 Physical and Mathematical Background or in component notation 1 Ω i,j ( vi σ ij + v ) ( ) j dω t i v i ds f i v i dω = 0 v [HΓ 1 x j x u (Ω)]. (.44) i Γ t i By expressing σ in (.43) by Hooke s law (.5) and using the symmetry of the elasticity tensor C, the weak formulation (.43) can be written as Ω ɛ(u) : C : ɛ(v)dω t vds f vdω = 0 v [HΓ 1 u (Ω)]. (.45) Γ t Ω The obvious symmetry and linearity of this equation in u and v allow us to write it in terms of the symmetric bilinear form a(u, v) = = Ω Ω ɛ(u) : C : ɛ(v)dω (.46) (µ ɛ(u) : ɛ(v) + λ divu divv) dω (.47) = µ ɛ(u), ɛ(v) L + λ divu, divv L, (.48) where we used the constant Lamé coefficients and (.31) as well as the L scalar product, and the linear form l(v) = Ω f vdω + t vds (.49) Γ t t vds. (.50) Γ t = f, v L + Therefore, the weak formulation can be expressed by a(u, v) = l(v) v [H 1 Γ u (Ω)]. (.51) We can now try to find a solution u ū + [H 1 Γ u (Ω)] which satisfies these equations. The symmetry and linearity will further lead to a symmetric stiffness matrix as we will see in Section 3.1. Equivalently to the above, we can try to find a solution by minimizing the total po- 11

22 Physical and Mathematical Background tential energy functional defined as (see [Bra07]) Π(u) = 1 Ω ɛ(u) : C : ɛ(u)dω t uds f udω min. (.5) Γ t Ω u ū+[hγu 1 (Ω)] The necessary condition for a minimum u requires that all directional derivatives for all directions v vanish. This leads to 0 = DΠ(u)[v] = d dδ δ=0π(u + δv) [ = d dδ 1 δ=0 ɛ(u) : C : ɛ(u)dω + δ 1 ɛ(u) : C : ɛ(v)dω Ω Ω + δ 1 ɛ(v) : C : ɛ(v)dω t uds δ t vds Ω Γ t Γ t ] f udω δ f vdω = Ω Ω Ω ɛ(u) : C : ɛ(v)dω t vds Γ t which is identical to the weak formulation (.45). Ω f vdω,.4 Fracture Modes There are three basic types of crack tip deformation ([And05]), see Figure.1: Mode I: Mode II: Mode III: opening mode - force is applied normal to the crack plane sliding mode / in-plane shear - force is applied parallel to the crack plane and normal to the crack front tearing mode / out-of-plane shear - force is applied parallel to the crack plane and parallel to the crack front 1

23 Physical and Mathematical Background Figure.1: The three basic fracture modes 1 These modes are used to fully describe general crack growth behavior through the superposition principle. Obviously, in D only Mode I and Mode II apply. Thus, we will consider mixed Mode I/Mode II fracture for our propagation..5 The Energy Balance Approach and the J-Integral The J-integral, introduced by Rice in [Ric68], is an important instrument in the analysis of crack propagation ([Ewa84], [And05]). It is based on an energy balance approach which we want to present first and that was introduced by Griffith [Gri1] and improved by Irwin [Irw97]. Consider a loaded elastic plate with an initial crack of length a. Its total energy U can be written as U = U 0 + U a + U γ F (.53) with

24 Physical and Mathematical Background U 0 : U a : U γ : F : total elastic energy of the loaded but uncracked plate (constant), change in the stored elastic energy due to the formation of the initial crack; introducing the crack causes the plate to lose stiffness and therefore potential energy, so U a < 0, change in the elastic surface energy due to the introduction of new surface along the crack; energy is consumed to disrupt the chemical bonds between the molecules of the material and stored in the new surface, so U γ > 0, work performed by external forces. Crack growth can only occur if an increment of the crack length a decreases the stored energy, which yields du da With the definition of U in (.53) and U 0 constant this means which can be rearranged to where 0. (.54) d da (U a + U γ F ) 0, (.55) d da (F U a) du γ da, (.56) = df da : du a da : ( df da du ) a : energy available for crack growth, da du γ da : energy provided by external work per crack increment da, elastic energy released by a potential crack increment da, surface energy of the crack surface per crack increment da. In other words: In order for the crack to grow the energy provided by external forces and released from the plate by a crack increment must exceed the energy needed to form the new crack surfaces. 14

25 Physical and Mathematical Background We now consider a two dimensional cracked body free of body forces (i.e. f = 0) and subject to a constant traction t and therefore to a displacement u. Definition.5.1. Strain Energy Density The strain energy density W as a function of ɛ is defined as W = W (ɛ) = i,j ɛij 0 σ ij (ɛ)dɛ ij. (.57) So for linear elasticity, where σ is a linear function of ɛ (see (.6)), the strain energy density is given by Using (.57), we can define the J-integral as follows. Definition.5.. J-Integral W = 1 σ ij ɛ ij. (.58) i,j For a cracked body (two dimensional, of linear or elastic-plastic material, in a state of equilibrium) free of body forces and subject to a constant traction t and a displacement u, assume that the crack is straight in a sufficiently small neighborhood around the tip such that we can choose the coordinate-system to be as follows: the origin is located in the tip, x 1 is aligned with the crack and x is perpendicular. Let Γ be an arbitrary but fixed counterclockwise path within the body and surrounding the crack tip. Then the J-integral is defined as J = = Γ Γ ( W dx t u ) ds x 1 ( ) W dx i t i u i x 1 ds (.59). (.60) An important attribute of the J-integral is given in the following theorem. Theorem.5.1. Path Independence of the J-Integral The J-integral is path independent. Proof. Let Γ be an arbitrary closed path counterclockwise around the crack tip and A the area enclosed by Γ. With the definition of the traction t: t i = j σ ijn j, where n is 15

26 Physical and Mathematical Background an outward normal to the surface, the J-integral (.59) can be written as J = Γ W dx u i σ ij n j ds. x i,j 1 Now consider the tangential vector dr = (dx 1, dx ) with length ds = dr. Then the outward normal to Γ is given by n = 1 ds (dx, dx 1 ). So we obtain nds = (dx, dx 1 ) and therefore the identities Using these and applying Green s theorem lead to J = A W x 1 i,j n 1 ds = dx (.61) n ds = dx 1. (.6) u i (σ ij ) dx 1 dx. x j x 1 Since we assume an equilibrium with no body force, (.18) provides j σ ij x j = 0 for all i. We assume u sufficiently smooth away from the crack and we do not integrate over the discontinuity at the crack and therefore conclude On the other hand we have i,j ( u ) i σ ij = x j x 1 i,j W x 1 = i,j σ ij W ɛ ij ɛ ij x 1 u i. x 1 x j = i,j σ ij ɛ ij x 1 16

27 Physical and Mathematical Background and with the symmetry of σ (.4) and the definition of ɛ ij given by (.30) W x 1 = i,j σ ij u i. x 1 x j By combining these results we see J = A i,j σ ij u i x 1 x j i,j σ ij u i dxdy x 1 x j = 0, and thus J vanishes on every closed path around the crack tip. Now consider two different paths: Γ 1 counterclockwise and Γ clockwise around the crack tip. Take two more paths Γ 3 and Γ 4 which connect the ends of Γ 1 and Γ but are oriented contrary along the crack, and connect all four so that together they build a closed curve (see Figure.). Figure.: Crack tip with integration paths: the union of Γ 1, Γ, Γ 3 and Γ 4 builds a closed path around the crack tip As we have seen previously, the integral along Γ 1 + Γ + Γ 3 + Γ 4 vanishes because it 17

28 Physical and Mathematical Background is a closed path by construction: ( 0 = W dx ) u i T i ds Γ 1 +Γ +Γ 3 +Γ 4 x i 1 ( = W dx ) ( u i T i ds + W dx Γ 1 x i 1 Γ i ( + W dx ) ( u i T i ds + W dx Γ 3 x i 1 Γ 4 i T i u i x 1 ds ) T i u i x 1 ds Since Γ 3 and Γ 4 are crack faces, we have t = 0 (crack faces are traction free) and dy = 0 (due to the choice of the coordinate system). It follows that ). 0 = Γ 1 ( W dx i T i u i x 1 ds ) ( + W dx Γ i T i u i x 1 ds ). Reversing the second integral yields Γ 1 ( W dx i ) u i T i ds = x 1 Γ ( W dx i T i u i x 1 ds ) and path independence is proven since Γ and Γ 1 are now oriented in the same way. With path independence we can evaluate J locally around the crack tip and so use near-tip equations for the calculation (see Section.6). Moreover, the J-integral is directly connected to the energy of the cracked body and therefore to crack growth: The potential energy of the body can be written as U p = U s F (.63) with U s being the "total strain energy contained in the body". Hence, by using the strain energy density defined in (.57) to express U s and furthermore representing F by F = Γ tds u, it follows that U p = A W dx 1 dx + Γ tds u. 18

29 Physical and Mathematical Background Since the traction is assumed constant, the derivative with respect to a is given by du p da = W A a dx 1dx + Γ t u a ds. Due to the choice of the coordinate system we can substitute a = x 1 du p da = W dx 1 dx + A x 1 Γ t u a ds. and obtain Using Green s Theorem on the first part yields du p da = W dx + Γ Γ t u a ds and by definition of the J-integral (.59) this means J = du p da. (.64) Equation (.64) demonstrates that the J-integral represents the energy release rate. As we have stated earlier, this is crucial in crack growth because the dissipated energy is the energy available for crack propagation. Thus, due to its path independence the J-integral provides a useful instrument in the calculation of crack propagation..6 Crack Propagation For a given crack and an equilibrium state there exist different criteria for crack propagation. We follow the maximum circumferential stress criterion used in [MDB99] to compute a direction and then propagate the crack by a fixed increment. For a mixed Mode I/Mode II loading the circumferential and shear stresses near the crack tip are given in local polar coordinates by ([And05]) σ θθ = σ rθ = K [ I 1 πr 4 K I 1 πr 4 ( θ ) + cos 3 cos [ ( θ sin + sin ) ( 3θ ( 3θ ) ] + K II 1 πr 4 ) ] + K II πr 1 4 [ ( θ 3 sin 3 sin ) [ cos ( θ ) + 3 cos ( 3θ ( 3θ ) ] (.65) ) ] (.66) with the stress intensity factors K I and K II for Mode I and Mode II respectively. The maximum circumferential stress criterion states that the crack will grow in a direction of an angle θ c which maximizes σ θθ and therefore minimizes σ rθ. So we compute 19

30 Physical and Mathematical Background θ c by setting (.66) equal to zero. We then use some trigonometric identities to obtain 1 ( θ cos (K I sin(θ) + K II [3 cos(θ) 1]) = 0. (.67) πr ) Because θ c cannot be of the form kπ, k Z for all Mode I/II loadings, (.67) leads to K I sin(θ) + K II [3 cos(θ) 1] = 0, (.68) which is solved by θ c = arctan 1 4 K I K II ± ( ) KI + 8. (.69) K II So the critical angle θ c = θ c (K I, K II ) is a function of the stress intensity factors. To compute K I and K II, we consider the J-integral and the relationship for linear elasticity (see [And05]) as given by J = K I E + K II E (.70) with E E = 1 ν plane strain E plane stress. (.71) We consider two different states of the cracked body: The first, characterized by the values of (σ (1) ij, ɛ(1) ij, u(1) i ), will be chosen as the actual present state of the material; the second, specified by (σ () ij, ɛ() ij, u() i ), is an auxiliary state which we are free to choose (see below for details). By using the superposition principle (see [And05]), equations (.58) and (.59) and substituting dx by n 1, the normal vector to x 1, the J-integral for the sum of these two states is given as ( J (1+) 1 = Γ i,j i,j (σ (1) ij + σ () ij )(ɛ(1) ij + ɛ () ij )dx (σ (1) ij + σ () ) (u(1) i + u () ij x 1 i ) n j ds ). (.7) 0

31 Physical and Mathematical Background Expanding and rearranging these terms leads to J (1+) = Γ + + ( 1 σ(1) ij ɛ(1) i,j ( 1 Γ Γ i,j ( i,j ij dx σ (1) ij σ() ij ɛ() 1 ij dx σ () ij [ (1) σ ij ɛ() ij + σ () ij ɛ(1) ij ) u (1) i n j ds x 1 u () i x 1 n j ) ds ] dx [ σ (1) ij For linear elasticity we observe the following symmetry i,j σ (1) ij ɛ() ij = i,j = i,j u () i + σ () ij x 1 ( ) µɛ (1) ij ɛ() ij + λ(ɛ (1) 1 + ɛ (1) )δ ijɛ () ij ( ) µɛ () ij ɛ(1) ij + λ(ɛ () 1 + ɛ () )δ ijɛ (1) ij ) u (1) ] i nj ds. x 1 = i,j σ () ij ɛ(1) ij. (.73) Therefore, we can write the above as J (1+) = J (1) + J () + I (1,), (.74) where J (1) is the J-integral for the first state, J () for the second respectively and the so-called interaction integral I (1,) is defined as I (1,) = where we use the definition Γ [( W (1,) δ 1j j i [ σ (1) ij u () i + σ () ij x 1 ) ] u (1) ] i n j ds, (.75) x 1 W (1,) = i,j σ (1) ij ɛ() ij. (.76) At the same time we can use the superposition principle again to write (.70) for the 1

32 Physical and Mathematical Background mixed mode case as J (1+) = (K(1) I + K () I ) E + (K(1) II + K () II ) E (.77) = (K(1) I ) E + (K(1) II ) E + (K() I ) E + E (K(1) I K () I + K (1) II K() II ). + (K() II ) E Recalling (.70) for the first and second state respectively leads to J (1+) = J (1) + J () + E (K(1) I K () I + K (1) II K() II By comparing the two expressions for J (1+), (.74) and (.78), we obtain ). (.78) I (1,) = E (K(1) I K () I + K (1) II K() II ), (.79) which we will use to determine K (1) I and K (1) II. Now we specify the two states. We choose state 1 to be the current state of the material in consideration. Since state is an arbitrary auxiliary state we can choose it to be a pure Mode I with K () I = 1 and K () II = 0. Then we get K (1) I = E I(curr,Mode I). (.80) Similarly, with state as pure Mode II, i.e. K () II = 1 and K () I = 0, it follows K (1) II = E I(curr,Mode II). (.81) In order to simplify the two needed evaluations of (.75) we multiply the integrand by a differentiable function q : Ω R with q(x) = 1 for all x in an open area B around the crack tip and q(x) = 0 for all x outside of an area A around the crack tip bounded by a contour Γ 0. Further, we assume again that the crack faces are traction free and that the crack is straight in A. For any closed path Γ within B around the crack tip consider the contour C constructed by the union of Γ, Γ 0 (which we interpret as a positively oriented path) and the connections between them along the crack faces (similar to Figure.). Then (.75)

33 Physical and Mathematical Background becomes I (1,) = C ( W (1,) δ 1j j i [ σ (1) ij u () i + σ () ij x 1 ) u (1) ] i q( m j )ds x 1 with m being the outward normal vector to C with m = n on Γ and A C the enclosed area. Gauss theorem leads to I (1,) = ( W (1,) δ 1j A C j i [ σ (1) ij u () i + σ () ij x 1 ) u (1) ] i x 1 q x j da. (.8) For the last step we used the following approach very similar to the one used for the path independence of the J-integral and look at the term j x j ( = i,j = i,j W (1,) δ 1j i ( σ (1) ij ( ɛ (1) ij σ () ij [ σ (1) ij u () i + σ () ij x 1 ɛ (1) ij ɛ () ij + σ (1) ɛ () [ ij ij x 1 x 1 ɛ (1) ij + σ (1) ɛ () ij ij x 1 x 1 [ σ (1) ij ) u (1) ] i x 1 σ (1) ij u () i + σ () ij x j x 1 u () i + σ () ij x j x 1 u (1) ] i x j x 1 u (1) ] i x j x 1 where we used the chain rule in the first and linear elasticity in the second step. Using the symmetry of σ (.4) and the definition of ɛ ij given by (.30) as well as the assumption that u is sufficiently smooth allow us to finally conclude j x j ( = i,j = 0. W (1,) δ 1j i ( σ () ij [ σ (1) ij u (1) i + σ (1) ij x 1 x j u () i + σ () ij x 1 u () i x 1 x j [ ) u (1) ] i x 1 σ (1) ij u () i + σ () ij x j x 1 ), u (1) ] i x j x 1 The last step is again justified because we do not integrate across the crack. ) ) Furthermore, we can shrink the inner curve Γ to the crack tip justified by the dominated convergence theorem using the following line of reasoning. Since Γ is arbitrary within B, the following simplification is reasonable: we assume Γ 3

34 Physical and Mathematical Background is the boundary of a ball B 1 B around the crack tip. Let r be the radius of B 1, then set B i = B(x crack tip, r i ) = {x R : x crack tip x < r i }. Also, for f = j ( W (1,) δ 1j i [ σ (1) ij u () i + σ () ij x 1 ) u (1) ] i x 1 q x j and A i := A \ B i define f(x) x A i f i (x) = 0 elsewhere with the obvious properties f i (x) f(x) x A and lim f i = f. i The limit f is integrable as sum of products of integrable functions. Then the dominated convergence theorem implies lim f i da = i A A lim f ida = i A fda. (.83) Equation (.83) allows us to rewrite (.8) and therefore we gain a volume form of the interaction integral (.75) which is more suitable for the computational evaluation in the finite element setting: I (1,) = A ( [ j i σ (1) ij u () i + σ () ij x 1 ) u (1) ] i W (1,) δ 1j x 1 q x j da. (.84) 4

35 Physical and Mathematical Background All that is left is to determine the needed variables for state. As mentioned earlier we choose state to be of pure Mode I with K () I = 1 or respectively pure Mode II with = 1. The analytic solutions for the near-tip displacement field u for these cases K () II are known ([And05]). In local polar coordinates with origin at the crack tip and x 1 -axis aligned with the crack (see Section.5) they are given by: For Mode I and for Mode II u I,1 (x, y) = K I r ( θ ) [ ( µ π cos κ 1 + sin θ ) ] (.85) u I, (x, y) = K I r ( θ ) [ ( µ π sin κ + 1 cos θ ) ] (.86) u II,1 (x, y) = K II r µ u II, (x, y) = K II µ ( θ ) [ ( π sin κ cos θ ) ] (.87) r ( θ ) [ ( π cos κ 1 sin θ ) ] (.88) where 3 4ν plane strain κ = 3 µ 1 + ν plane stress. From (.85) - (.88) the derivatives are computed by using the chain rule as: Mode I u I,1 = u I,1 x 1 r x 1 u I, = = K I µ π r + u I,1 θ x 1 θ x { 1 1 ( θ ) [ ( r cos κ 1 + sin θ ) ( ( cos θ ) 1 [ + ( θ r sin { K I µ π + ( θ r cos 1 r sin ( θ ) ] x 1 r κ 1 + sin ( θ ) [ ( κ + 1 cos θ [ ) ( ( sin θ ) + 1 ) ] x 1 r κ + 1 cos ( θ ) ]) } ( x ) r ) ]) } ( x ) r (.89) (.90) (.91) 5

36 Physical and Mathematical Background Mode II x 1 u II,1 = x 1 u II, = K II µ π { + ( θ r cos { K II µ π + ( θ r sin 1 r sin ( θ ) [ ( κ sin θ [ ) ( ( sin θ ) r cos ( θ ) ( ( cos θ ) + 1 ) ] x 1 r κ cos ( θ ) [ ( κ 1 sin θ [ ) ] x 1 r κ 1 sin ( θ ) ]) } ( x ) r ) ]) } ( x ) r (.9) (.93) where we used r(x 1, x ) = x 1 + x (.94) θ(x 1, x ) = atan(x, x 1 ) (.95) with ( y arctan ( x) y arctan + x) π ( y arctan atan(y, x) = x) π + π x > 0 x < 0, y 0 x < 0, y < 0 x = 0, y > 0 (.96) π x = 0, y < 0 0 x = 0, y = 0. Furthermore, the stress for these cases is given by ([And05]): Mode I σ I (1, 1) = σ I (1, ) = K ( I θ ) [ ( θ cos 1 sin sin πr ) K ( I θ ( θ cos sin cos πr ) ) ( 3θ ) ( 3θ ) ] (.97) (.98) σ I (, 1) = σ I (1, ) (.99) σ I (, ) = K ( I θ ) [ ( θ ( 3θ ) cos 1 + sin sin πr ) (.100) 6

37 Physical and Mathematical Background Mode II σ II (1, 1) = K ( II θ ) [ ( θ sin + cos cos πr ) σ II (1, ) = K ( II θ ) [ ( θ cos 1 sin sin πr ) ( 3θ ) ] (.101) ) ] (.10) ( 3θ σ II (, 1) = σ II (1, ) (.103) σ II (, ) = K ( II θ ( θ ( 3θ ) sin cos cos πr ) ) (.104) Instead of approximating the derivatives by finite differences, the above equations can be hard-coded into the implementation in order to compute the Interaction Integral (.84) and with it determine the direction of the crack growth. More implementation details will be presented in the next chapter. 7

38

39 3 Implementation In this chapter we explain the main concepts we use for our simulations. All the presented implementation is written in C++ as part of PhysBAM c. First we discretize the weak formulation derived above and then we explain the cutting algorithm we use to introduce the extra degrees of freedom that the system needs for the crack to separate away from the tip. Near the tip we use a different type of new degrees of freedom, which will be discussed next. Then we will introduce our integration scheme that is based on a hybrid mesh approach. Finally, we will give further remarks which turned out to be useful for the computational solution. 3.1 Discretization We discretize the weak formulation (.43) using the Galerkin method ([Bur07b]). We choose a subspace V h (Ω) H 1 Γ u (Ω) with basis {ϕ i } i=1,...,n and write the discrete displacement as a linear combination of this basis with parameter vectors u k u h (x) = n u i ϕ i (x), u i R (3.1) i=1 with the coordinate vector x = (x 1, x ) T. To derive a matrix notation (cf. [BW08]) we reinterpret the three independent components of the stress tensor as the vector σ = σ 11 σ σ 1 (3.) 9

40 3 Implementation and similarly ɛ = ɛ 11 ɛ. (3.3) ɛ 1 We can then express ɛ(u) as a matrix vector multiplication ɛ(u h ) = i B i u i (3.4) with With the use of D = B i = ϕ i x ϕ i x ϕ i x ϕ i x 1 µ + λ λ 0 λ µ + λ µ. (3.5) (3.6) and setting the test functions to for an arbitrary V v = V ϕ j j {1,..., n} (3.7) R, the discretized version of the weak formulation for linear elasticity given by (.45) can be written as which can be rearranged to i Ω V i (B j V ) T DB i u i dω = t V ϕ j ds Γ t Ω Ω f V ϕ j dω, (3.8) ( ) B T j DB i u i dω = V tϕ j ds fϕ j dω. (3.9) Γ t Ω 30

41 3 Implementation Since V is arbitrary, this leads to i Ω B T j DB i u i dω = tϕ j ds Γ t By applying this for all j = 1,..., n and using the solution vector U h = u 1... u n Ω fϕ j dω. (3.10) R n (3.11) we obtain the FEM system of linear equations K h U h = F h (3.1) with the stiffness matrix K h = K 11 K 1. K 1n K 1 K. K n R n n (3.13) K n1 K n. K nn and the components K ji = Ω B T j DB i dω R (3.14) as well as the force vector F h R n with components F h,j = tϕ j ds Γ t Ω fϕ j dω R. (3.15) Up to this point, we held the derivation general with an arbitrary f. For our simulations we use f = 0 which reduces (3.15) to F h,j = tϕ j ds R. (3.16) Γ t From (3.14) it is obvious that K ji = K T ij and thus K h is symmetric. Furthermore, K h is positive definite: Let u = i u iϕ i V h be an admissible displacement, then we 31

42 3 Implementation observe u T Ku = (B i u i ) T DB j u j dω i,j Ω = ɛ(u h )Dɛ(u h )dω Ω [ = µɛ 11 + λ(ɛ 11 + ɛ 11 ɛ + ɛ ) + µɛ + 4µɛ ] 1 dω Ω [ = µ ɛ(uh ) : ɛ(u h ) + λ(divɛ) ] dω, Ω which is always positive if the first term is. This property is provided by Korn s inequality (see [Bra07]): ɛ(u) : ɛ(u) c Korn u H (3.17) 1 Ω for a constant c Korn > 0. Therefore, we obtain u T Ku > 0 u V h \ {0}. (3.18) In general, we follow the standard idea and choose the ϕ i to be the affine functions over the nodes of the mesh. We denote these nodal functions as φ i, i {1,..., n}. However, in Section 3.3 we will introduce a new class of nonlinear functions that we will use to extend this function basis. The function space enlarged in this way will allow us to find a more accurate solution u h. 3. The Cutting Algorithm The cutting algorithm of Sifakis et al. ([SDF07]) provides the simulation mesh with the necessary degrees of freedom for the cracked material to split apart. This happens by duplicating vertices and the introduction of virtual nodes (see also [MBF04]) to build new mesh elements. The different material regions of a cracked element are then assigned to different copies of that element which allows it to open up. This means that those elements contain both material and empty regions but maintain their form and geometric properties. In two dimensions it works on a triangle mesh that represents the domain and with a segmented curve for the crack. The algorithm works as follows: In a first step the geometries are resolved. Intersections 3

43 3 Implementation within the crack curve are resolved to obtain a set of intersection-free cutting segments. Then all intersections between the crack curve and triangle edges are found, yielding an intersection-free segment-mesh. The material regions within each triangle, divided by the crack curve, are determined. For each of these distinct material regions one copy of the original triangle is created. For such a copy every node not included in the corresponding material region is a so-called virtual node (see Figure 3.1). These virtual nodes do not represent material nodes but provide the degrees of freedom that allow the crack to open. Figure 3.1: Cutting algorithm: on left, original mesh triangle; on right, duplicates with material regions (shaded) and virtual nodes (hollow blue circles) 1 In a next step the global material connectivity of the newly created mesh is determined. First consider the map C that takes a triangle T of the original mesh to the set of its copies C(T ) of the duplicated mesh. Then for every T in the duplicated mesh the algorithm finds T in the original mesh such that T C(T ). For every triangle T n neighboring T all nodes of all triangles in C(T n ) are checked if they share a material connection to T. In that case the corresponding material nodes are identified and their degrees of freedom are collapsed into a single one. The virtual nodes of T are processed in a similar fashion. Figure 3. shows an example geometry subject to the cutting algorithm. The completely cut element gets duplicated and the crack can separate. 1 Graphics from [RHS + ] 33

44 3 Implementation Figure 3.: Cutting algorithm: the mesh on the left is cut by the crack. The cutting algorithm provides duplicated elements and virtual nodes for the fully cut element (right). An important limitation of the virtual node algorithm (as well as the Heaviside representation of XFEM, see Section 3..1) is its inability to deal with partially cracked elements: because it detects only regions fully cut out of a node s one-ring, it needs a fully cracked element to create the needed degrees of freedom (see Figure 3. on the left). By one-ring of a node we denote the set of elements that form the support of the affine basis function associated to that node s degree of freedom. Accordingly, a triangle s one-ring is the union of the one-rings of the triangle vertices. One way to improve the problem mentioned above is to refine the mesh around the crack tip until the crack is fully resolved. But that is not favorable in order to maintain a fast algorithm, especially in the case of a complicated crack geometry. In the next section we employ an alternate approach that allows us to overcome this issue by the usage of so-called enrichment functions, but first we compare our approach to the Heaviside representation from [MDB99] Comparison to Heaviside Representation In [MDB99] the extra degrees of freedom for the crack opening are introduced by a representation using a Heaviside-type function. Considering only fully-cracked elements that representation is of the form u h (x) = i u i φ i (x) + n j b j φ j (x)h(x), (3.19) where H(x) is a jump function defined as 1 on the left and 1 on the right side of the crack (orientation towards the crack tip) and n j are the according nodes of fully-cracked triangles. We now show that in typical cases our cutting approach is equivalent to this method. Graphics from [RHS + ] 34

45 3 Implementation As an example consider the geometry shown in Figure 3.1 again. finite element space can be written as The corresponding u h = 3 φ i i=1 j=1 u j i χ j, (3.0) with j ranging over the two copies of the triangle, i.e. the u 1 i are the degrees of freedom of the first copy and u i those of the second; χ j denotes the characteristic function of the material region associated with triangle copy j. We can now use the function H as defined above to express the χ j as χ 1 = 0.5(1 + H) (3.1) χ = 0.5(1 H) (3.) and rewrite (3.0) into u h = i φ i ( 0.5(u 1 i u i )H + 0.5(u 1 i + u i ) ). (3.3) With the linear transformation of the degrees of freedom u 1 = (u u 1) = 0.5(x 1 + x 5 ), (3.4) u = (u 1 + u ) = 0.5(x 4 + x ), (3.5) u 3 = (u u 3) = 0.5(x 6 + x 3 ), (3.6) b 1 = (u 1 1 u 1) = 0.5(x 1 x 5 ), (3.7) b 1 = (u 1 u ) = 0.5(x 4 x ), (3.8) b 1 = (u 1 3 u 3) = 0.5(x 6 x 3 ) (3.9) equation (3.3) can be finally written in the form of (3.19). On the other hand there are crack geometries where the two approaches lead to different finite element spaces. One example is shown in Figure 3.3. The red line in the configuration on the left represents a crack cutting the three triangles. The Heaviside representation in the middle does not contain the same number of degrees of freedom as the result of the cutting algorithm on the right (see top row of Figure 3.3). The reason for the additional opening for the virtual nodes originates from the fact that the two copies with the material on the right side of the crack do not share a material connection 35

46 3 Implementation along an edge. They are still each connected to the third triangle on the right, so they link to that one. Figure 3.3: The crack example on the left produces different degrees of freedom for the Heaviside representation (center) and the cutting algorithm (right) 3 The virtual node technique used by the cutting algorithm provides at least the same amount of degrees of freedom as the Heaviside representation or, as we have seen, sometimes a few more. 3.3 Enrichment For mesh elements which contain a crack tip, i.e. are not entirely cracked and therefore not handled by the cutting algorithm, we follow the idea of the Extended Finite Element Method (XFEM, see, e.g., [BB99] and [MDB99]). We introduce new degrees of freedom for these elements and associate them with a set of enrichment functions, which are added to the finite element basis. We use the knowledge about the analytic near tip displacement fields for Mode I and II fracture and find a function basis in which they can be expressed. By reviewing (.85) - (.88) and using some trigonometric identities it is easy to see that one choice of these basis functions is given in polar coordinates (for more implementation details on the computation of (r, θ) see Section 3.6.3) locally around the crack tip by 3 Graphics from [RHS + ] 36

47 3 Implementation F 1 (r, θ) = ( θ r sin (3.30) ) F (r, θ) = ( θ r cos (3.31) ) F 3 (r, θ) = ( θ r sin sin(θ) (3.3) ) F 4 (r, θ) = ( θ r cos sin(θ). (3.33) ) In order to maintain a local support we multiply these enrichment functions with the nodal functions φ i and then include these products in the function space we use. The additional degrees of freedom that are associated to the newly created functions as well as the fact that F 1 is discontinuous at π and π, i.e. across the crack face, will allow even the half-cut tip element to open. See Figures 3.7 and 3.8 for an illustration of F 1. For an element containing a tip the displacement is represented as u h (x) = u i φ i (x) + c l k φ k(x)f l (r(x), θ(x)). (3.34) i=1 k=1 l=1 For an adjacent triangle the sum over k in (3.34) has to be adjusted to the actual indices of the enriched degrees of freedom, i.e. the set of summation indices is K {1,, 3}. Over the whole domain we can express the solution for one crack tip by u h = u i φ i + 4 c l k φ kf l, (3.35) i I k K l=1 where I is the index set of all nodal degrees of freedom, including the virtual nodes. K is given as K = {i I x crack tip ω i }, (3.36) where ω i denotes the one-ring of node i, i.e. the support of the nodal basis function φ i. This can easily be generalized to multiple crack tips by introducing additional sums for the additional tips, each with an index set similarly to (3.36). In doing so, we assume that all crack tips are located far enough from each other in order not to enrich the same nodes. 37

48 3 Implementation 3.4 Hybrid Mesh In this section we will describe the hybrid mesh approach which we use for our integration scheme. The general idea is to use the given geometry to create a second mesh by an easily performed refinement around the crack. This mesh is then embedded into the simulation mesh using so-called hard bindings ([SSIF07]) and only used for integration purposes. It does not introduce any new degrees of freedom to the system. Therefore, we deal with two meshes: the simulation mesh represents the degrees of freedom of the system and is created by cutting the original mesh with the crack geometry, the quadrature mesh is created by refining the initial mesh; the refined mesh is also cut by the cutting algorithm, and the result is finally embedded back into the simulation mesh (and therefore does not represent any degrees of freedom). The finer quadrature mesh is then used to perform all the integration needed for solving the equilibrium problem on the relatively coarse simulation mesh. All mentioned aspects are now presented in more detail Refinement We consider the uncut triangulated representation of the domain Ω and the elements which contain a part of the crack. We create a new mesh by a red-green refinement ([BSW83]) locally around the crack, i.e. on all triangles which contain either a crack tip or are fully cut by the crack as well as their one-rings. Each of those triangles is subject to red refinement. In the first level the midpoints of the edges of the original triangle are connected to build the vertices for four new triangles. Every further level is done similarly within the triangles created in the previous step. This creates a new mesh in which all the new refined triangles are geometrically similar to the original coarse one. We therefore maintain the triangle quality and do not introduce ill-posed elements in the critical region around the crack. This procedure introduces new points to each triangle that shares an edge with a refined element but is not refined itself. Therefore, in such a triangle T those points are not resolved by any mesh. To overcome this issue, we red refine T until all triangles within T contain at most one unresolved point. These triangles get green refined. The remaining unresolved point is the midpoint of one of the triangle s edges by construction. 38

49 3 Implementation This point is connected to the opposite vertex of that triangle. See Figure 3.4 for an example where two crack triangles are subject to a one level red refinement and the adjacent ones are green refined. Also, see Figure 3.5 on the top left for the same setting with two levels of refinement. Green refined triangles are of worse quality than their parents. However, they are never refined further. If we have to change the refinement, e.g. due to crack growth, we ignore any existing green refinement and apply a new red refinement on the parents. Figure 3.4: Red-Green-Refinement: on left, coarse mesh with crack; on right, one level of red-green refinement on crack triangles For each refined triangle we store its unrefined parent triangle in a map. Both the unrefined simulation mesh and the refined quadrature mesh are then subject to the cutting algorithm as described above, which creates the actual meshes with the extra triangles we need for the crack to open. Again we keep track of the newly created triangles and their predecessors they were created from. By combing the previous map information we gain a correspondence between the duplicated unrefined and the duplicated refined triangles. We can then use this final mapping to embed the refined, duplicated mesh into the coarse and duplicated simulation mesh by the concept of bound particles described in the next section. Figure 3.5 shows the same geometry as Figure 3., this time with a refined mesh embedded into the simulation mesh which allows a more accurate separation of the crack. 39

50 3 Implementation Figure 3.5: The refined mesh on the top left is subject to the cutting algorithm, resulting in the geometry on the bottom. The inset shows how the virtual nodes allow the crack to separate Bound Particles We will now describe the embedding of the refined mesh into the simulation mesh. For the simulation mesh denote the finite element space of nodal basis functions that are continuous and affine by M H. Let V H be the function space created by enriching M H as described in Section 3.3. Further, let X H and F H be the degrees of freedom corresponding to M H and V H respectively. Those degrees of freedom corresponding to the enrichment we call C H. Note that some of the affine functions of M H correspond to virtual nodes created by the cutting algorithm as described in Section 3.. This means we have to define basis functions that actually respect the crack geometry, as we did in equation (3.0). Let { φ i } be the usual affine hat functions and let ω i be the one-ring of the node associated with φ i. Then define a new basis of M H by φ i (x) = φ i (x) χ T (x), (3.37) T ω i where χ T denotes the characteristic function of the material region of triangle T. For a 4 Graphics from [RHS + ] 40

51 3 Implementation solution u H we then have the representation u H = i u i φ i + k c l k φ kf l (3.38) l with {u i } X H, {c l k } F H and X H, C H F H. For the refined mesh consider M h to be the finite element space and X h the degrees of freedom of the refined mesh. Because of the way we performed the refinement and by identifying those degrees of freedom corresponding to the same nodal position, we obtain the inclusion X H X h. To embed the refined mesh into the simulation mesh we use a technique called hard bindings ([SSIF07]). Bound particles are embedded in the simulation mesh, fully dependent on their parent particles and thus they do not represent any degrees of freedom at all. In order to bind the refined triangle vertices to their corresponding degrees of freedom in the simulation mesh we establish a fixed relationship between X h and F h. For any u i X h we enforce the linear constraint u i = j u j β j + k β k c l k F l(r(x i ), θ(x i )), (3.39) l where the {β j = φ j (x i )}3 j=1 are the barycentric coordinates of the refined triangle node x i associated with u i with respect to the unrefined parent triangle of x i, and (r(x i ), θ(x i )) are the local polar coordinates of x i with respect to the crack tip. Since the refined mesh is also subject to the cutting algorithm, x i does not necessarily correspond to a material node and might represent a virtual node. In that case we associate x i with that side of the crack on which the material for that triangle is located. Consequently, we have to manipulate the polar coordinates (r(x i ), θ(x i )) accordingly and adjust the sign of θ(x i ) to be consistent with the assigned angle of the material nodes of that same triangle by θ adjusted (x i ) = θ(x i ) sgn(θ(x i )) π (3.40) θ(x i ) = θ adjusted (x i ). That way θ(x) can take values outside of the usual [ π, π]. For an illustration see Figure

52 3 Implementation Figure 3.6: Adjusting the angle in order to bind a virtual node 5 After binding all refined mesh nodes into the simulation mesh we create the refined triangles based on the material connectivity as determined by the cutting algorithm Integration To assemble the stiffness matrix K H and the vector F H we use the embedded mesh. We project the basis function of our enriched space V H onto M h, the space of affine functions on the refined mesh. These projections are then used for the integrations needed to assemble the linear equation system. We will now go through the details more thoroughly. Assume first that there is no enrichment. Thus, the function space is M H and each node x associated with u X h is bound to its parents barycentrically by x = 3 β i x i, (3.41) i=1 again with {β k } 3 k=1 being the barycentric coordinates of x in the unrefined parent triangle T = {x 1, x, x 3 }. The nodal basis functions {φ j } can then be expressed using the nodal basis {φ i } of M h on the refined mesh φ j = i β j i φ i, (3.4) 5 Graphics from [RHS + ] 4

53 3 Implementation where β j i is the jth barycentric coordinate of the node x i in the corresponding triangle (taken to be 0 if x k is not in the support of φ j). We then use (3.4) to assemble the stiffness matrix K H. Since the φ j M h are linear, their derivatives are constant, i.e. the integrations needed to assemble K H are piecewise constant on each refined triangle. Of course, in the case described above with only nodal degrees of freedom this will result in exact quadrature. We now consider the function space V H whose basis includes the nonlinear enrichment functions {F j } 4 j=1 given by (3.30) - (3.33) and therefore has singular derivatives. The binding for a node x i in the refined mesh in this case is given by x i = j β j x j + k c l k F l(r(x i )θ(x i )), (3.43) l with the same notation as in equation (3.39) and the same special care for virtual nodes that leads to (3.40). The relation in (3.43) is still linear. It is extended by a linear interpolation of the nonlinear {F j } with the sampling points being the refined mesh nodes. See Figure 3.7 and 3.8 for such an interpolation of the enrichment function F 1 introduced in Section 3.3. Assembling the stiffness matrix in a manner similar to the one described above leads again to integrating constant derivatives in each refined triangle. Figure 3.7: Enrichment function F 1 = r sin(θ/) projected on a low resolution quadrature mesh 6 6 Graphics from [RHS + ] 43

54 3 Implementation Figure 3.8: Enrichment function F 1 = r sin(θ/) projected on a high resolution quadrature mesh 7 Although our integration scheme uses only an approximation of the singular derivatives, this approximation improves with increasing refinement of the quadrature mesh. Also, our sampling of the singular basis functions could occur at points near the singularity. However, unlike integration schemes based on Gauss quadrature, those evaluations will be additionally weighted by the area of the smaller refined triangle. Thus, no single value, possibly located near to the singularity at the tip, contributes disproportionally. Moreover, the more complicated the crack geometry, e.g. multiply jagged, the more expensive a remeshing which incorporates the crack faces becomes, but our scheme avoids that and remains easily performable. For simplicity and stability reasons we treat every cut triangle as if it was filled entirely with material and therefore we do not apply the multiplication with the characteristic functions in (3.37) and instead integrate over the usual hat functions. This modification becomes negligible under refinement of the quadrature mesh since the error in the support of the basis functions goes to zero. 3.5 Solving the FEM System CG Algorithm To solve the FEM system of linear equations derived in Section 3.1 and assembled as described in Section 3.4 we use the conjugate gradient (CG) method ([GL96]) as follows. 7 Graphics from [RHS + ] 44

55 3 Implementation The solution of the linear equation system Ax = b with a real, symmetric, positive definite matrix A and an initial guess x 0 can be estimated (ɛ 0) by k = 0 r 0 = b Ax 0 p 0 = r 0 while r k > ɛ α k = rt k r k p T k Ap k x k+1 = x k + α k p k r k+1 = r k α k Ap k β k = rt k+1 r k+1 r T k r k p k+1 = r k+1 + β k p k k = k + 1 end of while x = x k+1. As shown in Section 3.1 our stiffness matrix K h fulfills the requirements of the CG algorithm Condition of the Stiffness Matrix We also want to investigate how the use of the enrichment functions changes the condition of the stiffness matrix. We therefore study two simple examples. The first example consists only of one isosceles triangle with a centered crack on the symmetry axis, see Figure 3.9. As discussed in Section 3.3, all nodes in this setting are enriched, summing up to 30 degrees of freedom. The results in this section are computed in MATLAB R 8 for simplicity reasons and rounded to four decimal places. Figure 3.9: Setting of the triangle example considered to study the condition 8 The MathWorks, Inc., 45

56 3 Implementation We use a = and h = and consider two different cases: one or two nodes enforced by boundary conditions. Figure 3.10 shows the eigenvalues for both cases without enrichment as well as with enrichment for c = 1. Eigenvalues without enrichment with enrichment 1 node enforced nodes enforced 1 node enforced nodes enforced Figure 3.10: Eigenvalues of the stiffness matrix without enrichment and with enrichment for c = 1 in the triangle example Since the stiffness matrix is symmetric and positive definite, the -norm condition 46

57 3 Implementation number is given in terms of the eigenvalues η by k = η max η min. (3.44) The eigenvalues η = 1 in Figure 3.10 correspond to the degrees of freedom set by the boundary conditions. Further, the eigenvalues of order and 10 1 respectively can be interpreted as zero and correspond to the rotation possible in this example with only one node fixed. We ignore these eigenvalues and compute the relevant condition number according to (3.44). The results for other crack lengths c can be treated in a similar fashion. The corresponding condition numbers are stated in Figure Condition crack length 1 node enforced nodes enforced without enrichment c = c = c = c = c = c = c = Figure 3.11: Condition of the stiffness matrix for the triangle example without enrichment and with enrichment for different crack lengths As a second example we look at a square of edge length a =, which consists of four triangles. One triangle contains a central crack of length c. We consider three different configurations. In the first the crack remains centered with a variable crack length c. In the second we shift the crack away from the center by a length b while keeping the crack length constant at c = 0.5. For the third setup we rotate the centered crack by an angle ϑ, again with constant length c = 0.5. Figure 3.1 diagrams the settings. The three round nodes are the enriched ones, the other two correspond only to nodal degrees of freedom. These two are fixed by boundary conditions. 47

58 3 Implementation Figure 3.1: The three configurations for the square example: in red a centered crack of length c; in blue a crack shifted by b; in green a crack rotated by ϑ We proceed as described above and use the eigenvalues to compute the condition of the stiffness matrix. See Figures 3.13, 3.14 and 3.15 for the results corresponding to the three different configurations. Condition without enrichment c = c = c = c = c = Figure 3.13: Condition of the stiffness matrix for the square example without enrichment and with enrichment for different crack lengths 48

59 3 Implementation Condition b = b = b = b = b = Figure 3.14: Condition of the stiffness matrix for the square example with a crack of length c = 0.5 shifted away from the center by a length b Condition ϑ = ϑ = ϑ = ϑ = ϑ = Figure 3.15: Condition of the stiffness matrix for the square example with a centered crack of length c = 0.5 rotated by an angle ϑ As we see from both examples and the various configurations, the condition is worsened by the introduction of the enrichment and is further dependent on the crack position within the enriched element. These investigations show that some improvements in the solving process might be possible by using a preconditioned conjugate gradient solver (cf., e.g., [GL96] or [Bra07]), which we leave to future work. 3.6 Remarks Computation of the Interaction Integral In order to evaluate the interaction integral (.84) we use the following quadrature in a similar fashion as described above. We denote the integrand by f(x) as before and furthermore decompose the integral into integrals over the triangles T in the coarse simulation mesh. We then consider the corresponding refined triangles t T. Within every triangle t we sum over the values of 49

60 3 Implementation f at the vertices v i and weight the result by a third of the triangle area A t T f(x)dx t T A t 3 3 f(v i ). (3.45) i=1 Besides the quadrature described above we need to specify the numeric properties of the function q and the area A used in (.84). A is simply assembled as a k-ring around the crack tip triangle. Here we use the triangles in the coarse simulation mesh to maintain A independent of the chosen refinement levels. The only challenge is to choose an adequate size of the integration area. A and therefore k should contain enough elements to achieve accuracy but it should not be too large for the used asymptotic displacement fields to hold since they are near-tip solutions. Our experiments show that k = 1 and k = are appropriate choices. We use the two-ring for all numerical examples shown in Chapter 4. With A specified we can determine the function q occurring in equation (.84) to be constantly 1, and therefore q x j = 0 for j = 1,, in the tip triangle. Over the other triangles within A we linearly interpolate q using the nodal hat functions on the elements and thus gain constant derivatives on these triangles Computation of the Strain To compute the strain ɛ we assume a piecewise linear mapping φ from the undeformed to the deformed configuration, i.e. for every triangle φ(x) = F X + b. (3.46) By recalling (.1), u = φ id, and the definition of ɛ (.9) we obtain ɛ = F + F T I. (3.47) To compute F efficiently we use an idea from [TBHF03]. For a given triangle with vertices X 0, X 1 and X before and x 0, x 1 and x after deformation we define the edge 50

61 3 Implementation vectors d m1 = X 1 X 0 (3.48) d m = X X 0 (3.49) and d s1 = x 1 x 0 (3.50) d s = x x 0. (3.51) By using (3.46) we observe the relation d s1 = (F X 1 + b) (F X 0 + b) = F d m1 (3.5) d s = (F X + b) (F X 0 + b) = F d m. (3.53) So, defining the matrices D m = (d m1, d m ) (3.54) D s = (d s1, d s ) (3.55) leads to the identity D s = F D m (3.56) or F = D s D 1 m (3.57) and D 1 m can be precomputed and stored during the initialization Kinked Cracks We form the crack by a series of linear segments. Even an initially straight crack might develop kinks in the process of propagation. We therefore need to adapt the local polar coordinates used in the enrichment functions as well as in the formulas used in the computation of the propagation angle. We follow [BB99] and use a series of mappings that rotate the crack segments to align with the straight discontinuity of F 1, i.e. sin( θ ) 51

62 3 Implementation (see equation (3.30)), at π and π. We enumerate the segment vertices beginning with 1 at the tip, so (x tip, y tip ) = (x 1, y 1 ), and assume a kink at every vertex (otherwise we can fuse two segments together). We then define the mappings as follows. In the nth step for a point (x, y) Ω we consider the vector r from the kink in vertex (x n, y n ) in the crack segments to (x, y) and define the following angles (see Figure 3.16): θ R : the kink angle between the crack segments at (x n, y n ) (3.58) α : the angle between the straight discontinuity and r. (3.59) Since (x, y) Ω and therefore (x, y) is not located on the crack itself, θ R α is always guaranteed. Further, we assume π < θ R < 3π then define θ = 3π π θ R π θ R π and map (x, y) to new coordinates (x, y ) by from the physical nature of cracks. We (α θ R ) α > θ R (α θ R ) α < θ R (3.60) (x, y) (x, y ) = ( l r cos( θ), r sin( θ)), (3.61) where l = (x n x tip ) + (y n y tip ) (3.6) r = (x n x) + (y n y). (3.63) We continue this series of mappings until r cos(θ) < l or we reach the last segment. The new polar coordinates (r, θ) are then given by r = (x ) + (y ) (3.64) θ = atan(y, x ). (3.65) The series of mappings can be imagined as a virtual rotation of all crack segments to align with the tip segment (see Figure 3.16). 5

63 3 Implementation Figure 3.16: Mapping for kinked cracks: on top the setting of the mapping at step n with the previous segments aligned due to the (n 1) previous mappings, on bottom the mapped point (x, y ) used to determine the mapped polar coordinates 53

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65 4 Numerical Results In this section, we want to test our simulation on several example cases, for which the analytic solution for the stress intensity factors are known. Further, we present a propagation development as well as an example where our cutting algorithm handles a complicated crack geometry. For all examples we use the material parameters E = 100 [kpsi] and ν = 0.3 and assume a state of plane strain. 4.1 Opening Example For the first example we consider a pure Mode I case (i.e. K II = 0) with a rectangular plate of length L, width W and a centered crack of length a. A constant traction is applied to parts of the boundary, i.e. the two shorter ends, as illustrated in Figure 4.1. Figure 4.1: Setting for the opening example 1 1 Graphics from [RHS + ] 55

66 4 Numerical Results The exact solution is given by [Ewa84] as K I = Cσ aπ, (4.1) where C is the finite geometry correction factor ( a ) ( a ) ( a ) 3 ( a ) 4 C = (4.) W W W W For our experiments we normalize K I by an appropriate choice of σ and use L =, W = 1 and a = 0.3. We further test on different meshes with various numbers of elements. We also compare different refinement levels of the embedded quadrature mesh. See Figure 4. and 4.3 for the results. K I Levels 8x4 16x8 3x16 64x3 18x K II Levels 8x4 16x8 3x16 64x3 18x Figure 4.: Numerical results for opening example: K I and K II 56

67 4 Numerical Results x4 16x8 3x16 64x3 18x64 exact solution K I refinement levels x4 16x8 3x16 64x3 18x64 exact solution K II refinement levels Figure 4.3: Illustration of the results for K I and K II for the opening example We can see that additional levels of refinement of the quadrature mesh lead to a substantial improvement in the accuracy of the solution. 57

68 4 Numerical Results 4. Shear Example The setting for the second example is similar to the previous one. This time one end of the plate is subject to a constant shear τ = 1 and the other one is fixed to a zero displacement. See Figure 4.4 for a diagram. Figure 4.4: Setting for the shear example For W = 7, L = 16 and a = 3.5 the exact solutions are given by ([MDB99]) K I = 34.0 [psi in] (4.3) K II = 4.55 [psi in]. (4.4) Figures 4.5 and 4.6 show the results for varying refinement levels on different resolutions of the simulation mesh. Graphics from [RHS + ] 58

69 4 Numerical Results K I Levels 16x8 3x16 64x3 18x K II Levels 16x8 3x16 64x3 18x Figure 4.5: Numerical results for shear example: K I and K II 59

70 4 Numerical Results x8 3x16 64x3 18x64 exact solution K I refinement levels x8 3x16 64x3 18x64 exact solution K II refinement levels Figure 4.6: Illustration of the results for K I and K II for the shear example As in the previous example we observe a leap in the accuracy with additional quadrature refinement. We gain parts of the benefits of refining the simulation mesh by just refining the quadrature mesh which does not lead to additional degrees of freedom. 60

71 4 Numerical Results 4.3 Complicated Geometries To demonstrate the ability of our approach to handle rather complicated crack geometries we consider a branched crack and a refinement of three levels. In Figure 4.7 we can see the cutting geometry with the simulation mesh, in Figure 4.8 with the refined quadrature mesh. Figure 4.9 shows the crack opening as a result of a constant opening force similar to the one described in Section 4.1. Figure 4.7: Complicated crack geometry: branched crack with the relatively coarse simulation mesh 61

72 4 Numerical Results Figure 4.8: Complicated crack geometry: branched crack with the refined quadrature mesh Figure 4.9: Complicated crack geometry: branched crack under opening force 4.4 Example of Crack Propagation We also want to show a crack propagation. We therefore choose the setting of the example in Section 4.1 because then a straight crack path is obviously to be expected. We choose 6

73 4 Numerical Results a simulation mesh not too coarse in order to have smaller elements and therefore make the adaptive change in the refined mesh visible more clearly. The number of refinement levels is set to three. Figure 4.10: Crack propagation: initial configuration Figure 4.11: Crack propagation: after 1 iteration 63

74 4 Numerical Results Figure 4.1: Crack propagation: after 5 iterations Figure 4.13: Crack propagation: after 10 iterations 64

75 4 Numerical Results Figure 4.14: Crack propagation: after 15 iterations Figure 4.15: Crack propagation: after 16 iterations - the material is completely cut into two independent components Note how the refinement adaptively changes with the propagating crack. Further, we gain a smooth and straight crack path. 65