Model-independent approaches for the XFEM in fracture mechanics

Transcription

1 Model-independent approaches for the XFEM in fracture mechanics Safdar Abbas 1 Alaskar Alizada 2 and Thomas-Peter Fries 2 1 Aachen Institute for Computational Engineering Science (AICES), RWTH Aachen University, Schinkelstr. 2, Aachen, Germany 2 Chair for Computational Analysis of Technical Systems (CATS), RWTH Aachen University, Schinkelstr. 2, Aachen, Germany SUMMARY Applications of the XFEM to fracture mechanics involve enriching the regions near the crack-tips. The accuracy of such approximations largely depends upon these crack-tip enrichment functions. Such functions are based on the asymptotic fields in the near-tip region for the fracture model considered. Consequently, for each fracture model, a different set of enrichment functions is required. Herein, two model-independent approaches are presented that can be used independent of the fracture model considered. In the first approach, a set of crack-tip enrichment functions is proposed that is able to capture arbitrary high gradients at the crack-tip. A second approach is based on an adaptive mesh refinement in the near-tip region and the treatment of hanging nodes in combination with the step enrichment is of concern. Examples show the effectiveness of the proposed methods for brittle and cohesive cracks. Copyright c 2010 John Wiley & Sons, Ltd. key words: Extended finite element method (XFEM), high gradient solutions, fracture mechanics, hanging nodes, model-independent, cohesive fracture, adaptive refinement. 1. Introduction In fracture mechanics, various fracture models are used to analyze different types of fracture processes. In the case of brittle materials, concepts of linear elastic fracture mechanics (LEFM) are used. Cracks in these materials are characterized by a separation of crack faces without cohesion as soon as the fracture energy reaches the material strength. In contrast, for quasi brittle materials, the transition from the full material strength to zero material strength occurs in a smooth fashion. This transition is characterized by a softening law that takes into account the non-linear material behavior in the vicinity of the crack-tip. The finite element method (FEM) is the method of choice for cases where the solution is smooth. However, special care is needed when the solutions involve discontinuities and singularities. Suitable meshes need to be constructed and maintained during the simulation. For Correspondence to: fries@cats.rwth-aachen.de

2 MODEL-INDEPENDENT APPROACHES FOR THE XFEM IN FRACTURE MECHANICS 1 discontinuities, the element edges need to be aligned to the discontinuity. For singularities and other high gradients, a mesh refinement is needed for accurate results. In contrast, the extended finite element method (XFEM) [1, 2, 3] is able to account for these non-smooth features without the need for special meshes; discontinuities and singularities may be right within elements. The XFEM uses the partition of unity concept [4] to incorporate a priori knowledge of the solution into the approximation space. This is achieved by enrichment functions at specified nodes that reflect special solution characteristics. In one of the first applications of the XFEM, Belytschko and Black [1] used a cracktip enrichment function for fracture mechanics. The enrichment function is based on the asymptotic field at the crack-tip for the case of brittle fracture. Later, Moës et al. [2] added the jump enrichment function to capture the discontinuity along the crack path away from the crack-tip. The method was successfully applied in two dimensional [5, 6, 7] and three dimensional [8, 9, 10] elastic crack growth. Xiao and Karihaloo [11] and Liu et al. [12] added some higher order terms in their enrichment functions and were able to compute stress intensity factors directly. In [13], Xiao and Karihaloo studied the effect of some parameters in the cracktip enrichments, i.e. the region of enrichment, and made several suggestions to improve the accuracy of the crack-tip fields. In order to get better results it was suggested in [14] to enrich all nodes within a characteristic radius. In this work, the XFEM is applied to solve fracture problems in brittle and quasi-brittle materials. Linear elastic fracture mechanics (LEFM) is only applicable when the size of the fracture process zone (FPZ) is negligible compared to the size of the crack and the size of the specimen. Other models account for the effect of the FPZ; the cohesive crack model is the simplest of such models [15, 16, 17]. For a general overview of both, LEFM and cohesive models we refer to [18]. One of the first applications of the XFEM for cohesive cracks is found in [19]. Only the generalized Heaviside function was used on a triangular mesh and no special near crack-tip enrichments were included in the formulation. As such, a sufficiently fine mesh was required to capture the complex fields near the cohesive zone. Later on, in [20] the idea was extended to solve geometrically non-linear problems. In [21], the XFEM was used to solve cohesive crack propagation problems. At the crack-tip, a modified set of branch enrichments is used since the stresses at the crack-tip are not singular as for brittle fracture. Along the crack path, the Heaviside enrichment function is applied. The growth of the cohesive zone is based on the energetic approach requiring the stress intensity factor to vanish at the tip of the cohesive zone. An alternative approach to determine the crack growth is to require the stress projection in the normal direction of the crack to be equal to the tensile strength of the material [22]. Another crack-tip element for cohesive cracks is developed in [22]. The mesh is based on three-node triangular elements. The crack-tip element is a three noded linear element or a six noded quadratic element. In [23], instead of common branch enrichment functions used for LEFM models, the enrichments were developed by superposition of the standard nodal shape functions and nodal shape functions for a sub-triangle of the cracked element. In [24], the XFEM was applied for cohesive cracks in concrete. The crack-tip field was enriched by using the same enrichment function as [21, 25]. In addition to the enrichment procedure, a local refinement is used to get better results. All elements cut by the crack plus an additional layer of one element are refined. Such a procedure avoids hanging nodes in interface elements. Hanging nodes only appear in elements that are not cut by the interface.

3 2 S. ABBAS ET AL. Thus additional elements are refined, even though they do not have any information about the crack. Constrained approximation is applied for hanging nodes. Various criteria of the direction of a crack extension are shown: maximum circumferential tensile stress [26], maximum energy release rate [27] and minimum strain energy density criteria [28]. There are also some other interesting applications of XFEM to cohesive crack models. In [29, 30, 31, 32, 33] the advantage of using the XFEM to solve cohesive fracture problems was shown. However, if the enrichment function does not represent the true asymptotic nature of the crack-tip field, an adaptive mesh refinement is to be applied to get accurate results. In most of the above mentioned applications of the XFEM in fracture mechanics, different crack-tip enrichment functions are used respectively for each model in order to capture the different nature of the solution, see e.g. [25]. For someone who needs to concentrate on modeling issues, it becomes an additional task to create the appropriate enrichment functions. In the case where fracture models with unknown analytical solutions are considered, a model-independent approach is useful. In such an approach, instead of having an enrichment function based on the asymptotic crack-tip fields, the aim is to capture all the possible solution gradients in the near-tip region. One of the possibilities is a special set of enrichment functions following ideas of Abbas et al. [34] to capture high gradients in convection dominated problems. A set of high gradient enrichment functions is used such that all the gradients starting from a large gradient that can no longer be captured by a standard FEM up to the situation where the gradient is almost infinite can be captured. A similar set of crack-tip enrichment functions is proposed here that can be applied to the fracture problems irrespective of the fracture model, see Chapter 4. Another approach shown in this paper is to capture high gradients and singularities by using adaptive mesh refinement based on elements with hanging nodes, see Chapter 5. Based on heuristic criteria, the refinement is locally applied at the crack-tips and sometimes even along the crack-path. In contrast to the constraint approximation for meshes with hanging nodes [35], a special set of conforming shape functions [36], that has partition of unity property, is used. This approach allows to enrich hanging nodes in a straight forward manner in the frame of XFEM. Only the model-independent Heaviside enrichment function is used along the crack path. At the crack-tip the high gradients are captured by the refinement. The idea shown in [22] allows to capture the exact crack-tip position (which is otherwise difficult for a pure Heaviside enrichment) and is adapted for arbitrary quadrilateral meshes herein. The paper is organized as follows: The problem statement for linear elastic fracture mechanics and cohesive fracture mechanics is given in Section 2. The general form of XFEMapproximations for fracture mechanics is given in Section 3. In Section 4, the special modelindependent crack-tip enrichment functions are proposed. The procedure for finding suitable sets of enrichment functions is discussed. Section 5 discusses the adaptive mesh refinement algorithm and describes the conforming shape functions. A special procedure for the Heaviside enrichment is shown which captures the crack-tip exactly. Numerical examples from LEFM and cohesive fracture are given in Section 6. The paper concludes with a summary and outlook in Section 7.

4 MODEL-INDEPENDENT APPROACHES FOR THE XFEM IN FRACTURE MECHANICS 3 2. Problem statement Fracture occurs when the stress at the atomic level exceeds the cohesive strength of the material [37]. Stress concentrations in solids due to flaws such as cracks were first studied by Inglis [38]. He predicted an infinite stress at the tip of an infinitely sharp crack. An infinitely sharp crack in a continuum is a mathematical abstraction that is justified when the crack-tip radius is in the order of the atomic radius [37]. In an ideally brittle solid a crack can be formed merely by breaking atomic bonds. However, in other solids such as metals, additional energy is dissipated due to a dislocation motion that occurs in the vicinity of the crack-tip making the material behavior non-linear. Such non-linearity is related to plastic, viscoplastic or damage effects depending on the material. When such a non-linear behavior is negligible, the crack can be modeled by the concepts of the linear elastic fracture mechanics (LEFM); this is called brittle fracture and was first studied by Griffith [39]. However, when the non-linear effects cannot be ignored, the crack needs to be modeled by elastic plastic fracture mechanics (EPFM). One of such models is the strip-yield model or cohesive crack model introduced by Dugdale [15] and Barenblatt [16]. Both LEFM and cohesive crack model are briefly recalled below. Γ c n Γ c n f Γ Coh Γ t Ω Γ t Ω Γ u Γ u Figure 1. Cohesionless crack. Figure 2. Cohesive crack Linear elastic fracture Consider a domain Ω shown in Figure 1 containing a crack. The domain consists of traction boundary Γ t, displacement boundary Γ u and crack faces Γ c. Equilibrium and boundary conditions are as follows [2] σ = 0 in Ω, (2.1) σ n = F on Γ t, (2.2) σ n = 0 on Γ c, (2.3) where n is the unit normal vector, F are the prescribed tractions and σ is the stress field inside the domain expressed in terms of the linear elastic and isotropic constitutive law σ = C : ɛ. (2.4)

5 4 S. ABBAS ET AL. Here, C is the Hooke s tensor. The elastic strains ɛ are expressed in terms of kinematic equations under the assumption of small strains and displacements ɛ = 1 2 ( u +( u)t )=ɛ(u), (2.5) For the approximation with finite elements, the problem has to be stated in its discretized variational form. The displacement u must belong to the set U of kinematically admissible displacement fields depending on the solution regularity [21] u U = {v V : v =0on Γ u }. (2.6) The space V must allow for discontinuous displacements across Γ c. The weak form of equilibrium equations is given as σ : ɛ(v) dx = F v ds v U. (2.7) Ω Γ t The discretized weak form becomes Cɛ(u h ):ɛ(v h ) dx = Ω F v h ds Γ t v h U h. (2.8) This gives a linear equation to solve for the unknown displacements u h. For crack propagation problems, the direction of crack propagation θ may be found as the direction of maximum hoop stress through the formula θ = 2 arctan 1 4 ( KI /K II sign(k II ) (K I /K II ) 2 +8 ) (2.9) where K I and K II are stress intensity factors, other criteria are found in [40, 26, 28]. For details on computing stress intensity factors see [1] Cohesive fracture Cohesive fracture models are based on the fact that the molecular forces of cohesion are not negligible near the crack-tip where two crack surfaces are not far from each other. Intensity of these forces quickly attains a maximum value, i.e. the tensile strength of the material f t, and then starts to decrease as the crack faces move away from each other while the crack opens up. Barenblatt [16] showed two important results for cohesive cracks, i.e. 1. The normal stress at the crack-tip is finite. 2. The opposite faces of a crack close smoothly at the crack-tip. Cohesive forces start to develop after the tensile strength of the material is reached. The point where the tensile strength is reached first is called the mathematical tip of the crack. The crack then starts to open up but the cohesive forces are still acting across the cracked surfaces until the crack opening displacement w reaches a critical value w c. The region between the point where the normal stress is equal to the tensile strength of the material and the point at which the crack opening displacement is equal to the critical opening, is called the cohesive zone. The problem for a cohesive crack is described as follows.

6 MODEL-INDEPENDENT APPROACHES FOR THE XFEM IN FRACTURE MECHANICS 5 Consider a domain Ω shown in Figure 2 containing a crack. The domain consists of traction boundary Γ t, displacement boundary Γ u, cohesive faces Γ coh and cracked faces Γ c. Equilibrium and boundary conditions are as follows σ = 0 in Ω, (2.10) σ n = F = λf on Γ t, (2.11) σ n + = σ n = f + = f = f on Γ coh, (2.12) where n is the unit normal vector, n + and n are the unit normal vectors on the two respective faces of the cohesive zone, f is the cohesive traction from the cohesive traction-displacement law f = f(w) (2.13) where w is the crack opening displacement given as w =(u u + ) on Γ coh. (2.14) Various cohesive displacement laws can be used. In this study, a linear law as shown in Figure 3, f = f t kw, (2.15) is used. In (2.10)-(2.12), F are the prescribed tractions linearly dependent on a scalar parameter λ called the load factor and σ is the stress field inside the domain expressed in terms of the linear elastic and isotropic constitutive law, see Equations (2.4) and (2.5). For the approximation with finite elements, the problem has to be stated in its discretized variational form. The displacement u must belong to the set U of kinematically admissible displacement fields (2.6) and the weak form of the equilibrium equations is given as σ : ɛ(v) dx + f w(v) ds = λ F v ds v U. (2.16) Ω Γ coh Γ t Using the definition of cohesive tractions f from (2.15), the weak form can be expressed as σ : ɛ(v) dx + Ω (f t kw(u)) w(v) ds = λ Γ coh F v ds Γ t v U. (2.17) Introducing the XFEM formulation (3.1) and the constitutive law (2.4) gives the discretized weak form Cɛ(u h ):ɛ(v h ) dx + (f t kw(u h )) w(v h ) ds = λ F v h ds v h U h. (2.18) Ω Γ coh Γ t This is a non-linear equation which is to be solved for the unknown displacements u h as well as the load factor λ. The problem description for cohesive crack growth is completed as follows. For a given crack path and the position of the mathematical tip, find the load factor λ and displacements such that the stress at the mathematical tip is equal to the tensile strength f t. This condition is called the stress condition: n σ n = f t. (2.19)

7 6 S. ABBAS ET AL. In the context of the XFEM, this approach is adopted by Zi and Belytschko [22]. The other approach that is used by Moës and Belytschko [21] is to find λ such that the stress intensity factor (SIF) vanishes at the crack-tip K I =0. (2.20) In the current study, the stress condition is used due to its intuitive nature and ease of implementation. An algorithm similar to that suggested by Zi and Belytschko [22] is used that is based on non-linear Newton-Raphson iterations for the convergence of displacements u h and the load factor λ. Details of the algorithm can be found in [22]. The direction of crack growth is found through the LEFM approach as given in (2.9), i.e. the crack path is not effected by the cohesive zone [21]. In cohesive crack growth, the first step is to choose the cohesive law representing the energy release from uncracked to the cracked surface. In this study, a linear cohesive law is used as given in (2.15), see Figure 3. The material softening starts as soon as the tensile strength of the f t Stress G f w c Crack opening displacement Figure 3. Linear cohesive law. Fracture energy G f represents the area under the linear law. material f t is reached. The slope of the law is given by k and w c is the critical crack opening displacement when the intermolecular forces of attraction vanishes. The shaded area under the cohesive law gives the fracture energy of the material G f. 3. General XFEM formulation A standard XFEM approximation for fracture problems consists of three parts, (i) a continuous FE approximation for the overall domain, (ii) an enrichment that accounts for the discontinuities across the crack surface, and (iii) an enrichment that accounts for the high gradients at the crack front. The (shifted) XFEM approximation for displacements u h (x) in a

8 MODEL-INDEPENDENT APPROACHES FOR THE XFEM IN FRACTURE MECHANICS 7 d-dimensional domain Ω R d is given as u h (x) = N i (x)u i i I }{{} continuous approximation m + k=1 i Jk N i (x) + i I N i (x) [S(x) S(x i)] a i } {{ } discontinuous enrichment ] [ B k (r, θ) B k (r, θ i ) } {{ } crack-tip enrichment where N i (x) is the standard FE function for node i, u i is the unknown of the FE part at node i, I is the set of all nodes in the domain, Ni (x) is a partition of unity function of node i, S(x) is a step enrichment, a i is the unknown of the enrichment at node i, I is a set of nodes that are cut by the crack (and which are not in J ), m is the number of enrichment terms, B(r, θ) is the crack-tip enrichment function, b k i is the unknown of the enrichment term k at node i, J is a set of nodes of element containing the crack-tip. The functions Ni (x) are equal to N i(x) in this work although this is not necessarily the case. For S(x), a typical enrichment function is the Heaviside-enrichment: { 0:φ 1 (x) < 0, S(x) = (3.1) 1:φ 1 (x) 0, where φ 1 (x) is the level-set function specifying the crack-path. The crack-tip enrichment function B k (r, θ) exhibits the asymptotic displacement field in the near-tip region. In the case of brittle fracture, typically the following four functions are used for the crack-tip enrichment { ( ) r θ B(r, θ) = sin, ( ) θ r cos, ( ) ( ) θ r sin sin θ, ( ) ( )} θ r cos sin θ, (3.2) where (r, θ) are the local polar coordinates at the crack-tip. The branch enrichment functions in the above form have singular derivatives with respect to r at the crack-tip, i.e. at r = 0, thus represented the stress singularity at the crack-tip in the case of brittle fracture. In the case of cohesive cracks, the stresses at the crack-tip are finite thus the asymptotic character of the displacement field is not represented by the classical branch enrichment functions. The enrichment functions (3.2) are then no longer appropriate [25]. In case there are other fracture models with unknown analytical results in the near-tip region, an additional task is to find an appropriate enrichment function in the near-tip region. In order to circumvent the need for customized enrichment schemes, two model-independent approaches are presented in this work that can be used irrespective of the fracture model. In the first approach, an optimal set of model-independent crack-tip enrichment functions is proposed that is found in a similar fashion as in [34], see Chapter 4. Then, the classical branch enrichment function in (3.2) will be replaced by a set of high gradient enrichment functions. In the second approach, instead of using a crack-tip enrichment, an adaptive mesh refinement is realized in order to capture the high gradient at the tip. Applying the local mesh refinement introduces hanging nodes. As an alternative to the constrained approximation and other approaches for meshes with hanging nodes, special conforming shape functions with partition of unity property are used [36] and degrees of freedom are associated with the hanging nodes. b k i,

9 8 S. ABBAS ET AL. 4. Special crack-tip enrichment functions In the following, an enrichment scheme in the XFEM is proposed that is suitable to capture arbitrary gradients in the near-tip region. This involves the definition of a class of appropriate functions from which a set of four particularly useful functions is identified through an optimization procedure Choice of the high gradient enrichment functions The first step is the design of a set of model-independent enrichment functions that is able to capture any stress-gradient in the near-tip region. For the case of linear elastic fracture, stresses are singular at the crack-tip. This a priori knowledge about the solution behavior is incorporated in the classical branch enrichment functions which have infinite derivatives at the crack-tip. For other fracture models the stresses are non-singular at the crack-tip. For those models, the enrichment functions have finite derivates at the crack-tip. In both cases, the displacement field in the near-tip region does not have a high gradient whereas the stresses do have a high gradient in the near-tip region. In order to incorporate this a priori knowledge about the solution behavior into the approximation space, only those functions are acceptable for the enrichment that have high gradients in their derivatives rather than in the functions themselves. A possible candidate for such a function is G(r) =r α, α R (4.1) with derivative G = dg(r) = αr α 1 (4.2) dr where r is the distance from the crack-tip, i.e. r = 0 at the crack-tip. We distinguish three cases for the derivative G at r = 0 as shown in Figure 4. The derivative is infinite for the case where α< 1 and is finite otherwise but for the case α> 2 the derivative is not a high gradient function at r = 0. Thus a set of enrichment functions is acceptable where α varies from 1 to 2. Using this range of the values of α, the following optimization procedure is used to find optimal sets of 2, 3 and 4 enrichment functions Optimization procedure The aim is to cover the complete range of high gradients starting from the gradient that can no longer be represented well by the standard FEM approximation up to the case of almost a singularity (the gradient is extremely large). For that purpose, one enrichment function is not sufficient. In contrast, several enrichment functions have to be chosen. It is to be noted that in this case the derivatives of the enrichment functions should span the complete gradient range. For a given number of m functions G (r) ={G 1(r),..., G m(r)} = {α 1 r α1 1,...,α m r αm 1 }, (4.3) an optimization procedure is employed in order to determine the corresponding values α 1,...,α m. The aim is to minimize the largest point wise error ε(g ) = sup(u h (r) f(r)) (4.4)

10 MODEL-INDEPENDENT APPROACHES FOR THE XFEM IN FRACTURE MECHANICS dg / dr 1! < 1! > 1 &! < 2 0.5! > r Figure 4. Choices for the high gradient enrichment functions. of the following interpolation problem G u h (r) = Ω Ω G f(r) (4.5) where f can be any given function having a high gradient in the near-tip region, i.e. at r = 0. In the current study, the following function f(r) is chosen that is interpolated by the m functions of (4.2) f(r) =G (r) =αr α 1. (4.6) The approximate solution u h (r) of the interpolated function f(r) is given by a linear combination of interpolator G j and the solution uh j, i.e. m u h (r) = G j u h j. (4.7) j=1 An important point is that for each prescribed set of enrichment functions G, the gradient of the function f(r) is varied systematically between a minimum and a maximum gradient by varying the value of α in (4.6). For each set G, the largest value for ε is stored in ε total. The optimal set for each number m is then the one with the smallest ε total. In this way, optimal sets of two, three and four enrichment functions are found, see Table I. These functions can interpolate all the near-tip stress gradients with some accuracy. The accuracy of the approximation increases with the number of enrichment functions but it is also noted that the condition number for the stiffness matrix increases with an increasing number of enrichment functions. In order to have a discontinuity along the crack path another ingredient is added to the enrichment functions which is the polar coordinate θ. The polar coordinate θ is defined as an angle that varies smoothly from π to π and is discontinuous along the crack. Thus the enrichment functions have the following form. B(r, θ) =θg(r). (4.8) In order to study the efficiency of the optimal sets (with 2,3, and 4 functions), the following preliminary study is performed. Further numerical results are shown in Section 6.

11 10 S. ABBAS ET AL. (a) 2 enr. func. Enr. Func. α G G (b) 3 enr. func. Enr. Func. α G G G (c) 4 enr. func. Enr. Func. α G G G G Table I. Optimal sets of enrichment functions Edge crack in a square specimen An academic test case of the edge crack in a square specimen W H is considered, see Figure 5(a). The displacement boundary conditions are prescribed such that the well known analytic solution of the near-tip field is the exact solution in the entire domain. In this case, the displacement field is a linear combination of crack mode I and II scaled by the stress intensity factors K I = 2 and K II = 3. Material parameters are E = and ν =0.3. The crack length is a = 1 and the specimen size is H = W = 2. A cartesian mesh with n el d = {19, 39, 59, 79} elements per dimension is used. H/2 Branch enrichment Optimal set with 2 functions Optimal set with 3 functions Optimal set with 4 functions a = W/2 H L2 Norm 10 2 W (a) Geometrical parameters. element size h (b) Comparison of the L2-Norm Figure 5. Edge crack test case. The L2 norms are compared with the reference solution where the four classical branch enrichment functions (3.2) are employed in the crack-tip element, see Figure 5(b). It is observed that the accuracy is increased with an increasing number of enrichment functions. The accuracy is slightly worse than the classical branch enrichments for the case of two and three enrichment functions whereas the accuracy is slightly better than the classical branch enrichments for the case of four enrichment functions. The order of convergence for all solutions shown in the figure is the same. Please note that only the crack-tip element is enriched with the set of high

12 MODEL-INDEPENDENT APPROACHES FOR THE XFEM IN FRACTURE MECHANICS 11 gradient enrichment functions G dg/dr 1 r = r = 1.20 r = 1.57 r = r (a) Functions. 0.5 r = 1.02 r = 1.20 r = 1.57 r = r (b) Derivatives. Figure 6. The optimal set of functions Optimal set of enrichment functions (a) High gradient enrichment function. (b) Derivative of high gradient enrichment function. Figure 7. High gradient enrichment function in 2D. In this work, the optimal set of four functions is used as the enrichment functions in all following numerical examples. These functions are shown in Figure 6(a) and their derivatives in Figure 6(b). One can see that the derivatives vary in a gradient range from a mildly high to a very high gradient thus covering the complete range of gradients that need to be captured. Introducing the coordinate θ in the enrichment functions gives the following set of functions for enriching the displacement field in the near-tip region. { } B(r, θ) = r 1.02 θ, r 1.20 θ, r 1.57 θ, r 1.95 θ. (4.9)

13 12 S. ABBAS ET AL. These functions will replace the classical branch enrichment functions (3.2) in the XFEM formulation (3.1). 5. Adaptive mesh refinement with hanging nodes Many mechanical problems feature regions where discontinuities or singularities appear. A common strategy for capturing non-smooth solution features is (local) mesh adaptivity as an alternative to or in addition to the enrichment of the approximation. In crack problems, one may apply adaptive mesh refinement in the near-tip region in order to capture the high gradients at the tip and use the step-enrichment in order to capture discontinuities across the crack surface. Adaptive refinements may lead to conforming meshes [41, 42] or non-conforming meshes with hanging nodes. The generation of conforming meshes requires considerably more computational effort. The advantage, of course, is the use of standard FEM (or XFEM) solvers as no hanging nodes are present. Nevertheless, we prefer refined meshes with hanging nodes due to the efficiency of the refinement. The hanging nodes require special consideration in the simulation code. We distinguish two approaches based on whether degrees of freedom (DOFs) are associated with the hanging nodes or not. The latter is known as constraint approximation and is applicable in the context of the XFEM with special care [35]. Therefore, we consider the special shape functions of Gupta [36]. Then, DOFs exist at the hanging nodes and the combination with XFEM is straightforward. It is noted that these shape functions fulfill the partition of unity property in the whole domain. This approach was first used by Fries et al. in [35] in the frame of the XFEM in order to solve problems in linear elastic fracture and incompressible two-phase flows Adaptive mesh refinement The adaptive mesh refinement is applied near the crack-tips where high gradients appear. Introducing the irregularity index k, one can define the maximum difference of refinement levels between adjacent elements in the mesh (see Figure 9). In this paper, isotropic 1-irregular meshes are used. Thus, the difference of refinement levels between two adjacent elements is equal to one (see Figure 8). This is also called 2-to-1 property. This property is desirable, because it does not allow large jumps of the element sizes in the mesh and ensures the sparseness of the global system matrix (small bandwidth). In this work we use up to 5 refinement levels, which makes the size of the smallest element 2 5 = 32 times smaller than the size of the corresponding original element. We are interested in keeping the DOFs at the hanging nodes, so that the XFEM can be applied in the straightforward way. An important consequence is that in this strategy, hanging nodes may be enriched as can be seen in Figure Conforming shape functions The functions presented by Gupta in [36] construct a conforming finite element space on a 1-irregular mesh. It is easily shown that these functions fulfill the partition of unity property in the whole domain. The definition is based on a reference element Ω =[ 1, 1] [ 1, 1]

14 MODEL-INDEPENDENT APPROACHES FOR THE XFEM IN FRACTURE MECHANICS 13 Figure 8. 1-irregular mesh. Figure 9. k-irregular mesh with k>1. Figure 10. Example of the refined mesh with crack (thick line). The elements cut by the crack are enriched. Consequently, hanging nodes may be associated with enriched DOFs (black dots). with 4 regular nodes and 4 hanging nodes (see Figure 11(a)). Nodes 1 4 are regular nodes and nodes 5 8 are hanging nodes that are positioned at the center of the edges. First, the reference shape functions associated with the hanging nodes are defined: N5 = 1 (1 ξ )(1 η), (5.1) 2 N6 = 1 (1 + ξ)(1 η ), (5.2) 2 N7 = 1 (1 ξ )(1 + η), (5.3) 2 N8 = 1 (1 ξ)(1 η ) (5.4) 2 for all ξ, η Ω. If one or more hanging nodes are not present, the associated shape functions are set to zero. For the reference element in Figure 11(b), only one hanging node is present, thus N5 = N 7 = N 8 = 0. The original bi-linear shape functions at the regular nodes 1 4, i.e. Ni,i=1...4, are then

15 14 S. ABBAS ET AL. η η ξ 6 ξ (a) 1 2 (b) Figure 11. Reference elements: (a) with a maximum number of four hanging nodes, (b) with one hanging node. modified as: N 1 = Ñ (N 5 + N 8 ) (5.5) N 2 = Ñ (N 5 + N 6 ) (5.6) N 3 = Ñ (N 6 + N 7 ) (5.7) N 4 = Ñ (N 7 + N 8 ) (5.8) For the example in Figure 11(b), only N2 and N 3 need to be changed, since functions N5,N7,N8 have no contribution. It can be seen, that the shape functions ( ) reduce to the standard bi-linear shape functions if no hanging nodes are present, i.e. N5 8 = 0. The shape functions are labeled N i after the bi-linear mapping from the reference to the real element. A visualization of some shape functions for an example mesh is shown in Figure 12. The extension of the proposed shape functions to three dimensions is straightforward, see Morton et al. [43] Zero-valued enrichment functions In the presence of hanging nodes, the modified shape functions of the regular nodes N 1 4 may be zero over parts of the element. On the other hand, the shifted step enrichment (3.1), is also zero on one side of the crack. That is, in cut elements, the enrichment functions N i [S (x) S (x i )] may be zero everywhere in the element, see Figure 13 (the solid line represents the discontinuity). Consequently, it is crucial that these nodes, for which the resulting enrichment functions vanish, are removed from the set I. We find that the detection of such nodes is simple and does not pose any disadvantage on the proposed procedure.

16 MODEL-INDEPENDENT APPROACHES FOR THE XFEM IN FRACTURE MECHANICS 15 (a) ( ) (b) (c) Figure 12. Conforming shape functions at regular and hanging nodes. hanging node (a) Example of a cut element with one hanging node. (b) Conforming shape function of the upper right node. (c) Shifted global enrichment function of the upper right node. Figure 13. Special case where the local enrichment function of the upper right node vanishes in the cut element. The product of the functions shown in (b) and (c) is zero Special quadrature rule The standard FE shape functions are C -continuous in the elements. In contrast, the shape functions employed at the hanging nodes are C 0 -continuous inside the elements, in particular, at the ξ- and η-axes of the reference element. This has important implications for the numerical integration. The appropriate integration scheme depends on the fact whether an element has hanging nodes and whether the element is cut by the crack. In the case where no hanging nodes are present and the element is not cracked, standard Gauss points are used (see Figure 14(a)). If the element is cracked, an element decomposition as in standard XFEM-applications [2, 3] is realized (see Figure 14(b)). In the case where hanging nodes are present, the enrichment scheme has to account for the kinks at the ξ- and η-axes in the reference element no matter whether the element is cracked or

17 16 S. ABBAS ET AL. (a) Uncut element without hanging nodes. (b) Cut element without hanging nodes. (c) Uncut element with hanging nodes. (d) Cut element with hanging nodes. Figure 14. Integration points in reference element. not. In an uncut case, four subcells [ 1, 0] [ 1, 0], [0, 1] [ 1, 0], [0, 1] [0, 1], [ 1, 0] [0, 1] with standard Gauss points are introduced as shown in Figure 14(c). If the element is cracked then subcells need to account for the crack surface as well (see Figure 14(d)). Applying this special quadrature rule ensures the correct integration and appropriately accounts for the discontinuities Exact crack-tip treatment When the pure Heaviside enrichment is employed for problems with cracks, it is well-known that the enrichment is virtually extended to the next edge of the crack-tip element, i.e. the crack-tip is not correctly captured. In [22], a procedure is proposed for triangular elements to capture the crack-tip exactly. Here, this idea is applied and extended for arbitrary quadrilateral elements. Let us consider an arbitrary quadrilateral cracked element that contains the crack-tip, see Figure 15. The aim is to match the crack-tip exactly using the Heaviside enrichment. As an input for the procedure, the position of the crack-tip, the corner nodes, and the edge cut by the crack path is needed. The procedure is then described as follows (see Figure 16): 1. The crack-tip is mapped into the reference element.

18 MODEL-INDEPENDENT APPROACHES FOR THE XFEM IN FRACTURE MECHANICS Figure 15. A partly cracked element with the crack-tip (star). The crack path (solid line) is virtually extended (dashed line) to the next element edge if the Heaviside-enrichment is employed only (rather than a special crack-tip enrichment). 2. The reference element is divided into two parts. The line that divides the reference element goes through the crack-tip and is parallel to the cut edge. 3. The points of intersection of this line and the edges are mapped into the real element and the line is reconstructed in the real element. In general, the mapped line is still straight and goes through the crack-tip. Thus the original crack-tip element is virtually divided into a cracked and an uncracked part. Both subelements remain quadrilateral. 4. Referring to Figure 16, the enrichment functions for node 1 and 2 are now evaluated in the cracked subelement as if this were a standard completely cracked quadrilateral element. Of course, the enrichment functions are zero in the uncracked part. The standard FE shape functions, however, are evaluated in the original element ( ) uncracked cracked 2 real element reference element real element Figure 16. The algorithm for constructing the exact crack-tip enrichment. Applying this algorithm allows to capture the crack-tip exactly. This procedure can be applied to arbitrary quadrilateral crack-tip elements. In case of Cartesian meshes, the cut line is parallel to the cut edge.

19 18 S. ABBAS ET AL Refinement schemes for crack propagation For crack propagation, the mesh refinement must follow the movement of the crack tip. Two approaches for adaptive refinement may be distinguished. In approach 1, the original coarse mesh is refined only in the vicinity of the current crack tip (see figure 17(a)). In this case the propagation increment a is constant and of the same order (or greater) than the element size h of the original coarse mesh. (a) Approach 1. (b) Approach 2. Figure 17. Refinement schemes for crack propagation. In approach 2, the mesh refinement is based on the previously refined meshes, i.e. the mesh is refined in the vicinity of previous and current crack tip positions (see Figure 17(b)). This approach has the potential advantage that the propagation increment can be in the order of the size of the refined elements at the crack-tip. That is, a h/2 n ref rather than a h, where n ref is the number of refinement levels. Thereby, the resolution of the crack can be considerably improved. It is found in the numerical results of Section 6 that the two approaches achieve almost identical results for the considered crack geometries. Approach 1 is, therefore, preferred as it requires considerable less elements than approach Numerical examples In order to show the effectiveness of the proposed model-independent techniques, two benchmark test cases are considered in brittle fracture and three test cases are considered in cohesive fracture. The results from the classical branch enrichment functions are compared with the results from the special high gradient enrichment scheme as well as from the adaptive mesh refinement technique. In the case of the special high gradient enrichment function, only

20 MODEL-INDEPENDENT APPROACHES FOR THE XFEM IN FRACTURE MECHANICS 19 the element that contains the crack-tip is enriched with the crack-tip enrichment functions Linear elastic fracture In this section, test cases related to quasi static linear elastic fracture mechanics (LEFM) are presented. The test cases considered are a double cantilever beam and a single edge notch beam Double cantilever beam The first test case considers crack growth in a double cantilever beam. The geometry is shown in Figure 18, where L = 11.8in, h =3.94in, a =3.94in. Plane stress conditions are assumed with elastic modulus E = psi, P = 197 lbs and Poissons ratio ν =0.3. The crack is extended by an initial perturbation of a =0.3in with an initial angle dθ =5.71. The crack increment is da =0.1in in each step with the direction of increment computed using Equation (2.9). The stress intensity factors are computed using the domain integral technique [1] within a square domain with side lengths 1.2in centered at the crack-tip. For the case of the high gradient enrichment functions, a mesh as shown in Figure 19 is used. In the case of adaptive mesh refinement, a uniform coarse mesh with elements is taken and then the refinement is applied at the crack-tip element. P dθ h a a L Figure 18. Quasi static crack growth in a double cantilever beam. Figure 19. Mesh for the double cantilever beam.

21 20 S. ABBAS ET AL. y tip high grad. enr. branch enr. adapt. mesh ref x tip Figure 20. Comparison of the crack path. The crack paths obtained for all the three methods show excellent agreement, see Figure 20. Moreover, as already suggested in Section 5.6, the refinement at the crack-tip can be applied after each crack propagation step to resolve the crack path and to vary the propagation increment. In Figure 21, both refinement approaches for the double cantilever beam problem are shown. We kept the propagation increment constant, since the considered crack geometry is simple. Both refinement approaches give the same results for the considered problem Single edge notch beam The second test case is a quasi static crack growth in a single edge notch beam taken from [44]. This test case comprises of two types of cases as shown in Figure 22. The difference is that in the second case there is an additional constraint near the top left corner. These two cases result in two different crack paths for the same initial notch and applied loading. Plane stress conditions are assumed with elastic modulus E = 38 GPa and Poissons ratio ν =0.18. The crack increment is da = 5 mm in each step with the direction of increment computed using Equation (2.9). The stress intensity factors are computed using domain integral with a square domain with side lengths 6 mm centered at the crack-tip. For the case of the high gradient enrichment functions, meshes are shown in Figure 23. In the case of adaptive mesh refinement, a uniform coarse mesh with elements is taken and then the refinement is applied at the crack-tip element. In both cases, the crack paths obtained with the high gradient enrichment function and the adaptive mesh refinement show very good agreement with the crack path obtained using the classical branch enrichment function, see Figures 24 and 25.

22 MODEL-INDEPENDENT APPROACHES FOR THE XFEM IN FRACTURE MECHANICS (a) Only crack-tip element is refined (b) Refinement is applied at each crack propagation step. Figure 21. Different refinement schemes for the double cantilever beam problem.

23 22 S. ABBAS ET AL. 50 q (a) 50 q (b) Figure 22. Quasi static crack growth in a single edge notch beam: (a) Boundary conditions type 1, (b) Boundary conditions type 2. (a) (b) Figure 23. Mesh for single edge notch beam: (a) Boundary conditions type 1, (b) Boundary conditions type 2.

24 MODEL-INDEPENDENT APPROACHES FOR THE XFEM IN FRACTURE MECHANICS y tip 50 high grad. enr. branch enr. adapt. mesh ref x tip Figure 24. Quasi static crack growth in a double cantilever beam (Type 1) y tip high grad. enr. branch enr. adapt. mesh ref x tip Figure 25. Quasi static crack growth in a double cantilever beam (Type 2).

25 24 S. ABBAS ET AL Cohesive fracture In the following examples, three test cases related to cohesive fracture are considered. The consideration of the fracture process zone affects the load-displacement curve only and the crack path stays unaffected. We do no longer show results obtained by the classical branch enrichment (3.2) as this is no longer appropriate for cohesive cracks. b a d 2b 2b Figure 26. Three point bending test Load/f t b high grad. enr. Carpinteri and Colombo (1989) adapt. mesh ref Deflection/b x 10 3 Figure 27. Load-point displacement curve for three point bending test G f = 50N/m Three point bending test The first test case is a three point bending test with the parameters shown in Figure 26 with b = 150 mm. The material parameters E = MPa, ν =0.18 and tensile strength f t =3.19 MPa. The beam is fixed in x-direction at the top

26 MODEL-INDEPENDENT APPROACHES FOR THE XFEM IN FRACTURE MECHANICS Load/f t b high grad. enr. Carpinteri and Colombo (1989) adapt. mesh ref Deflection/b x 10 4 Figure 28. Load-point displacement curve for three point bending test G f = 10N/m. middle point of the beam, i.e. at (0, b/2). The initial mesh is a cartesian mesh. In the case of the high gradient enrichment functions, only the crack-tip element is enriched. In the case of adaptive mesh refinement, the mesh is 5 times refined at the crack-tip in order to capture the high gradients. The crack is initiated at the bottom of the beam when the tensile stress reaches the tensile strength f t. The loading and boundary conditions are such that the crack follows a straight path. Two cases are considered here. In the first case the fracture energy is G f = 50 N/m, whereas in the second case a more brittle specimen is considered with the fracture energy G f = 10 N/m. The second test case is considered to show the snap-back phenomena in the case of brittle materials. The load-displacement curves are compared with the benchmark results of Carpinteri and Colombo [45]. Both model-independent approaches conform very well with the benchmark solution, see Figure 27. A snapback phenomena is observed for the case of low fracture energy. Figure 28 shows the load displacement curve for the case of fracture energy G f = 10 N/m. A very good agreement is shown with the benchmark solution of [45] Double cantilever beam with straight crack In the second test case, a double cantilever beam is considered as shown in Figure 29 with L = 400 mm and h =0.5L. The material parameters being E = MPa, ν = 0.18, f t = 3.19 MPa with fracture energy G f = 50 N/m. The geometry and loading conditions are symmetric, thus, the crack goes along a straight path. In this case a cartesian mesh of is used. Normal stress profiles along the crack path are observed and stresses are found to be finite, i.e. n σ n = f t, see Figure 30. The load displacement curves for both approaches conform to the reference solution in [22],

27 26 S. ABBAS ET AL. P L h Dimensionless stress P L Distance [mm] Figure 29. Double cantilever beam. Figure 30. Stress profiles. see Figure 31. It is noted that the adaptive mesh refinement performs slightly better than the high gradient enrichment scheme Load [KN] high grad. enr. Zi and Belytschko (2003) adapt. mesh ref Deflection [mm] Figure 31. Load-point displacement curve for straight crack in a DCB specimen Double cantilever beam with curved crack In the third test case the same double cantilever beam is considered as shown in Figure 29 with a difference that the crack is given

28 MODEL-INDEPENDENT APPROACHES FOR THE XFEM IN FRACTURE MECHANICS 27 an initial perturbation of a = 2 mm with an initial angle of dθ =4.8. This makes the crack follow a curved path. In this case, a cartesian mesh of is used. The load displacement curves for both approaches conform to the reference solution in [22], see Figure 32. This shows the efficiency of the proposed approaches in the case of straight as well as curved cohesive cracks high grad. enr. Zi and Belytschko (2003) adapt. mesh ref. 40 Load [KN] Deflection [mm] Figure 32. Load-point displacement curve for curved crack in a DCB specimen. 7. Conclusions Two distinct model-independent approaches to handle fracture mechanics problems in the frame of XFEM are presented. In the first approach, the classical branch enrichment functions for the crack-tip are replaced by a specially designed set of model-independent high gradient enrichment functions. The optimal set of four functions span the complete range of high gradients starting from the gradient that can no longer be represented well by the standard FEM approximation up to the case of almost a singularity. Only the crack-tip element is enriched with the special set of high gradient enrichment functions. One of the advantages of this approach is that there is no need for any mesh manipulation. This approach can be easily implemented in an existing XFEM code as only enrichment functions need to be changed. In an alternative approach, adaptive mesh refinement is used at the crack-tip. Hanging nodes have degrees of freedom and thus can be enriched. A special procedure to capture the cracktip exactly is developed for arbitrary quadrilateral elements. Special features of this approach