Abdel Wahab, Magd

Nguyen-Xuan, H

Tran Vinh, Loc

English

Ghent University Academic Bibliography

 International Journal of Fracture Fatigue and Wear, Volume 3 57  Proceedings of the 4th International Conference on  Fracture Fatigue and Wear, pp. 57-64, 2015 EXTENDED ISOGEOMETRIC ANALYSIS IN MODELLING CRACKED STRUCTURES  L. Tran1, H. Nguyen-Xuan2 and M. Abdel Wahab1 1 Department of Mechanical Construction and Production, Faculty of Engineering and Architecture, Ghent University, Belgium 2 Department of Computational Engineering, Vietnamese – German University, Vietnam  Abstract: In this paper, we successfully extended eXtended IsoGeometric Analysis (XIGA) in simulation of stationary and propagating cracks. In this method, IsoGeometric Analysis (IGA) utilizing the Non-Uniform Rational B-Spline (NURBS) functions is incorporated with enrichment functions through the partition of unity. The Heaviside function is enriched to capture the discontinuous phenomenon at the crack faces while the asymptotic functions from analytical solution are incorporated with NURBS to perform the singular field at the crack tips. Based on the NURBS’s characteristics, this approach allows us to achieve easily the higher approximation order and continuity of the basic functions. As a result, XIGA can improve accuracy and higher convergence rate as compared with traditional finite element. Keywords: eXtended IsoGeometric Analysis; Linear Fracture Elastic Mechanics; crack propagation 1 INTRODUCTION In service, the cracks generated and grown from the defects under a cyclic loading cause a reduction of the load carrying capacity of the structures. Therefore, study of fracture mechanics is virtually important in guarantee of the performance and safety of the structures. The prediction and analysis of the crack problems are the attractive topics for many scientists in general and for computational experts in particular. As a result, a numerous numerical methods have been developed including finite element method (FEM) [1], boundary method [2], meshfree method [3], extended finite element method (XFEM) [4], etc. In these methods, approximated geometries introduce some error in the solutions because different shape functions are utilized in describing geometry and analysis. To overcome this issue, Hughes et al. proposed isogeometric analysis (IGA) [5] which fulfils a seamless bridge link between computer aided design and finite element analysis based on using the same B-Spline or non-uniform rational B-Spline (NURBS) functions in describing the exact geometry of problem as well as constructing finite approximation for analysis. IGA provides a flexible way to make refinement and degree evaluation. It allows us to achieve easily the higher approximation order and continuity of the basic functions as compared with the traditional finite element method. Up-to-now, IGA has been extensively and successfully studied for various fields of engineering and science, including fracture mechanics. For the crack problems, the enrichment functions through the partition of unity method are incorporated into IGA to capture the discontinuous phenomenon at the crack faces and singular filed at the crack tips. Recently, De Luycker et al. [6] has been used XFEM incorporated with IGA for linear fracture of mode I crack. He found that this method gains the greater accuracy and convergence rate. Then, Gorashi et al. [7] kept developing this approach (in short name XIGA) to perform the behaviour of the crack structures in 2D. XIGA also successfully applied in bi-material body with curved interface [8], assessment of collapse load of cracked plate [9], vibration of crack plate [10], cracked thin shell structures [11]. In this paper, we extended XIGA for simulation of stationary and propagating cracks. In this method, the  Heaviside function is utilized for model the discontinuous phenomenon at the crack face while the asymptotic functions from analytical solution is used to capture the singular fields at the crack tips. Based on the NURBS, this approach allows us to achieve easily the higher approximation order and continuity of basic functions. As a result, XIGA can improve accuracy and higher convergence rate as compared with XFEM.   International Journal of Fracture Fatigue and Wear, Volume 3 58  2 GOVERNING EQUATIONS OF CRACKED STRUCTURE Consider a linear elastic solid defined in a domain Ω with a boundary Γ  such that ,t cΓ = Γ ∪ Γ ∪ Γu  t cΓ ∩ Γ ∩ Γ = ∅u where , ,t cΓ Γ Γu are the Dirichlet and Neumann boundary and crack surfaces, respectively. The body subjected to body forces b  and to surface tractions t  on the free portion tΓ . The governing equations for this problem are 0∇ + =.σ b  in Ω      subjected to on0 onontcu Γ=  Γ = Γtσ .nu u (1) in which u  is the displacement field which satisfies the compatibility relation s= ∇ε u  in Ω      where / 0 /0 / /Tsx yy x∂ ∂ ∂ ∂ ∇ =  ∂ ∂ ∂ ∂  (2) and the Cauchy stress tensor σ is obtained from the constitutive relation based on Hooke’s law =σ Dε  in Ω       (3) where D  is the elastic constant matrix. The weak form of the equilibrium equations can be expressed as d d dtTΩ Ω ΓΩ = Ω + Γ∫ ∫ ∫D bu tuε ε  (4) 3 DISCRETIZATION 3.1 A brief of B-spline/NURBS functions A knot vector { }1 2 1, ,..., n pξ ξ ξ + +=Ξ is a non-decreasing sequence of parameter values iξ , 1,...i n p= + , where i Rξ ∈  called ith knot lies in the parametric space, p  is the order of the B-spline and n  is number of the basis functions. Using Cox-de Boor algorithm, the univariate B-spline basis functions are defined recursively start with order 0p =   ( ) 1,010 otherwisei iiifNξ ξ ξξ +< <=  (5) as 1p ≥ the basis functions are obtained from ( ) ( ) ( )1, , 1 1, 11 1 i pii p i p i pi p i i p iN N Nξ ξξ ξξ ξ ξξ ξ ξ ξ+ +− + −+ + + +−−= +− − (6) The multivariate B-spline basis functions are generated by a simple way - tensor product of the univariate B-splines ( ) ( ),1dA i pN N α ααα == ∏ξ ξ      (7) where 1, 2, 3d =  is  dimensional space. Fig. 1 illustrates an example of bivariate B-spline basis function from tensor product of two univariate B-splines { }1 1 34 2 40,0,0,0, , , ,1,1,1,1=ψ  and { }1 2 3 3 45 5 5 5 50,0,0, , , , , ,1,1,1=Ξ  in ξ  and η direction, respectively.   International Journal of Fracture Fatigue and Wear, Volume 3 59           To present exactly some conic sections, e.g., circles, cylinders, spheres, etc., non-uniform rational B-splines (NURBS) with an additional weight value 0Aζ > for each control point is used.  ( ) ( ), / ,m nA A A A AAR N Nξ η ζ ξ η ζ×= ∑   (8) 3.2 Extended isogeometric finite element method In order to capture the local discontinuous and singular fields, the enriched displacement approximation is introduced according to idea of XFEM as follow: ( ) ( ) ( )enrh std enr enrI I J JI S J SR R∈ ∈= +∑ ∑u x ξ q ξ q   (9) Here, the NURBS basis functions are utilized instead of the Lagrange polynomials to create a new numerical procedure – so-called eXtended IsoGeometric Analysis (XIGA) [7]. enrJR are the enrichment functions associated with node J located in enriched domain enrS  which is splitted up two parts including: a set Sc for crack faces enriched control points and a set Sf for crack tips enriched control points as shown in Fig. 2.                           Heaviside enriched control pointsCrack tip enriched control pointsCrack pathPhysical mesh Control points Fig. 2 Illustration of the nodal sets Sc, Sf for a quadratic NURBS mesh. To model the discontinuous displacement at crack faces, the Heaviside function, which otherwise becomes +1 if physical coordinate is above the crack and -1, is incorporated in the enriched functions: ( )( ) ( ) ( ) ( )enrJ J JR R H H= −ξ ξ x x   (10)  Fig. 1 B-splines basic functions.  Control point of set Sc Control point of set Sf  International Journal of Fracture Fatigue and Wear, Volume 3 60  The analytical displacement field of linear elastic fracture problem at crack tips in polar coordinate ( , )r θ  is expressed as below [12]: ( )( )222 2sin 1 2coscos 1 2sin 2 222 2 2 2sin 1 2cos cos 1 2sin2 2 2I IIu K Kr rvθ θθ κκ θθµ pi µ pi θ θκ θ κ    + +− +          = +           + −− −          (11) where IK  and IIK are the stress intensity factors of mode I and mode II, respectively.  From the analytical displacement, the enriched functions for crack tips are chosen as [13]: ( )41( ) ( ) ( , ) ( , )enrJ J J JLR R r rψ θ ψ θ= = −  ∑ξ ξ   (12) where ( , ) sin cos sin sin cos sin2 2 2 2r rθ θ θ θψ θ θ θ =     (13) 3.3 Discretization Now, substituting Eq. (9) into Eq. (2), the strain can be expressed follow the nodal displacement as 1m nTI II×== ∑B qε  (14) where the strain matrix B  is given by std enrI I I  B = B B  (15) in which stdB  and enrB are the standard and enriched part of matrix B  defined in the following form , ,, ,00Tx yy xR RR R =    B     (16) where R  can be either the NURBS basic functions ( )R ξ or enriched functions enrR . Substituting Eq.(14) into Eq.(4), the linear static equation for crack problem is obtained Kq = F  (17) with the global stiffness and force vector d and d dtT R RΩ Ω Γ= Ω = Ω + Γ∫ ∫ ∫K DB F b tΒ  (18) 4 NUMERICAL RESULTS In this section, we study linear elastic fracture mechanics in some numerical examples with rectangular geometry such as: an infinite plate under in-plane tension, the static crack and crack propagation of an edge cracked plate subjected to shear stress. In all numerical examples, plane strain state is assumed. Herein, we illustrated the present method using meshing of cubic elements.   International Journal of Fracture Fatigue and Wear, Volume 3 61  4.1 An infinite plate subjected to tension Firstly, let us consider an isotropic infinite plate with material parameters of E = 107 N/mm2, ν = 0.3 containing a centre crack of length 2a subjected to a remote uniform stress σ = 102 N/mm2. A unit shaded domain, as shown in Fig. 3, with crack length of 0.5 mm is modelled. To evaluate the present method, Fig. 4 reveals the comparison between XFEM and XIGA in investigation of the relative error and convergence rate of displacement and energy which are given by ( ) ( ) 1/2/Th h Td dΩ Ω = − − Ω Ω  ∫ ∫u u u u u u u  (19) ( ) ( ) 1/ 2/Th h Td dΩ Ω = − − Ω Ω  ∫ ∫e ε ε σ σ ε σ  (20) It is noted that as p = 1, XIGA formulation is identical to XFEM. As compared to XFEM, it is observed that present method archives supper accuracy and higher convergence rate in displacement error norm as well as energy error norm. Fig. 5 displays the contour plots of displacements and stress distributions in x and y directions, respectively.  10−110−410−310−2Size of elementRelative displacement error norm  XFEM (r = 0.98)XIGA (r = 2.25)   10−110−210−1Size of elementRelative energy error norm  XFEM (r = 0.48)XIGA (r=0.55) Fig. 4 Comparison of relative error norm of: (a) displacement, (b) energy.      (a) DisplacementxU  (b) Displacement yU  (c) Axial stress xxσ  (d) Axial stress yyσ  Fig. 5 Contour plot of the displacements and stresses distribution.       Fig. 3 Infinite crack plate in tension. (b) (a)  International Journal of Fracture Fatigue and Wear, Volume 3 62   4.2 An edge cracked plate under shear stress In next example, we consider a rectangular plate subjected to a shear stress τ = 1 N/mm2 as shown in Fig. 6 with material parameters of E = 3x107 N/mm2,ν = 0.25. Fig. 7a and b show the relations between relative error of stress intensity factor (SIF) via number of degrees of freedom (DOFs) and CPU time, respectively. It is again seen that present method archives more accurate than XFEM. Indeed, to get the accuracy of SIF IK  with error lower than 0.1%, XIGA needs approximate 4300 DOFs while XFEM uses more than 25000 variables with nearly two time computational cost. Fig. 8 plots the distribution of axial stress, shear stress and displacement along y direction. For comparison purpose of displacement yU , 3D elastic solid is also modelled by XIGA with meshing of 13x27x3 elements. The same contour plot of displacement is observed in Fig. 8c and d. 103 1040246810Number of degrees of freedomRelative error of KI (%)   XFEMXIGA 0 150 300 450 6000246810CPU tine (s)Relative error of KI (%)   XFEMXIGA Fig. 7 Comparison between XIGA and XFEM.   Fig. 8 Contour plot of the stresses and displacements: (a)xxσ , (b) xyσ , yU in 2D (c) and 3D (d).  4.3 Crack propagation of an edge cracked plate under mixed mode loading In this section, we want to show the capacity of present method in crack growth simulation without re-meshing. For this problem, two important parameters need to be specified during the crack growth   Fig. 6  Edge crack plate under shear (c) (b) (a) (d) (b) (a)  International Journal of Fracture Fatigue and Wear, Volume 3 63  procedure: crack growth direction cθ  and incremental crack length a∆ . The crack growth angle can be identified follow to several available criteria such as: the critical plane approach, the maximum circumferential stress, the maximum energy release rate, the minimum strain energy density (see review in [14]). In this present work, we adopt the maximum circumferential stress to evaluate the crack growth direction [15] ( )( )22atan 2( / ) / (1 1 8 / )c II I II IK K K Kθ = − + +  (21) The incremental crack length is commonly determined from Paris’s law. However, in this example, for simple, a constant crack increment length of a∆ = 0.3 mm is set for each step. Fig. 9 shows the meshes at the step number #1 and #8 with the axial stress distributions xxσ , respectively. It is observed that, during the crack growth, the mesh is unchanged while the position of crack tip downwards as oblique path. It leads to change the enrichment status of the control points, for instance, from being tip-enriched to Heaviside enriched or some standard control points becomes the enriched ones. (a)   (b)     Fig. 9 Propagation of an edge crack under shear stress after step 1 (a) and step 8 (b). 5 CONCLUSIONS In this paper, an eXtended IsoGeometric Analysis (XIGA) is extended to simulate the stationary and propagation of the crack problems. Based on idea of XFEM, the enrichment functions through the partition of unity method are integrated to capture the local discontinuous and singular fields. The advantage of present method is based on the NURBS which allows us to achieve easily the higher approximation and continuity in order of basic functions as compared with the traditional FEM. As a results, XIGA gains better accuracy and higher convergence rate than XFEM. Furthermore, the stresses and displacement are also plotted smoothly and continuous through the element boundaries.  6 REFERENCES [1] Barsoum RS. On the use of isoparametric finite elements in linear fracture mechanics. International Journal for Numerical Methods in Engineering. 1976;10:25-37. [2] Chang C, Mear ME. A boundary element method for two dimensional linear elastic fracture analysis. International Journal of Fracture. 1985;74:219-51. [3] Belytschko T, Lu Y, Gu L. Crack propagation by element-free Galerkin methods. Engineering Fracture Mechanics. 1995;51:295-315. [4] Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. Int J Numer Meth Engng. 1999;46:131-50. [5] Hughes TJR, Cottrell JA, Bazilevs Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering. 2005;194:4135-95. [6] De Luycker E, Benson D, Belytschko T, Bazilevs Y, Hsu M. 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Extended isogeometric analysis in modelling cracked structures

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Extended isogeometric analysis in modelling cracked structures

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