DELAMINATION PROPAGATION ANALYSIS OF COMPOSITE LAMINATE USING X-FEM

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16 TH NTERNATONAL ONFERENE ON OMPOSTE MATERALS DELAMNATON PROPAGATON ANALYSS OF OMPOSTE LAMNATE USNG X-FEM Toshio Nagashima, Hiroshi Suemasu Sophia Universit Kewords: Etended FEM; omposite laminate;

1 16 TH NTERNATONAL ONFERENE ON OMPOSTE MATERALS DELAMNATON PROPAGATON ANALYSS OF OMPOSTE LAMNATE USNG X-FEM Toshio Nagashima, Hiroshi Suemasu Sophia Universit Kewords: Etended FEM; omposite laminate; Delamination; Energ release rate Abstract The etended finite element method (X-FEM), which utilies interpolation functions based on the concept of the partition of unit, has been applied to evaluate fracture parameters such as stress intensit factors and crack etension simulations. X-FEM can simplif the modeling of continua containing several cracks, and can thus be used to perform effective stress analses related to fracture mechanics. X- FEM is applied to model interlaminar delamination, which ma propagate in composite laminate. The present paper describes a numerical procedure for modeling the delamination and the propagation of composite laminate. As numerical eamples, threedimensional stress analses for DB test specimens were performed b X-FEM, and fatigue delamination propagation analsis was demonstrated. 1 ntroduction The etended finite element method (X-FEM) [1][2], which utilies interpolation functions based on the concept of the partition of unit [3], has been applied to evaluate fracture parameters such as stress intensit factors and crack etension simulations. The X-FEM can simplif the modeling of continua containing several cracks, and thus can be used to perform effective stress analses related to fracture mechanics. n the X-FEM, the basis functions, which can approimate the properties of the displacement field, are added locall to the interpolation functions used in the conventional FEM. Such an added function is referred to as an enrichment function. Stress analses of threedimensional bodies containing planar and/or nonplanar cracks having comple geometries [-6] were performed more effectivel in conjunction with the level set method [7][8]. Therefore, to effectivel perform design analses and damage evaluation of composite structures, we have applied the X-FEM to model interlaminar delamination, which ma propagate in composite materials [9][10]. The authors performed X-FEM analsis of the specimens used in double cantilever beam (DB) and end notched fleure (ENF) fracture tests and evaluated the energ release rate at the front tip of the delamination in a composite laminate [11]. As delamination can be modeled independentl of the finite element mesh in the X-FEM analsis, the X- FEM can be used to perform stress analsis of a composite laminate with interlaminar delamination much easier than the conventional FEM in conjunction with the automatic mesh generation method. The present paper describes a numerical procedure for analing the delamination propagation of a composite laminate and presents numerical results for the DB test specimen. 2 X-FEM 2.1 nterpolation function n the present stud, interlaminar planar delamination in a composite laminate is assumed. As shown in Fig. 1, the analed domain is defined using artesian coordinates (,, ), and the planar delamination is assumed to be in an plane parallel to the - plane. n the X-FEM, the approimate displacement function u h of the distributed displacement u near a delamination is epressed as follows: h u ( ) = N 8 = 1 ( ) N k = 1 8 ( ) u k γ ( f ( ), ) a k (1.1) N ( ) H ( ) b f ( ) = N ( ) f ( ) (1.2) = 1 1

2 TOSHO NAGASHMA, H. Suemasu where N is the interpolation function used in the formulation of the conventional FEM, and denote the node set considering the asmptotic solution and the discontinuit of displacement near a crack, respectivel, and u, a k, and b denote the vectors of freedoms assigned to each node. Here, = φ is satisfied. n addition, γ i (i = 1,2,3,) are the near-tip functions, which consider the discontinuit near the crack tip, H() is the Heaviside function used to epress the discontinuit of the displacement on a delamination, and f() is a level set function introduced in order to epress the shape of the front tip of the delamination using nodal information. The level set function f() is described below. n the present stud, the level set function f() used in Eq. (1.2) is defined as follows: T f ( ) = min sign ( n( ) ( )) (2) Γ where Γ represents the curved front line of the delamination, is a point on the curved line Γ, and n() denotes the vector orthogonal to the curved line Γ at point. This function is called the signed distance function, and the absolute value of f is the distance between the point and Γ. n the present stud, the near-tip function γ i, which is determined from the asmptotic solution of a crack in a homogeneous isotropic material, is defined as follows: γ = 1 γ = 3 r cos, γ 2 = r sin, r sin sin θ, γ = r cos sin θ (3) where r and θ are the polar coordinates in a plane near the crack tip, and r 2 2 = f (.1) θ = arctan( / f ) (.2) The near-tip function given in Eq. (3) can be used in the X-FEM analsis for a crack in a homogeneous isotropic elastic material. Since γ 2 in Eq. (3) is discontinuous on the crack line (θ = ±π), it can epress the discontinuit displacement on the crack line. f Delamination Evaluation point r Front tip of a delamination θ f Fig. 1 X-FEM modeling of a delamination. 2.2 Nodal propert Using the X-FEM, a delamination can be modeled independentl of the finite element mesh much easier than would be possible using the conventional FEM. An eample distribution of the enriched node used for three-dimensional X-FEM analses is illustrated in Fig. 2. n the figure, the node labeled has an additional freedom for the near-tip function, and the node labeled has an additional freedom for the Heaviside function, as shown, respectivel, b the second and third terms on the right-hand side of Eq. (1.1). Front of delamination Delamination surface : Enriched node with asmptotic basic function : Enriched node with the Heaviside function Fig. 2 Distribution of enriched nodes for modeling of delamination. 2.3 Numerical integration Since the interpolation function in the element, which is constructed from enriched nodes, includes high-order functions, the normal order of the Gauss numerical integration ma be inadequate. n the X-FEM analsis, if an element includes a discontinuit displacement field such as a crack surface, then the element is partitioned into multiple 2

3 DELAMNATON PROPAGATON ANALAYSS OF OMPOSTE LAMNATE USNG X-FEM domains, and numerical integration is performed for each domain [-6].However, in the present stud, a planar delamination is assumed to be located just on the element surface, and an element never contains a discontinuous surface. Therefore, a higher order Gauss numerical integration is used rather than the partitioned domain method. A higher order of integration is adopted in elements with enriched nodes. 2. Evaluation of energ release rate n the finite element models obtained b X- FEM analsis, the geometr of the crack front does not alwas match the element boundar. Therefore, the virtual crack closure method, which is used in conjunction with conventional FEM analsis, cannot be applied to evaluate the energ release rate. The domain form of the energ release rate [12] can be used to compute the energ release rate in conjunction with X-FEM. f we define the local artesian coordinates ~, ~, and such that the direction of ~ is parallel with the crack etension, then the domain form of the energ release rate G in three-dimensional problems can be epressed as follows: ~ ~ q G = ( σ ~ w V τ ~~ τ ~ ) ~ ~ q ( τ ~~ ~ ~ ) σ τ ( τ ~ ~ ~ τ G = L G αds ~ q σ dv ) (5.1) (5.2) where V is the volume, including the evaluation point on the front tip of the delamination, α is a scalar field that is unit at the evaluation point and vanishes at the surface of the volume V, L c is the delamination front, and σ ij, u j, and w are the stress, displacement, and strain energ densit, respectivel. The clindrical region can be used as an integrated domain for the conventional FEM, and this region can be used for special cases of the X-FEM analsis, where the geometr of the front tip matches the element boundar. However, in the X-FEM analsis, the geometr of the front tip does not alwas match the element boundar. n such a case, a rectangular parallelepiped region defined in local artesian coordinates ( ~, ~, ), as shown in Fig. 3, is used, the center of which corresponds to the evaluation point at the front tip [-6]. The energ release rate G is calculated using the stress, strain, and displacement, which are evaluated at integration points in the domain. The weight function q takes the value of one at the center of the rectangular parallelepiped and vanishes at the surface. The weight function q is epressed as follows: 2 ~ 2 ~ 2 q( ~, ~, ) = (6) L~ L~ L where L ~, and L are the lengths of the edges of, L~ parallelepiped in the ~, ~,and directions, respectivel. Equation (5.1) can be integrated numericall using several integration points inside the rectangular parallelepiped. Moreover, Eq. (5.2) can be evaluated as follows: G G G = αd L / 2 Lc Rectangular parallelepiped ~ L L~ L~ ~ ~ Numerical ntegration m m m ntegration point (7) Fig. 3 Volume for the domain integral used to evaluate the energ release rate in 3D X- FEM analsis. 3. Development sstem 3.1 X-FEM Solver A structural analsis program based on the X-FEM and denoted as XSOLD has been developed. XSOLD is outlined in Table 1. We can easil perform stress analsis of composite laminate with a delamination using XSOLD. The procedure used to perform X-FEM analsis in the present stud is shown in Fig.. First, a three-dimensional finite element model for a composite laminate that has no delaminations is generated using mesh generation software. n addition, the geometr of a planar 3

4 TOSHO NAGASHMA, H. Suemasu delamination in the laminate is modeled b defining lines, which can be obtained b connecting the points on the front tip of the delamination. Second, the enrichment tpe for each node near the delamination is determined automaticall using development software. onsequentl, the finite element model data for X-FEM analsis is prepared. The elastostatic X-FEM analsis is performed, and the energ release rate at the delamination front is computed. Moreover, the crack etension is evaluated for each point on the delamination front using the crack etension law, and the geometr of the delamination front is updated. This procedure provides the delamination propagation analsis. Table 1 Outline of the development code XSOLD. ode name Development Language Discretiation method Analsis tpe Element tpe XSOLD Etended Finite Element Method Elastostatic, Natural frequenc, Buckling 8-node heahedral element Method to solve sstem Direct method: equation Skline method (in-house) PARDSO (ntel Math Kernel Librar) terative method: onjugate Gradient Method Eigenvalue analsis scheme Subspace method FEM Modeling 3DFEM Data Geometr of delamination Delamination Modeling X-FEM Data X-FEM Analsis X-FEM Result Delamination Etension Updating front tip Energ Release Rate Fig. Procedures for the stress analsis of a delaminated composite laminate b X-FEM. 3.2 rack propagation analsis sstem n order to perform the procedure described in the previous section, the control program RAK3D, which controls the XSOLD code, has been developed (see Fig. 5). n the development sstem, both the mesh data file denoted as ORG and the control data file denoted as NT are required. The ORG data file contains the finite element mesh information without the crack geometr, and the NT data file describes the crack geometr and analsis conditions. The user can perform the stress analsis and the crack propagation analsis easil b preparing ORG and NT data files. The control program RAK3D eecutes the XSOLD code as another process and obtains the value of the energ release rate at the delamination front from the log file denoted as, which is output file from the XSOLD code. Using this sstem, we can perform delamination propagation analsis more efficientl than with the conventional FEM using mesh generation software. ORG NT DAT write read read read RAK3D XSOLD eecute visualie GLview read read OUT VTF write Fig. 5 Development sstem for delamination propagation simulation using X-FEM. Numerical Eamples As a numerical eample, elastostatic and fatigue delamination propagation analses of a DB specimen, as shown in Fig. 6, were performed. The DB specimen analed herein has a length of L = 150 mm, a width of B = 25 mm, a thickness of H = 3 mm, and an initial delamination. The emploed finite element mesh, which is shown in Fig. 7, has 150 divisions in the length direction, 25 divisions in the width direction, and eight divisions in the thickness direction. The fine mesh division was used near the one, where the delamination front eists. The material of the DB specimen is assumed to be fiber-reinforced, where the elastic properties of the unidirectional pl of the fiber reinforced material are E L = 12 GPa, E T = 10.8 GPa, G LT = 5.9 GPa, G TT = 3.71 GPa, ν LT = 0.3, and ν TT = 0.5. n the present calculation, an eight-node heahedral isoparametric linear element was used. The 6th-order Gauss integration is adopted for the evaluation of the local stiffness containing an enriched elements..1 Elastostatic analsis The DB test [13] [1], as shown in Fig. 6, is the standard eperiment used to evaluate the mode- interlaminar toughness of carbon fiber-

5 DELAMNATON PROPAGATON ANALAYSS OF OMPOSTE LAMNATE USNG X-FEM reinforced plastic composite materials. n the DB test, the delamination provided as the starter crack is etended b displacing the opposite sides of the two beams, and the relationship between the load and the opening displacement between the two beams is measured. The energ release rate is evaluated using the domain integral method as post-processing of numerical stress analsis b X-FEM and FEM. The 3 mm 3 mm 2.25 mm parallelepiped region with equall distributed numerical integration points are used for domain integration. n the present analsis, delamination with either a linear or curved front tip, as shown in Fig. 8, is modeled. The curved front tip is assumed to be a parabolic curve. The distribution of the energ release rate at the linear front tip is shown in Fig. 9, as compared with that obtained b the conventional FEM and the reference solution provided b the beam theor. omparison with the reference value reveals that the evaluated distribution of the energ release rate at the front tip gives appropriate results. Net, the analsis of the delamination with the curved front tip was performed. The conventional FEM analsis using the double node was also performed for comparison. The distribution of the energ release rate was evaluated as shown in Fig. 10. The distribution of the energ release rate for the curved front is somewhat flatter than that for the linear front. Although the results obtained b the X- FEM and the conventional FEM differ slightl, the are similar..2 Fatigue delamination propagation analsis The fatigue crack propagation law for the delamination front in the composite laminate is assumed as follows: da / dn = ( ΔG) m (8) where a is the crack etension and N is the load ccle, ΔG is the range of the energ release rate, and and m are empirical parameters, which are constant and depend on the material and the environment. The delamination propagation analsis for the DB specimen is performed using the development sstem. The parameters and m for the crack propagation law in Eq. (8) are set as = and m = 8.8, respectivel [15]. The elastostatic analsis using the X-FEM was performed several times in succession. The calculations are performed for various increments of the load ccle ΔN. The relationship between the load ccle and the maimum etension of the delamination are shown in Fig. 11. Both the initial geometr and the final geometr of the delamination fronts in the composite laminate in case of ΔN =1000 are depicted in Fig. 12. t was shown that the developed sstem using the X-FEM can adequatel solve the crack propagation analsis. a: delamination length L=150 mm Delamination (starter crack) B=25 mm H=3 mm Fig.6 DB (Double antilever Beam) test Number of nodes:35,33 Number of elements:30,000 Fig.7 Finite element meshes utilied in X-FEM analses. (a) Linear front 0.5 mm (b) urved front Fig.8 Delamination front tip with various geometr. 5

6 TOSHO NAGASHMA, H. Suemasu Energ release rate [N/mm] X-FEM FEM Beam-Theor Location[mm] Fig.9 Distribution of energ release rate at the linear front tip of DB test specimen. Energ release rate [N/mm] X-FEM FEM X-FEMusingFEMmesh Location[mm] Fig. 10 Distribution of energ release rate at the curved front tip of DB test specimen. Etension of delamination front [mm] dn=1000 dn=2000 dn= Load ccle Fig.11 Etension of delamination front vs. applied load ccle. (a) (b) Fig. 12 Geometr of delamination front in the propagation simulation (a) initial state (b) final state (ΔN=1000). 5 oncluding remarks The present paper described the numerical procedure based on the X-FEM in order to perform the delamination propagation analsis of composite laminates and presented numerical results for the DB specimen with interlaminar planar delamination. n the future, the method proposed herein will be applied to composite laminates with more realistic delaminations and cracks. Moreover, the near-tip functions for orthotropic materials should be improved because the near-tip functions used in the present analsis cannot reconstruct the asmptotic displacement solutions of a crack in an orthotropic material. References [1] Beltschko, T. and Black, T., Elastic crack growth in finite elements with minimal remeshing, nt. j. numer. methods eng., Vol. 5, pp ,1999. [2] Moës, N., Dolbow,. and Beltschko, T., A finite element method for crack growth without remeshing, nt. j. numer. methods eng., Vol. 6, pp , [3] Babuska,. and Melenk,. M., The partition of unit methods, nt. j. numer. methods eng., Vol. 0, pp , [] Sukumar, N., Moës, N., Moran, B. and Beltschko, T., Etended finite element method for threedimensional crack modeling, nt. j. numer. methods eng., Vol. 8, pp , [5] Moës, N., Gravouil, A. and Beltschko, T., Nonplanar 3D crack growth b the etended finite element and level sets-part : Mechanical model, nt. j. numer. methods eng., Vol. 53, pp , [6] Gravouil, A., Moës, N. and Beltschko, T., Nonplanar 3D crack growth b the etended finite element and level sets-part : Mechanical model, nt. j. numer. methods eng., Vol. 53, pp , 2002 [7] Sethian, H. A., Level Set Methods and Fast Marching Methods: evolving interface in computational geometr fluid mechanics computer vision and material science, ambridge, UK: ambridge Universit Press, [8] Sukumar, N., hopp, D. L., Moës, N. and Beltschko, T., Modeling holes and inclusions b level sets in the etended finite element method, omput. Methods Appl. Mech. Engrg., Vol. 190, pp , [9] Nagashima, T., Stress analsis of composite materials using X-FEM, Proceedings of 3rd AAA/ASME/ASE/AHS/AS Structures, Structural Dnamics and Materials onference (SDM), Denver, olorado, [10] Nagashima, T., Suemasu, H., Buckling Analsis of 6

7 DELAMNATON PROPAGATON ANALAYSS OF OMPOSTE LAMNATE USNG X-FEM omposite Laminates with Delaminations using X- FEM, Proceedings of 2nd nternational onference on Structural Stabilit and Dnamics (SSD-2002), Singapore, [11] Nagashima, T., Suemasu, H., Stress Analsis of omposite Laminate with Delamination using X- FEM, nternational ournal of omputational Methods (accepted, in print). [12] Moran. B., Shih,. F., rack tip and associated domain integrals from momentum and energ balance, Engineering, Fracture Mechanics, Vol. 27, pp ,1987. [13] Friedrich, K. (Editor), Application of fracture mechanics to composite materials, omposite Materials Series, Vol. 6, Elsevier, [1] apanese ndustrial Standards ommittee, Testing Methods for nterlaminar Fracture Toughness of arbon Fiber Reinforced Plastics, S K7086,1993. [15] Hojo, M., Gustafson, -G., Tanaka, K. and Haashi, R., Mode and Mied-Mode Propagation of Delamination Fatigue racks in FRP Laminates, ournal of the Societ of Materials Science, apan, Vol.36, ,

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