Analysis of Cracked Orthotropic Media Using the eXtended IsoGeometric - PDF document

1 Analysis of Cracked Orthotropic Media Us
Analysis of Cracked Orthotropic Media Using the eXtended IsoGeometric Analysis (XIGA)S.Sh. Ghorashi, N. Valizadeh, S. Mohammadi University of Science and Culture, Tehran, Iran, sh.ghorashi@usc.ac.ir 1111cfctiijjkkijkRRHRQ¦¦¦¦(1) R is the vector of NURBS basis functions, is the vector of additional degrees of freedom which are related to the modeling of crack faces, is the vector of additional degrees of freedom for modeling the crack tip, is the number of nonzero basis functions for a given knot span, is the number of basis functions that have crack face (but not crack tip) in their support domain and is the number of basis functions associated with the crack tip in their influence domain. H is the Heaviside function, which becomes +1 if , X (physical coordinates corresponding to the parametric coordinates , ) is above the crack and –1, otherwise and 1,2,3,4 are crack tip enrichment functions which are defined for orthotropic materials in the next section. 3 Orthotropic enrichment functions Asadpoure and Mohammadi [2] proposed the following orthotropic crack tip enrichment functions for XFEM: 123412121212,,,,(),(),(),()2222rQQQQrcosgrcosgrsingrsing (2) 1,2ssinarctanjcosssin(3)  1,2jjxjygcosssinssinj(4) are the local polar coordinates at the crack tip. and are real and imaginary parts of characteristic roots (1,2) , respectively, which are calculated by solving the following characteristic equation, 4321113123323222220cscsccscsc(5) In the above equation, ,1,2,3cijare the components of 2D orthotropic compliance matrix . It is noted that the roots of above equation are always complex or purel

2 y imaginary ,1,2jjxjyssisj and
y imaginary ,1,2jjxjyssisj and occur in conjugate pairs as s s and s s In order to evaluate the efficiency and validity of the proposed approach, an orthotropic disk with an inclined central crack subjected to double point loads (figure 1(a)) is solved. The material properties and material orthotropy axes are displayed in figure 1(a). 841 control points and 441 elements are used for modeling the problem which are illustrated in figures 1(b) and 1(c), respectively. The order of NURBS functions in both parametric directions and are considered three ( ), all knot vectors are open and uniform without any interior repetition and 2×2 Gauss quadrature and sub-triangles technique with 13 Gauss points in each sub-triangle are used for integration. The Lagrange multiplier method has been adopted for imposition of essential boundary conditions [11]. For determining the fracture properties, stress intensity factors (SIFs) are obtained by means of the interaction integral method which developed by Kim and Paulino [12]. Figure 1: An orthotropic disk with inclined central crack: (a) Geometry and boundary conditions; (b) Control points; (c) Elements. This problem has been solved by Kim and Paulino [1], Asadpoure and Mohammadi [2] and Ghorashi et al. [4] before. Table 1 compares the stress intensity factors reported before by those obtained using the present approach for the case of = 30°. Table 1: Stress intensity factors for an inclined central crack in an orthotropic disk subjected to point loads (Method DOFs Elements Cells Kim and Paulino [1] (MCC) 5424 999 - 16.73 11.33 Kim and Paulino [1] (M- integral) 5424 999 - 16.75 11.38 Asadpoure and Mohammadi [2] 1960 920 - 17.08 11.65 Ghorashi et al. [4] 1507 - 641 16.98 11.95 Proposed Method 1223 441 - 16.68 11.53 XIGA is used for analysis of this problem considering different inclination angles of the

3 crack. The computed values of mixed mode
crack. The computed values of mixed mode SIFs alongside those of reported by Asadpoure and Mohammadi [2] and Ghorashi et al. [4] are illustrated in figure 2. The results of XIGA are in good agreement with the others. Note that similar to XFEM and EFG method, XIGA enables us to analyze the problem of all crack inclinations applying only one discretization. 5 Conclusion In this paper, the recently developed XIGA has been further extended to analyze cracked orthotropic media. The orthotropic crack tip enrichment functions, applied in the XFEM, have been successfully adopted in XIGA to impose singular stress fields near the crack tip in the solution field. An orthotropic disk with an inclined central crack has been solved by the proposed approach. Results of mixed-mode stress intensity factors (SIFs) have been compared with the reference results and proved the validity, accuracy and efficiency of the proposed approach. (a)(b) Figure 2: SIF values corresponding to different central crack angles in the orthotropic disk: (a) mode I SIF; (b) mode II SIF. References [1] J.H. Kim and G.H. Paulino, Mixed-Mode Fracture of Orthotropic Functionally Graded Materials Using Finite Elements and the Modified Crack Closure Method, Engineering Fracture Mechanics, 69: 1557-1586, 2002 [2] A. Asadpoure and S. Mohammadi, Developing New Enrichment Functions for Crack Simulation in Orthotropic Media by the Extended Finite Element Method, International Journal for Numerical Methods in Engineering, 69(10): 2150-2172, 2007 [3] S.Sh. Ghorashi, S.R. Sabbagh-Yazdi and S. Mohammadi, Element Free Galerkin Method for Crack Analysis of Orthotropic Plates, Computational Methods in Civil Engineering (Iranian Journal), 1(1): 1-13, 2010 [4] S.Sh. Ghorashi, S. Mohammadi and S.R. Sabbagh-Yazdi, Orthotropic Enriched Element Free Galerkin Method for Fracture Analysis of Composites, Engineering

4 Fracture Mechanics, Available online: DO
Fracture Mechanics, Available online: DOI:10.1016/j.engfracmech.2011.03.011, 2011 [5] D. Motamedi and S. Mohammadi, Dynamic Crack Propagation Analysis of Orthotropic Media by the Extended Finite Element Method, International Journal of Fracture, 161: 21-39, 2010 [6] D. Motamedi and S. Mohammadi, Dynamic Analysis of Fixed Cracks in Composites by the Extended Finite Element Method, Engineering Fracture Mechanics, 77: 3373-3393, 2010 [7] S. Esna Ashari and S. Mohammadi, Delamination Analysis of Composites by New Orthotropic Bimaterial Extended Finite Element Method, International Journal for Numerical Methods in Engineering, Available online: DOI: 10.1002/nme.3114 , 2011 [8] T.J.R. Hughes, J.A. Cottrell and Y. Bazilevs, Isogeometric Analysis: CAD, Finite Elements, NURBS, Exact Geometry and Mesh Refinement. Computer Methods in Applied Mechanics and Engineering, 194: 4135–4195, 2005 [9] E. De Luycker, D.J. Benson, T. Belytschko, Y. Bazilevs and M.C. Hsu, X-FEM in Isogeometric Analysis for Linear Fracture Mechanics, International Journal for Numerical Methods in Engineering, Available online: DOI: 10.1002/nme.3121, 2011 [10] S.Sh. Ghorashi, N. Valizadeh and S. Mohammadi, Extended Isogeometric Analysis (XIGA) for Analysis of Stationary and Propagating Cracks, International Journal for Numerical Methods in Engineering, submitted for publication, 2011 [11] N. Valizadeh, S.Sh. Ghorashi, S. Mohammadi, S. Shojaee and H. Ghasemzadeh, Imposition of Essential Boundary Condition in Isogeometric Analysis Using the Lagrange Multiplier MethodApplied Mathematical Modelling, submitted for publication, 2011 [12] J.H. Kim and G.H. Paulino, The Interaction Integral for Fracture of Orthotropic Functionally Graded Materials: Evaluation of Stress Intensity Factors, International Journal of Solids and Structures, 40: 3967-4001, 2003 0153045 [2[ EnrichedEFG 0153045 XFEM Enriched (a)