Computers and Structures

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1 Computers and Structures 47 (05) Contents lsts avalable at ScenceDrect Computers and Structures journal homepage: T-splne based XIGA for fracture analyss of orthotropc meda S.Sh. Ghorash a,, N. Valzadeh b, S. Mohammad c, T. Rabczuk b, * a Research Tranng Group 46, Bauhaus-Unverstät Wemar, Wemar, Germany b Insttute of Structural Mechancs, Bauhaus-Unverstät Wemar, Wemar, Germany c School of Cvl Engneerng, Unversty of Tehran, Tehran, Iran artcle nfo abstract Artcle hstory: Accepted 9 September 04 Avalable onlne October 04 Keywords: Orthotropc meda Crack Extended sogeometrc analyss (XIGA) Orthotropc enrchment functons T-splne bass functons Local refnement Fracture analyss of orthotropc cracked meda s nvestgated by applyng the recently developed extended sogeometrc analyss (XIGA) (Ghorash et al., 0) usng the T-splne bass functons. The sgned dstance functon and orthotropc crack tp enrchment functons are adopted for extrnscally enrchng the conventonal sogeometrc analyss approxmaton for representaton of strong dscontnuty and reproducng the stress sngular feld around a crack tp, respectvely. Moreover, by applyng the T- splne bass functons, XIGA s further developed to make the local refnement feasble. For ncreasng the ntegraton accuracy, the sub-trangle and almost polar technques are adopted for the cut and crack tp elements, respectvely. The nteracton ntegral technque developed by Km and Paulno (003) s appled for computng the mxed mode stress ntensty factors (SIFs). Fnally, the proposed approach s appled for analyss of some cracked orthotropc problems and the mxed mode SIFs are compared wth those of other methods avalable n the lterature. Ó 04 Cvl-Comp Ltd and Elsever Ltd. All rghts reserved.. Introducton Orthotropc materals such as compostes have been ncreasngly appled n many engneerng applcatons e.g. aerospace, automoble and marne structures because of ther hgh strength and stffness to weght ratos. Consderng ther strength, they are appled n thn shell forms wth the possblty of crack ntaton and propagaton. As a result, fracture analyss of such meda has been the center of attenton for many researchers n the past few decades. Analytcal soluton of stress and dsplacement felds for an orthotropc plate wth a crack has already been obtaned by Sh et al. [3]. Some other analytcal nvestgatons on fracture behavor of compostes can be found n [4 0]. As the analytcal methods are not feasble n resolvng practcal engneerng problems, numercal methods are better alternatves. The remeshng requrement and the exstence of sngular felds around crack tps n smulaton of crack propagaton problems led to the development of several computatonal approaches such as meshfree methods [ ] and the extended fnte element Correspondng authors at: Cvl Engneerng Department, Bauhaus-Unverstät Wemar, Wemar, Germany. E-mal addresses: shahram.ghorash@un-wemar.de (S.Sh. Ghorash), navd. valzadeh@un-wemar.de (N. Valzadeh), smoham@ut.ac.r (S. Mohammad), tmon.rabczuk@un-wemar.de (T. Rabczuk). method (XFEM) [ 4]. Problems nvolvng wth movng dscontnutes such as crack propagaton can be analyzed by these methods wthout the requrement of remeshng or rearrangng of the nodal ponts. Some applcatons of these methods can be found n [5 3]. In XFEM, a pror knowledge of the soluton s locally added to the approxmaton space. Ths enrchment allows for accurate capture of partcular features such as dscontnutes and sngulartes whch are present n the soluton. In order to analyze the problem of cracked orthotropc bodes dfferent approaches have been appled such as the hybrddsplacement fnte element method [3], the boundary element method (BEM) [33], fnte elements and the modfed crack closure method [34]. Asadpoure et al. [35 37] succeeded n developng three sets of orthotropc enrchment functons for dfferent types of compostes usng the analytcal solutons and mplemented them wthn an XFEM framework. The general form of orthotropc enrchment functons [37] have also been adopted n the enrched element free Galerkn (EFG) method [38]. Further developments have been reported for dynamcs and movng cracks n orthotropc meda [39,40] and delamnaton analyss of compostes [4]. Although cracked orthotropc meda have been studed by several dfferent methods, the new developments n the promsng computatonal approach of the extended sogeometrc analyss (XIGA) [4,] have attracted new nvestgatons to further apply t for such problems /Ó 04 Cvl-Comp Ltd and Elsever Ltd. All rghts reserved.

2 S.Sh. Ghorash et al. / Computers and Structures 47 (05) XIGA takes the advantages of ts two orgns: the extended fnte element method (XFEM) and the sogeometrc analyss (IGA) [43]. IGA ntegrates the computer aded desgn (CAD) nto the fnte element method (FEM) usng the concept of soparametrc elements, n whch the same shape functons are used to represent the geometry and to approxmate the soluton. Its superortes over conventonal FEM are: capablty of exact representaton of complex geometres regardless of the mesh coarseness, smplfcaton of the refnement process and mprovement of the soluton accuracy. A large varety of problems [44 46] have already been solved by IGA. XIGA has been successfully appled for smulaton of statonary and propagatng cracks n D lnear-elastc sotropc meda []. Ghorash et al. have further enhanced XIGA for analyss of curved cracks [47]. In addton, more results have been reported on crack detecton usng the XIGA [48]. Recently, the XIGA has been also appled for analyss of materal nterface problems and the technques for achevng optmal convergence rates have been addressed [49]. The current XIGA method adopts the conventonal NURBS bass functon whch has a consderable drawback. The local refnement cannot be defned wthn t because t s based on a tensor-product structure whch requres the control ponts to le topologcally n a rectangular grd. In other words, when a control pont needs to be added, several superfluous control ponts should be defned. One soluton s to use multple patches, whch has also some lmtatons. The compatblty of adjacent NURBS patches on ther nterfaces has to be mantaned. Generally, the refnement extends from one patch to another unless the compatblty between patches are enforced n a dfferent way. One can weakly enforce t by the varatonal formulaton applyng the dscontnuous Galerkn formulaton. Another alternatve s to enforce C 0 contnuty on the nterfaces between patches [50] by utlzng constrant equatons for the control ponts and varables. IGA formulatons based on trangular splnes [5] mght be an nterestng alternatve as they smplfy the mesh generaton and adaptve refnement procedure. A more enchantng soluton for the aforementoned problem s to use the so-called T-splnes [5 54], whch are a generalzaton of NURBS, by allowng a row of control ponts to termnate before reachng the patch boundary. Ths feature enables the truly local refnement wthout extendng the entre row of control ponts. Furthermore, by usng the T-splnes several NURBS patches that have dfferent knot vectors can be effcently merged nto a sngle gapfree model of C 0 or hgher order contnuty [53]. Recently, a subset class of T-splnes called analyss-sutable T-splnes, whch are lnearly ndependent, has been ntroduced n the IGA framework [55] and a hghly localzed refnement algorthm whch meets the demands of both desgn and analyss has been presented [56]. Another alternatve to T-splnes are PHT-splnes or RHT-splnes that are based on herarchcal T-meshes [57]. Contnuum and structural element formulatons based on PHT- and RHT-splnes have been developed n [58 60], respectvely. In ths contrbuton, XIGA s further extended and the T-splne bass functons whch belong to the analyss-sutable T-splnes are adopted to make the local refnement for feasble adaptve procedure. Furthermore, based upon the work of Ghorash et al. [6], the orthotropc enrchment functons [37] are adopted to nvestgate cracked orthotropc bodes. In Secton, mportant formulatons of orthotropc materals are ntroduced. Bass functons ncludng NURBS and T-splnes are then descrbed n Secton 3. Thereafter, the proposed XIGA method ncludng orthotropc enrchment functons s presented n Secton 4. The results obtaned from the present orthotropc XIGA and those avalable n the lterature are compared n Secton 5 to demonstrate the accuracy and effcency of the proposed approach. Ths s acheved by mplementng the nteracton ntegral technque, developed by Km and Paulno [], for computaton of mxed mode stress ntensty factors. Ths secton s followed by some concludng remarks n Secton 6.. Fracture mechancs n orthotropc meda The stress stran law n an arbtrary lnear elastc materal can be wrtten as e ¼ cr where e and r are the stran and stress vectors, respectvely, and c s the complance matrx, E m E m 3 3 E m E E m 3 E m 3 c 3D E ¼ m 3 E E ðþ G G G where E; m and G are Young s modulus, Posson s rato and shear modulus, respectvely. For a plane stress case, the complance matrx s reduced to the followng form: E m 3 E m E E c D ¼ ð3þ G and for a plane stran state, c D j ¼ c 3D j c3d 3 :c3d j3 c 3D 33 for ; j ¼ ; ; 6 Now assume an ansotropc body s subjected to arbtrary forces wth general boundary condtons and a crack. The global Cartesan coordnate ðx ; X Þ, the local Cartesan coordnate ðx; yþ and the local polar coordnate ðr; hþ, defned on the crack tp, are llustrated n Fg.. A fourth-order partal dfferental equaton wth the followng characterstc equaton can be obtaned from the equlbrum and compatblty condtons [5], c s 4 c 6 s 3 þ ðc þ c 66 Þs c 6 s þ c 0 ¼ 0 ð5þ where c j ð; j ¼ ; ; 6Þare the components of c D. Accordng to [5], the roots of Eq. (5) are always complex or purely magnary s k ¼ s kx þ s ky ; k ¼ ; and occur n conjugate pars as s ; s Fg.. An arbtrary orthotropc cracked body subjected to body force b and tracton t. ðþ ð4þ

3 40 S.Sh. Ghorash et al. / Computers and Structures 47 (05) and s, s. The two-dmensonal dsplacement and stress felds n the vcnty of the crack-tp can then be derved as [3] Mode I rffffffffff u I r ¼ K I Re s p p s s cos h þ s sn h s p cos h þ s sn h rffffffffff v I r ¼ K I Re s p q s s cos h þ s sn h s q cos h þ s sn h w I ¼ 0 r I xx ¼ K I s s s s pffffffffffffffff Re pr s s cos h þ s sn h cos h þ s sn h r I yy ¼ K I s pffffffffffffffff s Re pr s s cos h þ s sn h cos h þ s sn h r I xy ¼ K I s pffffffffffffffff s Re pr s s cos h þ s sn h cos h þ s sn h Mode II rffffffffff u II r ¼ K II Re p p s s cos h þ s sn h p cos h þ s sn h rffffffffff v II r ¼ K II Re q p s s cos h þ s sn h q cos h þ s sn h w II ¼ 0 r II xx ¼ r II yy ¼ r II xy ¼ K II s s pffffffffffffffff Re pr s s cos h þ s sn h cos h þ s sn h K II pffffffffffffffff Re pr s s cos h þ s sn h cos h þ s sn h K II s pffffffffffffffff s Re pr s s cos h þ s sn h cos h þ s sn h where Re denotes the real part of the statement and K and K II are stress ntensty factors for modes I and II, respectvely. p and q can be defned by p ¼ c s þ c c 6 s ; ð ¼ ; Þ ð0þ q ¼ c s þ c s c 6 ; ð ¼ ; Þ ðþ 3. Bass functons 3.. NURBS Non-unform ratonal B-splnes (NURBS) are a generalzaton of pecewse polynomal B-splne curves. The B-splne bass functons are defned n a parametrc space on a knot vector N. A knot vector n one dmenson s a non-decreasng sequence of real numbers: N ¼ n ; n ;...; n nþpþ ðþ where n s the th knot, s the knot ndex, ¼ ; ;...; n þ p þ ; p s the degree of the B-splne, and n s the number of bass functons. The half open nterval ½n ; n þ Þ s called the th knot span and t can have zero length snce knots may be repeated more than once, and the nterval n ; n nþpþ s called a patch. In the sogeometrc analyss, always open knot vectors are employed. A knot vector s called open f t contans p þ repeated knots at the two ends. ð6þ ð7þ ð8þ ð9þ Wth a certan knot span, the B-splne bass functons are defned recursvely as, N 0 ðþ¼ n f n 6 n < n þ ð3þ 0 otherwse and N p ðþ¼ n n n N p n þp n ¼ ; ; 3;... ðþþ n n þpþ n N p n þpþ n ðþp n þ þ where ¼ ; ;...; n. A B-splne curve of degree p s defned by: CðnÞ ¼ Xn ¼ N p ðþp n ð4þ ð5þ where N p ðþs n the th B-splne bass functon of degree p and fp g are the control ponts, gven n d-dmensonal space R d. The non-unform ratonal B-splne (NURBS) curve of degree p s defned as: CðnÞ ¼ Xn ¼ R p ðþp n ð6þ R p ðþ¼ n Np ðþw n P n ð7þ j¼ Np j ðþw n j where R p are the NURBS bass functons and w s the th weght that must be non-negatve. In the two dmensonal parametrc space ½0; Š, NURBS surfaces are constructed by tensor product n o through knot vectors N ¼ n ; n ;...; n nþpþ and n o N ¼ n ; n ;...; n mþqþ. It yelds to: N p R p;q ;j n ; n n M q j n w;j ¼ P n m ð8þ k¼p l¼ Np k n M q l n wk;l For more detals on NURBS, refer to [6]. 3.. T-splnes T-splnes functons are a generalzaton of NURBS whch allow termnaton of a row of control ponts before reachng the patch boundary [5 54]. T-splne bass functons are defned n a trmmed ndex space enttled T-mesh. Trmmed ndex space s smlar to the ndex space representng the NURBS where the frst and last n o knots of the global knot vectors are neglected. n o equals to ðp þ Þ= and p= for odd and even polynomal degrees, respectvely. T-mesh s a trmmed ndex space where T-junctons, whch are vertces connectng three edges, are allowed. An example of T-mesh s llustrated n Fg.. It s noted that each lne n the mesh corresponds to a knot value of the trmmed global knot vector. Then, anchors are defned on the T-mesh to dentfy the locaton of each bass functon. In each parametrc drecton, anchors are located on the lnes f the correspondng polynomal degree s odd. Otherwse, ther locatons are assumed n the mddle of the lnes. Regardless of the degree, an anchor locaton s at the center of the support of a functon n the ndex space. For defnton of T-splnes, local knot vectors are defned nstead of usng the global knot vectors snce each bass functon has the compact support of ðp þ Þðq þ Þ knots. As llustrated n Fg. 3, local knot vectors n each drecton are defned by horzontally or vertcally marchng from the anchors backward and forward [53]. Afterwards, each bass functon T p;q a n ; n can be defned usng Eqs. (3), (4) and (8)ts correspondng local knot vectors N a and N a. In order to refne the mesh, the knot nserton process s performed. It conssts of addng new knots to the present (T-) mesh

4 S.Sh. Ghorash et al. / Computers and Structures 47 (05) Fg.. A sample of T-mesh. Fg. 4. Geometry and loadng of the orthotropc rectangular plate wth an edge crack. Control ponts Elements whole vew Fg. 3. Schematc vew of defnng local knot vectors for the anchor S a: (a) quadratc polynomal degrees: N a ¼ n ; n 3 ; n 6 ; n 7 and N a ¼ n 4 ; n 5 ; n 6 ; n 8 ; (b) cubc polynomal degrees: N a ¼ n ; n 3 ; n 6 ; n 7 ; n 8 and N a ¼ n 5 ; n 6 ; n 8 ; n 9 ; n 9. and correspondngly, modfyng and addng some control ponts. For more nformaton about T-splne and local refnement, readers are referred to [5 54]. The T-splnes appled n ths paper are analyss-sutable T-splnes [55]. Analyss-sutable T-splnes are a set of T-splnes whose T-mesh satsfes some topologcal condtons. It can contan around the crack tp Fg. 5. Dscretzaton of the globally refned model of the rectangular orthotropc plate. T-junctons but L-junctons, I-junctons and solated nodes are not allowed and ther extenson graphs should be empty [55,56]. Satsfyng the above condtons, assures lnear ndependence of T-splne s blendng functons [55]. Readers are referred to [55,56] for more detals.

5 4 S.Sh. Ghorash et al. / Computers and Structures 47 (05) Control ponts Physcal mesh Elements whole vew around the crack tp Fg. 6. Dscretzaton of the locally refned model of the rectangular orthotropc plate. 4. Extended sogeometrc analyss The extended sogeometrc analyss (XIGA) [4,] s a newly developed computatonal approach whch uses the superor concepts of the extended fnte element method (XFEM) [,3] wthn the sogeometrc analyss method [43]. As a result, a crack can be modeled ndependently of the mesh and can propagate wthout the necessty of remeshng snce the computatonal doman dscretzaton s defned ndependently of the crack locaton. Whle applyng the NURBS bass functons has some lmtatons n the local refnement, adoptng the T-splne bass functons n the XIGA allows for effcent ntroductons of local refnement and avodng refnng the unnecessary parts [63]. The geometry descrpton s then defned as, X nen X n ; n ¼ ¼ T p;q n ; n P ð9þ where n en s the number of nonzero bass functons for a gven knot span and T p;q n ; n are the T-Splne bass functons of degrees p and q n n and n drectons, respectvely, at the pont n ; n n the parametrc space ½0; Š½0; Š. The soluton feld approxmaton s extrnscally enrched by the Heavsde and branch functons for representng the crack face and the sngular stress feld around the crack tp, respectvely, X nen u h n ; n ¼ ¼ þ Xn Q T p;q n ; n u þ Xn H k¼ X 4 T p;q k n ; n a¼ j¼ Q a b a k T p;q j n ; n Haj ð0þ The frst term n the rght hand sde s the standard T-splne based sogeometrc analyss approxmaton. a j are the vectors of addtonal degrees of freedom whch are related to the modelng of crack faces, b a k are the vectors of addtonal degrees of freedom for modelng the crack tp, n Q s the number of n en bass functons whch are selected as branch enrched bass functons. They can be selected usng the topologcal enrchment strategy or the geometrcal enrchment approach. In the topologcal enrchment scheme, the bass functons whch contan the crack tp n ther nfluence domans are selected as the branch enrched bass functons whle n the geometrcal enrchment method, branch enrched bass functons consst of the bass functons chosen from the prevous strategy and the ones whch are selected accordng to consderng a constant doman around the crack tp. In ths contrbuton, the topologcal enrchment method s adopted. n H s the number of n en bass functons that have crack face n ther support domans and have not been selected as branch enrched bass functons. H s the generalzed Heavsde functon [3], HðXÞ ¼ þ f ðx XÞ e n > 0 otherwse ðþ where e n s the unt normal vector of crack algnment n pont X on the crack surface whch s the nearest pont to X n ; n. In Eq. (0), Q a fa ¼ ; ; 3; 4gare the crack tp enrchment functons whose roles are to reproduce the sngular feld around the crack tps. In ths paper, the followng orthotropc crack tp enrchment functons, developed by Asadpoure and Mohammad [37], whch were defned based on the analytcal soluton (Eqs. (6) (9)), are adopted,

6 fq a g 4 a¼ ¼ pffff h pffffffffffffffffffffff pffff h pffffffffffffffffffffff pffff h pffffffffffffffffffffff r cos g ðþ h ; r cos g ðþ h ; r sn g ðþ h ; r r sn h pffffffffffffffffffffff g ðþ h ðþ S.Sh. Ghorash et al. / Computers and Structures 47 (05) ntegraton of other elements. For determnng the fracture propertes, the stress ntensty factors (SIFs) are obtaned by means of the nteracton ntegral method, prevously developed by Km and Paulno []. wth h h ¼ arctan s y sn h cos hþs x sn h ; ð ¼ ; Þ ð3þ qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff g ðþ¼ h ðcos h þ s x sn hþ þ s y sn h ; ð ¼ ; Þ ð4þ where s x and s y are real and magnary parts of s computed by Eq. (5). It s noted that the thrd and fourth functons n the rght-hand sde of the Eq. () are dscontnuous across the crack faces whle the frst and second ones reman contnuous. 5. Numercal example In order to evaluate the effcency and valdty of the proposed approach, two numercal examples are consdered. The frst one s a rectangular plate wth an edge crack subjected to dstrbuted tenson loads and the other one s a crcular shape contanng an nclned central crack. Cubc and quadratc bass functons are adopted for the frst and second examples, respectvely. The effects of change n orentaton of materal elastc axes and crack nclnaton angle are nvestgated n the frst and second examples, respectvely. Both problems are analyzed usng both the NURBS and T-splne based XIGA. In order to ncrease the ntegraton accuracy, the sub-trangles technque wth 3 Gauss ponts s adopted for ntegraton over elements cut by the crack, whle the almost polar technque wth 7 7 Gauss ponts s utlzed for elements that contan the crack tp (see []). Standard 4 4 Gauss quadrature ponts are used for 5.. Rectangular plate wth an edge crack under tenson The proposed method s appled for analyss of a fnte rectangular orthotropc plate wth an edge crack subjected to un-axal tenson. The plate s consdered n the plane stress state and several orentatons of materal elastc axes are studed. The proportons of wdth to heght and crack length to wdth are equal to 0.5 (see Fg. 4). The plate s composed of a graphc-epoxy materal wth the followng orthotropc propertes: E ¼ 4:8 GPa; E ¼ :7 GPa; G ¼ 9:66 GPa; m ¼ 0: Local refnement around the crack tp s of nterest. 96 control ponts and 089 elements are used for modelng the problem n a global refned form usng NURBS, as llustrated n Fg. 5. By applyng the T-splne bass functons, 960 control ponts and 85 elements are utlzed for dscretzaton of the locally refned model (see Fg. 6). Note that the elements used for ntegraton are projecton of the unon of knot lnes and contnuty reducton lnes n the physcal space, whle the physcal mesh s defned by projecton of knot lnes n the physcal space (see [53] for more detals). The Fg. 8. Geometry and boundary condtons of the orthotropc dsk wth an nclned central crack. Control ponts Elements Fg. 7. The effect of varous nclnatons of elastc materal axes on the mxed mode SIFs: (a) normalzed mode I SIF ( K p Iffffffff), (b) normalzed mode II SIF ( K p IIffffffff). ro pa ro pa Fg. 9. Dscretzaton of the globally refned model of cracked orthotropc dsk.

7 44 S.Sh. Ghorash et al. / Computers and Structures 47 (05) Control ponts Physcal mesh Elements Fg. 0. Dscretzaton of the locally refned model of cracked orthotropc dsk. Table Stress ntensty factors for an nclned central crack n an orthotropc dsk subjected to pont loads (a ¼ 30 ). Method DOFs Elements Cells K I K II MCC [34] M-ntegral [34] XFEM [37] Enrched EFG [38] NURBS based XIGA T-splne based XIGA mnmum and maxmum szes of elements are ½w hš=7 and ½w hš=7 around the crack tp and far from t, respectvely. Cubc bass functons are adopted for the analyss. For comparng the obtaned results wth those avalable n the lterature, the Stress Intensty Factor (SIF) s calculated by the technque developed by Km and Paulno []. Effects of changng the materal elastc angle on mxed mode SIFs n the plate are probed. Comparson of the results of the proposed method and the results of the enrched element free Galerkn (EFG) [38], the extended fnte element method (XFEM) [37] and the boundary element method (BEM) [33], s shown n Fg. 7. It s observed that the results are n good agreement wth those obtaned by other methods. The results show that the trend of mode I SIF changes around b ¼ 45. It has an ncreasng trend n the span of b ¼ 0 to b ¼ 45 and then decreases n the span of b ¼ 45 to b ¼ 90 and reaches a value around ts ntal value,.e. when b ¼ 0. The turnng pont for the mode II SIF s about b ¼ 30. It should be emphaszed that the same level of accuracy s obtaned wth T-splne XIGA by much lower number of DOFS, control ponts and elements. 5.. Dsk wth an nclned central crack subjected to pont loads Consder an orthotropc dsk wth an nclned central crack subjected to double pont loads, as depcted n Fg. 8. The materal orthotropy axes and correspondng materal propertes are defned n Fg. 8. For dscretzaton purpose, the area around the crack s refned and both global (Fg. 9) and local (Fg. 0) refnements usng quadratc degree NURBS and T-splne bass functons are employed. Table compares the stress ntensty factors reported by Km and Paulno [34], Asadpoure and Mohammad [37] and Ghorash et al. [38] wth those obtaned usng the present approach for the case of a ¼ 30. XIGA s also used for analyss of dfferent nclnaton angles of the crack. The computed values of mxed mode SIFs alongsde those reported by Asadpoure and Mohammad [37] and Ghorash et al. [38] are llustrated n Fg.. The results of XIGA are n good agreement wth the others. Note that smlar to XFEM and EFG Fg.. SIF values correspondng to dfferent central crack angles n the orthotropc dsk: (a) mode I SIFs; (b) mode II SIFs. methods, XIGA s capable of analyzng several crack nclnaton problems on only one dscretzaton. 6. Concluson In ths contrbuton, the newly developed XIGA has been extended to analyss of cracked orthotropc plates. The recently proposed crack-tp orthotropc enrchment functons have been employed n the XIGA method to ncrease the approxmaton accuracy near the crack-tp. Furthermore, by adoptng the T-splne bass functons, the proposed approach allows for effcent local refnements. The proposed approach have been utlzed to analyze some numercal examples. Results of mxed-mode stress ntensty factors (SIFs) have been compared wth the reference results and proved the accuracy and effcency of the method. By adoptng an approprate error estmator n the developed scheme, whch s n progress by the authors, effcent adaptve procedures can now be followed.

8 S.Sh. Ghorash et al. / Computers and Structures 47 (05) Acknowledgement Ths research s manly supported by the German Research Insttute (DFG) va Research Tranng Group Evaluaton of Coupled Numercal Partal Models n Structural Engneerng (GRK 46), whch s gratefully acknowledged by the frst author. The second and fourth authors would lke to acknowledge the fnancal support of the Framework Program 7 Intal Tranng Network Fundng under Grant No Integratng Numercal Smulaton and Geometrc Desgn Technology. References [] Ghorash SSh, Valzadeh N, Mohammad S. Extended sogeometrc analyss (XIGA) for smulaton of statonary and propagatng cracks. Int J Numer Methods Eng 0;89: [] Km JH, Paulno GH. The nteracton ntegral for fracture of orthotropc functonally graded materals: evaluaton of stress ntensty factors. Int J Solds Struct 003;40: [3] Sh GC, Pars PC, Irwn GR. On cracks n rectlnearly ansotropc bodes. Int J Fract Mech 965;: [4] Muskelshvl NI. Some basc problems on the mathematcal theory of elastcty. 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