Mixed-mode delamination in 2D layered beam finite elements

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 215; 14: Publshed onlne 26 May 215 n Wley Onlne Lbrary (wleyonlnelbrary.com) Mxed-mode delamnaton n 2D layered beam fnte elements Leo Škec, Gordan Jelenć*, and Nkola Lustg Faculty of Cvl Engneerng, Unversty of Rjeka, Radmle Matejčć 3, 51 Rjeka, Croata Dedcated to the memory of our dear colleague and frend Nkola, who sadly passed away whle the paper was under revew. SUMMARY In ths work, a 2D fnte element (FE) formulaton for a mult-layer beam wth arbtrary number of layers wth nterconnecton that allows for mxed-mode delamnaton s presented. The layers are modelled as lnear beams, whle nterface elements wth embedded cohesve-zone model are used for the nterconnecton. Because the nterface elements are sandwched between beam FEs and attached to ther nodes, the only basc unknown functons of the system are two components of the dsplacement vector and a cross-sectonal rotaton per layer. Damage n the nterface s modelled va a b-lnear consttutve law for a sngle delamnaton mode and a mxed-mode damage evoluton law. Because n a numercal ntegraton procedure, the damage occurs only n dscrete ntegraton ponts (.e. not contnuously), the soluton procedure experences sharp snap backs n the force-dsplacements dagram. A modfed arc-length method s used to solve ths problem. The present model s verfed aganst commonly used models, whch use 2D plane-stran FEs for the bulk materal. Varous numercal examples show that the mult-layer beam model presented gves accurate results usng sgnfcantly less degrees of freedom n comparson wth standard models from the lterature. Copyrght 215 John Wley & Sons, Ltd. Receved 11 September 214; Revsed 14 Aprl 215; Accepted 15 Aprl 215 KEY WORDS: mult-layer beam; mxed-mode delamnaton; damage; fnte element analyss; modfed arc-length 1. INTRODUCTION Layered structures n many engneerng applcatons as well as n nature provde an extremely effectve means of optmsng functonal and structural performance of dverse mechancal systems. Partcularly, research and applcaton of layered composte structures n many areas of engneerng has been a topc of undmnshed nterest n the computatonal mechancs communty over the last few decades. Composte structures are made of two or more components from dfferent materals combned n such a way that each of them fulfls the functon for whch ts materal characterstcs are best suted. Due to ths optmsed performance of ther components, the composte systems are economcal and have a hgh load-bearng capacty. The mechancal behavour of these structures largely depends on the type of connecton between the layers, whch s usually not completely rgd and allows for nterlayer slp or/and uplft. Therefore, a partal nteracton has to be taken nto consderaton n the mechancal analyss of mult-layered structures. *Correspondence to: Gordan Jelenć, Faculty of Cvl Engneerng, Unversty of Rjeka, Radmle Matejčć 3, 51 Rjeka, Croata. E-mal: gordan.jelenc@unr.hr Deceased Copyrght 215 John Wley & Sons, Ltd.

2 768 L. ŠKEC, G. JELENIĆ AND N. LUSTIG Delamnaton s one of the most prevalent and severe falure modes n layered composte structures, dffcult to detect durng routne nspectons and one whch presents some of the bggest safety challenges that the aerospace ndustry has faced n the last decades [1]. When ntally proposed by Barenblatt [2], cohesve zone models (CZMs) provded a radcally new approach to the phenomenon of crack propagaton, fundamentally dfferent from that of Grffth [3] n that they allowed the fracturng process to be governed by the stress dstrbuted over a fnte regon around the crack tp, typcally named the process zone, rather than the stress concentrated at the crack tp. Ths model allowed the transfer of stresses over the crack provded t remaned suffcently narrow and could be justfed by a varety of physcal phenomena takng place n materals durng fracture [4]. Ever snce Hllerborg et al. [5] made ther frst fnte element (FE) mplementaton of the model, CZMs have contnued to generate much nterest wthn the computatonal mechancs research communty reflected by the mmense lterature n ths feld publshed n the last two decades (see e.g. [6 8] and the references theren). Obvously, t can be apprecated that to model complex layered structures numercally, along the lnes of the cohesve-zone theory, very sophstcated and computatonally ntense numercal procedures are needed, whch are often too computatonally expensve to be applcable as everyday desgn tools n engneerng practce. To brdge the gap between such expensve computatonal procedures and a desre of the structural analyst to have more effectve and engneer-orented desgn tools, n ths work, a fnte element formulaton for a mult-layer beam wth nterconnecton s presented. Here, the processes of crack occurrence and propagaton, damage-type materal softenng and eventual delamnaton are modelled usng beam-type FEs stemmng from Ressner s beam theory [9] to descrbe structural layers and nterface elements wth b-drectonal stffness [1]. Beam elements are more ntutve than sold elements, and n geometrcally lnear analyss, Ressner s theory corresponds to the wellknown Tmoshenko theory, whch forms a part of every engneerng educaton, and ther behavour s expected to be more famlar to the analyst. More mportantly, they make use of a smaller number of degrees of freedom eventually reducng the overall computatonal burden. Fnally, beam elements can be used wth very good accuracy for problems such as double cantlever beam (DCB) and peel tests [11], whch are wdely used to characterse fracture as dscussed earler. In spte of all these arguments, research n damage and delamnaton usng beam FEs has been rather scarce and, to the best of authors knowledge, has not addressed the mxed-mode delamnaton. In partcular, Sankar [12] proposed a geometrcally lnear lamnated shear deformable beam FE dvded nto two sublamnates connected by damage struts. Roche and Accors [13] developed a geometrcally lnear FE for lamnated beams based on smplfed knematc assumptons wth an addtonal nodal degree of freedom, whch s actvated when the element experences delamnaton. Ejo et al. [14] proposed a beam model for mode II delamnaton n geometrcally lnear lamnated beams assumng an sotropc non-lnear materal behavour and a pecewse lnear (zgzag) dsplacement functons to ntroduce the nterlayer slp nto the dsplacement feld. In the work of Kroflč et al. [15], geometrcally exact two-layer beam FE wth uncoupled non-lnear laws of nterlayer contact n both tangent and normal drectons s presented. In a more theoretcal ven, the ssues of damage and delamnaton n contnua subject to beam-lke knematc constrants have been nvestgated very recently by de Moras, who proposed an analytcal soluton for mode II [16] and mode I delamnaton [17] n geometrcally lnear beams wth blnear cohesve law and by Harvey and Wang who presented analytcal theores for the mxed-mode parttonng [18] of one-dmensonal delamnaton n lamnated composte beams wthn the context of both Euler and Tmoshenko beam theores [19]. In the present work, we show that modellng delamnaton usng beam FEs, rather than 2D plane stran FEs, s an alternatve that should be serously consdered.

3 MIXED-MODE DELAMINATION IN 2D LAYERED BEAM FINITE ELEMENTS PROBLEM DESCRIPTION 2.1. Poston of a layer of the composte beam n the materal co-ordnate system An ntally straght mult-layer beam of length L n whch the layers are allowed to move wth respect to one another dependng on the propertes of the nterconnecton s consdered. The beam s composed of n layers and n 1 nterconnectons. An arbtrary layer s denoted as, whle an arbtrary nterconnecton, placed between layers and C1, s denoted as (Fgure 1). The heght of an arbtrary layer s constant and denoted as h and the cross-sectonal area s denoted by A,where 2 Œ1; n. The dstance from the bottom of a layer to the layer s reference axs s denoted as a. Layers are made of lnear elastc materal wth E and G actng as Young s and shear modul of each layer s materal. Each layer has ts own materal co-ordnate system defned by an orthonormal trad of vectors E 1; ; E 2; ; E 3;, wth axes X 1; ;X 2; ;X 3;.AxesX 1; concde wth reference axes of each layer, whch must be parallel wth the layer s edges but otherwse may be chosen arbtrarly (they can pass through the correspondng layer but also fall outsde of t), that s, E 1 D E 1; and X 1 D X 1;. Note that ths freedom to choose the reference axes arbtrarly does not mply that the choce made wll not affect the fnal result n a numercal mplementaton. We wll get back to ths pont n the numercal secton. The cross-sectons of the layers have a common vertcal prncpal axs X 2 defned by a base vector E 2 D E 2; (a condton for a planar deformaton). However, for any chosen pont on the beam, the co-ordnate X ;2 s dfferent for each layer. AxesX 3; are mutually parallel but do not necessarly concde wth the horzontal prncpal axes of the layers cross-sectons, thus X 3 D X 3; and E 3 D E 3;. The frst and the second moment of area of the cross-secton A wth respect to axs X 3; are defned as Z Z S D X 2; da; I D.X 2; / 2 da: (1) A A The heght and the wdth of an nterconnecton are denoted as s and b, respectvely. Thus, t s assumed that the nterconnectons have rectangular cross-sectons, whle the layer s cross-sectons are arbtrary but wth a common prncpal axs X Poston of a layer of the composte beam n the spatal co-ordnate system The reference axes of all layers n the ntal undeformed state are defned by the unt vector t 1, whch closes an angle wth respect to the axs defned by the base vector e 1 of the spatal coordnate system (Fgure 2). The poston of a materal pont n the layer ;T.X 1 ;X 2; /, n the undeformed ntal confguraton s defned by the vector x ;.X 1 ;X 2; / D r ;.X 1 / C X 2; t 2 ; (2) Fgure 1. Poston of a segment of a mult-layer beam wth nterface n the materal co-ordnate system.

4 77 L. ŠKEC, G. JELENIĆ AND N. LUSTIG Fgure 2. Poston of a layer of the composte beam n undeformed and n deformed state (R.A. - reference axs). where r ;.X 1 / s the poston of the ntersecton of the plane of the cross-secton contanng the pont T and the reference axs of the layer n the undeformed state. Vector t j s defned as cos sn t j D ƒ e j D sn cos e j ; (3) where j D 1; 2. Durng the deformaton of the beam, the cross-sectons of the layers reman planar but not necessarly orthogonal to ther reference axes (Tmoshenko beam theory wth the Bernoull hypothess). The materal base vector E 3 remans orthogonal to the plane spanned by the spatal base e 1 ; e 2. Orentaton of the cross-secton of each layer n the deformed state s defned by the base vectors cos sn t ;j D cos sn e sn cos j D D sn cos. C / sn. C / sn. C / cos. C / cos e j D ƒ e j ; (4) where ndex denotes a layer and j D 1; 2. Rotaton of the cross-secton of each layer, denoted as, s entrely dependent on X 1, thus D.X 1 /. For the geometrcally lnear case (sn and cos 1), ƒ becomes ƒ D 1 1 ƒ : (5) The poston of a materal pont T n the deformed state from Fgure 2 can be expressed as x.x 1 ;X 2; / D r.x 1 / C X 2; t ;2.X 1 /; (6) where r.x 1 / s the poston of the ntersecton of the plane of the cross-secton contanng the pont T and the reference axs of layer n the deformed state. The dsplacement between the undeformed and the deformed state s defned for each layer wth respect to ts reference axs, thus r.x 1 / D r ;.X 1 / C u.x 1 /; (7) where u.x 1 / s the vector of dsplacement of the layer s reference axs.

5 MIXED-MODE DELAMINATION IN 2D LAYERED BEAM FINITE ELEMENTS GOVERNING EQUATIONS Governng equatons of the model consst of assembly equatons, whch defne how the layers and the nterconnectons are assembled nto a mult-layer beam and the knematc, consttutve and equlbrum relatons for the layers as well as for the nterconnecton. The dervaton of the governng equatons s explaned n detal n the followng sectons Assembly equatons Undeformed and deformed state of a segment of the mult-layer beam s shown n Fgure 3. The followng relatonshps between the dsplacements of the layers can be deduced from Fgure 3: u T; D u C1 C.t 2 t C1;2 /a C1 ; (8) u B; D u C.t ;2 t 2 /.h a /; (9) where u and u C1 denote the dsplacements of the reference axes of the layers lyng above and below the nterconnecton, whle u T; and u B; denote the dsplacements of the top and the bottom of the nterconnecton. Accordng to Fgure 3, vector (a drected stretched thckness of nterconnecton ) can be expressed usng (8) and (9) as D s t 2 C u T; u B; D (1) D u C1 u C a C1.t 2 t C1;2 / C.h a /.t 2 t ;2 / C s t 2 : 3.2. Governng equatons for layers Knematc equatons. Non-lnear knematc equatons accordng to Ressner s beam theory [9] read ² ³ D D ƒ T r E 1 D ƒ T t1 C u E 1 ; (11) D ; (12) where ; ; are the axal, shear and rotatonal stran (nfntesmal change of the cross-sectonal rotaton) at the reference axs of layer, respectvely. These quanttes are functons of only X 1 and the dfferentaton wth respect to X 1 s denoted as./. For the geometrcally lnear case (sn and cos 1) expresson (11) reduces to D ƒ T u t 2 ; (13) showng that, n ths case, Ressner s beam theory concdes wth Tmoshenko s. Fgure 3. Undeformed and deformed state of a mult-layer beam wth nterconnecton segment.

6 772 L. ŠKEC, G. JELENIĆ AND N. LUSTIG Consttutve equatons. The axal stran of a fbre at the dstance X 2; from the reference axs of the layer s computed as " D ".X 1 ;X 2; / D.X 1 / X 2;.X 1 /; (14) and, from here, the normal stress for a lnear elastc materal follows as D.X 1 ;X 2; / D E ".X 1 ;X 2; /; (15) where E s Young s modulus of the materal of layer. From (14) and (15), t s obvous that the dstrbuton of normal stresses over the layer s heght s lnear. The stress resultants read Z N D da; (16) A T D G k A ; (17) Z M D X 2; da; (18) A where N ;T ;M are the axal force, shear force and bendng moment wth respect to the reference axs of layer, respectvely. G s the shear modulus of materal of layer and k s the shear correcton coeffcent [2]. Combnng relatons (14 18), we fnally obtan 8 < : N T M 9 = ; D 2 4 E A E S G k A E S E I 3 8 < 5 : " 9 = ; ; or ² ³ ² ³ N D C M ; (19) where N T DhN T T ; T Dh" T ;S and I are the frst and the second moment of area of the cross-secton of layer and C s the consttutve matrx of layer Equlbrum equatons. Equlbrum equatons are derved from the prncple of vrtual work as: V L V nt V ext D Z L. N C M / dx 1 u./ F ;./W ; u.l/ F ;L.L/W ;L ; Z L u f C w dx 1 (2) where V L s the vrtual work of the layer composed of the vrtual work of nternal forces V nt and the vrtual work of external forces V ext actng on layer. and denote the vrtual strans, whle u and denote the vrtual dsplacements and rotatons, whch are all functons of X 1.The dstrbuted external forces and moments over the beam s length are denoted as f and w, whle the correspondng pont loads concentrated on the beam ends are denoted as F ;j and W ;j ;j D ; L. Accordng to (13) and (12), for a geometrcally lnear problem, the vrtual strans become ² ³ D and expresson (2) can be wrtten as " ƒ T T 1 d dx 1 t 2 T d dx 1 # ² ³! u D L.Dp /; (21)

7 MIXED-MODE DELAMINATION IN 2D LAYERED BEAM FINITE ELEMENTS 773 V L D Z L ² ³ ² ³ ² ³ ² ³.Dp / T L T N p M T f dx w 1 p T F./ ; p W T F.L/ ;L : (22) ; W ;L The resultng expresson s non-lnear n terms of the basc unknown functons (u and )and cannot be solved n a closed form. Followng the standard FE approach, the shape of vrtual (test) functons (u and ) s chosen n advance assumng that for a fnte number of nodes N; the vrtual dsplacements and rotatons are known at the nodes (u ;j and ;j ;j 2¹1; N º) and nterpolated between them as p : D NX j D1 ² ³ u;j j.x 1 / D ;j NX j.x 1 /p ;j ; (23) j D1 where j s the matrx of nterpolaton functons of dmensons 3 3. Further, 8 p D ı 1 I ı 2 I ::: ı n I ˆ< ˆ: p 1 p 2 : p n 9 >= : D ı 1 I ı 2 I ::: ı n I X N >; j D1 NX NX D ı1 j ı 2 j ::: ı n j pg;j D P ;j p G;j ; j D1 j D1 j 8 ˆ< ˆ: p 1;j p 2;j : p n;j 9 >= D >; (24) where p G;j D T p 1;j p 2;j :::; p n;j s the nodal global vector of vrtual unknown functons and ı j s the Kronecker delta defned as Now, expresson (22) becomes 8 NX < V L D p T G;j : j D1 Z L ı j D ² 1 f D j; otherwse: ² ³ ² ³ T DP;j L T N f dx M w 1 9 ² ³ ² ³ = P T F ;j./ ; P W T F N ;j.l/ ;L ; W ;L ; D X j D1 p T G;j gl ;j ; (25) (26) where g L ;j t the nodal vector of resdual forces for the layer, whch wll be later ntroduced to the global equlbrum equaton of the mult-layer beam wth nterconnecton Governng equatons for the nterconnecton Interface fnte elements by Alfano and Crsfeld [1] wth embedded CZM are adopted n the present mult-layer beam model. The nterface s a zero-thckness.s D / layer wth a non-lnear consttutve law allowng for delamnaton n modes I and II ncludng a mxed-mode delamnaton. Thus, dependng on the condtons on the nterface, the connecton between layers can be lnear-elastc, and after the softenng of the nterconnecton materal, a complete damage may occur Knematc equatons. For a zero-thckness nterconnecton, from Fgure 3, the vector of relatve dsplacements between the upper and the lower edge of the nterconnecton follows as D u T; u B;, from where the vector of the local relatve dsplacements s defned as

8 774 L. ŠKEC, G. JELENIĆ AND N. LUSTIG d D ² ³ d1; D ƒ D ƒ.u d T; u B; /; (27) 2; where d 1; and d 2; are relatve dsplacements of the nterconnecton n tangental and normal drecton, respectvely, whle ƒ s an orentaton that has to be defned based on the orentatons ƒ and ƒ C1. In a geometrcally lnear settng, ƒ D ƒ and d D ƒ.u T; u B; /, but n a geometrcally non-lnear settng, ƒ depends on the defnton of the normal and tangental drectons (wth correspondng delamnaton modes I and II), whch s non-unque due to large dsplacements and rotatons. Followng the dea n [15], these drectons may be defned by an algorthmc orentaton Q taken to be somewhere between C and C C1,thats, Q D. C / C.1 /. C C1 / D C C.1 / C1 ; (28) where represents the weght wth a value between and 1, gvng ƒ D cos Q sn Q sn Q cos Q : (29) By selectng D :5, the mean angle between layers and C 1 s obtaned as Q D C. C C1 /=2. In the present work, the emphass wll be placed on mplementaton of the materally nonlnear damage consttutve law enablng delamnaton, and the prevously mentoned geometrcally non-lnear aspects of the formulaton wll not be mplemented Consttutve equatons. A cohesve-zone model, embedded n the nterface FEs by Alfano and Crsfeld [1], s used n the present work and shown n Fgure 4 for an arbtrary nterconnecton. Accordng to (27), ndex 1 s assocated wth tangental delamnaton (mode II), whle for normal delamnaton (mode I), ndex 2 s used. The current state of damage s expressed usng a parameter that combnes delamnaton n both modes as jd1;. ˇ. /j hd2;. / 1 / D C 1; (3) d 1; d 2; where d 1; and d 2; are the relatve dsplacements at the nterconnecton correspondng to the start of the softenng process n modes II and I (drectons 1 and 2), respectvely, s the pseudo-tme varable and h s the McCauley bracket [1]. In the present, work D 2 s used. Expresson (3) determnes the current state of delamnaton for sngle-mode (d 1; or d 2; equals zero) as well as for the coupled, mxed-mode delamnaton, where the overall damage at the nterconnecton s affected by both modes. Damage of the nterconnecton s rreversble; thus, for a pseudo-tme parameter, the maxmum rate of delamnaton n the pseudo-tme hstory s expressed as Fgure 4. Consttutve law for the nterconnecton: (a) mode II (drecton 1) and (b) mode I (drecton 2).

9 MIXED-MODE DELAMINATION IN 2D LAYERED BEAM FINITE ELEMENTS 775 ˇ./ D max 6 6 ˇ. /: (31) The tractons at the nterconnecton! D T! ;1! ;2 are calculated accordng to the followng consttutve law: ² S! D d f ˇ 6 ; (32) ŒI G S d f ˇ >; where S D S1; ; S S ; D! ; ; 2; d ; μ d c; ˇ g ; D mn 1; d c; d ; 1 C ˇ G D D 1; 2: g1; ; hsgn.d 2; /g 2; (33) The constant! ; represents the contact tracton at the nterconnecton correspondng to the start of the softenng process n drecton, whle d co; represents the relatve dsplacement correspondng to the total damage of the nterconnecton n drecton. The case ˇ 6 corresponds to the lnear-elastc behavour of the nterconnecton, whle ˇ >ndcates the ongong delamnaton and damage process at the nterconnecton. Parameter g ; 2h; 1 ndcates the degree of the damage, where g ; D 1 means that total damage of the nterconnecton has occurred and the connecton between layers s completely lost (total delamnaton -! D ) Equlbrum equatons. Equlbrum equatons for the nterconnecton are agan derved from the prncple of vrtual work as V C D b Z L d! dx 1 ; (34) where V C denotes the vrtual work of nternal forces of the nterconnecton. Accordng to (27), vrtual relatve dsplacements of the nterconnecton n a geometrcally lnear settng become d D ƒ u C1 u C C1 a C1 t 1 C.h a /t 1 D ƒ B p C; ; (35) where B D It 1.h a / It 1 a C1 ; pc; D ² p p C1 ³ ; D : (36) Note that for a geometrcally non-lnear case, expresson (35) changes because ƒ n (27) s not constant but depends on and C1. Expresson (34) can be now wrtten as V C D b Z L p T C; BT ƒt! dx 1 : (37) Because expresson (37) s non-lnear n terms of the basc unknown functon, the vrtual functons are nterpolated and accordng to (24), the followng expresson s obtaned p C; D ² p p C1 ³ : D whch transforms expresson (37) nto NX j D1 P ;j p P G;j D C1;j NX R ;j p G;j ; (38) j D1

10 776 L. ŠKEC, G. JELENIĆ AND N. LUSTIG V C D NX j D1 p T G;j b Z L R T j; BT ƒt! dx 1 D NX j D1 p T G;j gc ;j ; (39) where g C ;j s the nodal vector of resdual forces for the nterconnecton, whch wll be later ntroduced to the global equlbrum equaton of the mult-layer beam wth nterconnecton. 4. SOLUTION PROCEDURE To solve the system of governng equatons for the mult-layer beam wth nterconnecton, the vector of resdual forces and the tangent stffness matrx have to be determned. The problem s then solved numercally, whch s explaned n detal n the followng sectons Vector of resdual forces and tangent stffness matrx Total vrtual work for the mult-layer beam analysed s composed by the vrtual work of n layers (26) and the vrtual work of n 1 nterconnectons (39), and t can be wrtten as V TOT D nx D1 V L C.1 ı n /V C D N X nx p T G;j j D1 D1 g L ;j C.1 ı n /g C ;j : (4) Note that the same counter s used for the n layers and the n 1 nterconnectons, where D and ı n s defned n (25). Because the total vrtual work for the mult-layer beam must equal zero.v TOT D / and the choce of the test parameters p G;j s arbtrary, t follows that g j D nx D1 g L ;j C.1 ı n /g C ;j D q nt j qext j D ; (41) where g j s the nodal vector of resdual forces for the mult-layer beam, whch s composed of the nodal vector of nternal forces q nt j and the nodal vector of external forces qext j defned as q nt j q ext j D D D1 D1 Z L Z L P T ;j ² ³.DP ;j / T L T N dx M 1 C.1 ı n /b Z L 1.ƒ B R ;j / T! dx 1 A ; 1 ³ A : W ;L ² ³ ² ³ ² f dx w 1 C P T F ;j./ ; C P W T F ;j.l/ ;L ; By lnearsng the nodal vector of resdual forces (41), the nodal tangent stffness matrx can be obtaned, and, because the unknown functon are contaned only n the nodal vector of nternal forces g j D q nt j. The unknown functons are contaned n N and M, whch are lnearsed as ² ³ ² ³ ² ³ N ƒ T D C M D C u t 2 D C L.D p /; (43) where L and D are gven n (21) and p D u T and n the vector of contact tractons!, whch s lnearsed as! D U d D U ƒ B N X kd1 (42) R ;k p G;k ; (44)

11 MIXED-MODE DELAMINATION IN 2D LAYERED BEAM FINITE ELEMENTS 777 where 8 < S f ˇ 6 ; U D.I G : / S f ˇ >and ˇ < ˇ; I G J d v T S f ˇ >and ˇ D ˇ; (45) p G;k D p 1;k p 2;k ::: p n;k T ; (46) J D 1 v T D 1; hsgn.d 2; / 2; ; j; D d 1; jd1; j 1 d 1; d 2; hd2; d 2; d cj; The lnearsed nodal vector of resdual forces fnally becomes d cj; d j; sgn.1 g j; /.1 C ˇ/ C1 ;jd 1; 2; (47) : (48) where NX g j D K j;k p G;k ; (49) kd1 K j;k D nx Z L D1 H T ;j C H ;k C.1 ı n /b T ;j U ;k dx1 (5) s nodal tangent stffness matrx wth H ;l D L DP ;l and ;l D ƒ B R ;l. The global vector of resdual forces g D¹g j º accordng to (41) and (42), global tangent stffness matrx K D K j;k accordng to (5) and global vector of ncrements of the unknown functons p D¹ p G;k º are assembled usng the standard FE assembly procedure [21] to solve the followng system: p D K 1 g: (51) For ntegraton n (42) and (5), Gauss quadrature wth N 1 ntegraton ponts s used for the beam parts (layers), and Smpson s rule wth N C1 ntegraton ponts s used for the nterconnecton parts. It has been shown n [22] that, for lnear elements, the applcaton of Gauss quadrature results n couplng between the degrees of freedom of dfferent node sets, whch causes oscllaton of the tracton profle for hgh values of the tracton gradents. Wth a Newton Cotes ntegraton rule (lke Smpson s rule), ths effect dsappears Soluton algorthm The algorthm presented has been mplemented wthn the computer package Wolfram Mathematca. Because the governng equatons of the problem are non-lnear n terms of the basc unknown functons, the soluton of the problem s obtaned teratvely usng the Newton Raphson soluton procedure. For each FE and each nterconnecton, the relatve dsplacements are calculated, and then at each ntegraton pont of the nterconnecton, the current stage of delamnaton s determned (lnear-elastc behavour, softenng, unloadng and reloadng wth a reduced stffness or total damage). The total loss of adherence at an ntegraton pont wll lead to very sharp snap-backs n the load-dsplacement dagram [1, 23], whch s a behavour that cannot be captured nether wth standard load-control or dsplacement-control methods n the Newton Rapshon soluton procedure nor wth the standard arc-length procedure [24]. In the present work, the modfed arc-length method proposed by Hellweg and Crsfeld [25] s used. The reasonng behnd ths dea, ts relatonshp to the standard cylndrcal arc-length method [24], and a detaled exposton of the algorthm are gven n [26].

12 778 L. ŠKEC, G. JELENIĆ AND N. LUSTIG 5. NUMERICAL EXAMPLES In ths secton, the presented model s tested for mode I, mode II and mxed-mode delamnaton Mode I delamnaton example Ths example s the standard test for mode I delamnaton known as the DCB test. The specmen [1] s shown n Fgure 5 wth the correspondng geometrcal propertes (the wdth of the beam s 2 mm), boundary condtons and the loadng, whch cause the notch to open vertcally and then propagate to the left-hand sde of the beam as the nterconnecton (the regon between the clamped end and the begnnng of the notch) delamnates n pure mode I. In the orgnal example [1], orthotropc materal data are gven, wth two Young s modul, one shear modulus and two Posson s coeffcents, whch for the beam consttutve model (19) s reduced only to E D 135:3 GPa, G D 5:2 GPa, (the beam s modelled as two-layered,.e. D 1; 2). For the nterconnecton (ndex s omtted because there s only one nterconnecton) crtcal energy release rate G cj D :28 N/mm, d j D 1 7 mm and! j D 57 MPa, j D 1; 2. From these values, the penalty stffness parameters are computed accordng to (33) as S j D! j =d j D 5:7 1 8 N/mm 3, whle the relatve dsplacement at whch complete delamnaton occurs s d cj D 2G cj =t j D 9: mm. The notch s modelled as a zero-strength nterconnecton. The beam and the nterconnecton are dvded n FEs of equal length. Thus, for 1 beam FEs, 7 nterconnecton FEs wth the above mentoned propertes are embedded, and ths number has been doubled three tmes untl the fnest mesh of 8 beam FEs and 56 nterconnecton FEs Fgure 5. Test specmen for the double cantlever beam test. Fgure 6. Double cantlever beam test results for varous fnte element meshes.

13 MIXED-MODE DELAMINATION IN 2D LAYERED BEAM FINITE ELEMENTS 779 has been obtaned. Lnear two-noded mult-layer beam FEs are used n ths example. A sngle Gauss pont s used for the ntegraton of the beam bulk, whle the three-pont Smpson s rule s used for the ntegraton of the nterconnecton. The results of the analyss for varous FE meshes are presented n Fgure 6, where the relaton between the appled force and the vertcal dsplacement at the free end has been shown. The total number of degrees of freedom s 66 for the 1-element mesh, 126 for the 2-element mesh, 246 for the 4-element mesh and 486 for the 8-element mesh. The lner-elastc part of behavour of the system can be clearly observed n Fgure 6. However, sgnfcantly before the peak of the dagram, a certan amount of softenng occurs, ndcatng that the damage process has already started, whch can be dstngushed graphcally n Fgure 6 as a devaton of the solutons usng nterface elements wth respect to the analytcal soluton where ths softenng does not occur (see [26] for detals). The peak s reached when the nterconnecton at the frst ntegraton pont (the one nearest to the notch) s completely lost (total damage). As the crack propagates from the notch to the clamped end, a decrease n overall stffness of the system can be observed, whch after a specfc pont (F 22 Nandv.L/ 9 mm), stablses meanng that the nterconnecton s almost completely damaged, and the stffness of the system approxmately equals the stffness of the completely separated beam layers, whch s gven by the analytcal soluton (see [26] for detals). In Fgure 6, t can be clearly notced that for a 1 mult-layer beam elements (wth 7 nterconnectons), the number of sharp snap backs corresponds to the number of completely damaged nterconnectons wthn elements, whch s 67 (not all the nterconnectons are completely damaged at the end of the test). The same behavour can be notced also for the fner meshes, but n these cases, the ampltudes of the nstabltes become reduced wth an ncrease n number of the FEs as the nodes and ntegraton ponts are gettng closer. A closer look at the peak of the dagram s shown n Fgure 7, where t can be observed that for all meshes, a certan amount of oscllatons around the exact soluton s obtaned, but the phenomenon s reduced by ncreasng the number of ntegraton ponts through an ncreased number of FEs. Ths phenomenon s mentoned earler and s caused by the dscretsaton n numercal ntegraton, whch s obvously mesh dependent. The senstvty of the results wth respect to the mesh used dscussed n [1] s, therefore, also vsble n the present model. Comparng the results from Fgures 6 and 7 to those from Fgure 14, n [1], t shows that the present 2-element results are comparable to the 4 2-element results n [1], whle the reference results n [1] usng a rectangular mesh of 4 4 quadrlateral eght-node plane-stran (Q8) elements and 28 sx-node nterface elements (INT6) results n 12, 82 degrees of freedom, whch gves accuracy somewhere between those of the present 4-element and 8-element meshes. Fgure 7. Mesh dependence of the double cantlever beam test results for varous fnte element meshes.

14 78 L. ŠKEC, G. JELENIĆ AND N. LUSTIG Table I. Total number of degrees of freedom for a mult-layer beam model and a 2D Q4 plane-stran element model for varous meshes. No. of rows/columns of elements (mult-layer beam) (Q4) (Q4) (Q4) , 2 Usng sgnfcantly less degrees of freedom, the beam model thus gves satsfactory accuracy even for such a demandng value of the penalty-stffness parameter (see [1] for a detaled senstvty analyss of the orgnal plane-stran formulaton). Fgure 6 shows that both the reference soluton [1] and the mult-layer beam solutons usng fner meshes show very good agreement wth the analytcal soluton (see [26] for more detals). Usng the FE formulaton proposed n [1], the results of ths DCB test are also presented n [27] usng dfferent meshes wth quadrlateral four-node and eght-node plane-stran elements n conjuncton wth four-node and sx-node nterface elements (Q4-INT4 and Q8-INT6). An addtonal comparson s shown n Fgure 7, where t can be clearly notced that a mesh wth 8 8 Q4 elements n conjuncton wth 56 INT4 elements (resultng n 16, 2 degrees of freedom) [27] gves the results that are better than the results of the mult-layer beam model wth 4 elements (246 degrees of freedom) but worse than the results of the mult-layer beam model wth 8 elements (486 degrees of freedom). At ths pont, t can be concluded that the mult-layer beam model n ths example gves better results than the plane-stran 2D FE formulaton usng sgnfcantly less degrees of freedom. The number of the degrees of freedom of the mult-layer beam model and the 2D Q4 plane-stran element model for dfferent meshes s shown n Table I where the effectveness of the mult-layer beam model regardng the total number of the degrees of freedom can be clearly seen. It can be also mentoned that, optonally, each arm of the DCB test may be modelled usng the mult-layer beam model wth rgd nterfaces (presented n [28]), whch s capable of capturng the cross-sectonal warpng effect wth less degrees of freedom than the 2D plane-stress FEs Mode II delamnaton example The example presented next was proposed by M et al. [29] and ts geometrcal propertes are shown n Fgure 8, wth wdth of the beam b D 1 mm ( D 1; 2, beam s agan modelled as twolayered) and a varable notch length a. Materal propertes of the beam are E D 135:3 N/mm 2, G D 54:12 N/m 2, whle for the nterconnecton G cj D 4: N/mm,! j D 57 N/mm 2 and d j D 1 7 mm, j D 1; 2. In the same manner as n the prevous numercal example, d cj D :14 mm and S j D 5:7 1 8 N/mm 3 are obtaned. The force F causes the two layers of the beam to slp aganst each other causng the pure mode II delamnaton at the nterconnecton. Obvously, penetraton between the layers along the notch must be suppressed, whch s n the mxed-mode nterface FEs performed by provdng zero stffness n mode II and a hgh penalty stffness S 2 n mode I. The results presented n Fgure 9 show the load mdspan deflecton relatonshp for a D 15 mm and a D 3 mm. Two-noded mult-layer beam FEs are used. Mesh dependence n ths example s less pronounced than n the mode I example, and t can be observed that even for rather coarse meshes the results are very close to the converged Fgure 8. Test specmen for the mode II delamnaton test.

15 MIXED-MODE DELAMINATION IN 2D LAYERED BEAM FINITE ELEMENTS 781 Fgure 9. Load mdspan deflecton dagram for the mode II delamnaton test. FEA, fnte element analyss. ones. The reason for ths les n the fact that n mode II delamnaton, mesh-dependent numercal oscllatons occur n the longtudnal drecton and can be thus hard to dstngush n Fgure 9, where the vertcal dsplacement at the mdspan s plotted. Smlar behavour as n the prevous example can be noted n the lnear elastc range, wth a decrease n overall stffness after the peak load has been reached and subsequent hardenng eventually leadng to a lnear-elastc behavour wth a completely damaged nterconnecton. The results presented n ths work show a very good accordance wth the results presented n [29], whch are obtaned usng the 2D plane-stran FEs for the bulk materal. It can be also notced that the results obtaned usng the mult-layer beam model, compared wth the numercal results presented n [29], naturally show better agreement (almost concde) wth the analytcal results, whch are obtaned usng the prncples of lnear-elastc damage mechancs n conjuncton wth the lnear-elastc Bernoull beam theory. These results are obtaned for the stuatons n whch the crack propagates from the left-hand end of the beam to the beam centre (a <L) and further on to the rght hand end of the beam (a >L)(for more detal see [29]). Because n [29], the mesh data for the fnte element analyss soluton has not been provded, the comparson between the total number of degrees of freedom for the beam and the plane FE model s not presented Mxed-mode delamnaton examples In ths secton, the presented mult-layer beam model s tested on two mxed-mode delamnaton examples. In the frst example, mxed-mode delamnaton occurs along a sngle nterconnecton, whle n the second example, the delamnaton occurs along two nterconnectons smultaneously Sngle-nterconnecton mxed-mode delamnaton. Ths example, proposed by M et al. [29], too, s very smlar to the mode II delamnaton example and s shown n Fgure 1. Geometrcal propertes, as well as the materal propertes for the nterconnecton, are the same as n the mode II delamnaton example, except that the results are now gven only for a D 3 mm. The materal propertes for the bulk materal n [29], however, are gven as for the orthotropc materal, wth two Young s modul, one shear modulus and two Posson s coeffcents, whle n the present model, only the Young s modulus n the longtudnal drecton and the shear modulus n the correspondng transverse drecton are used as E D 1353 N/m 2 and G D 52 N/mm 2, D 1; 2: In ths example, two forces F 1 D :4535F 2 and F 2 are appled to the system. The force F 2,asnthe prevous example, causes a mode II delamnaton at the nterconnecton, whle the force F 1 causes a

16 782 L. ŠKEC, G. JELENIĆ AND N. LUSTIG Fgure 1. Test specmen for the mxed-mode delamnaton test. Fgure 11. Load vertcal dsplacement at the left-hand sde dagram for the mxed-mode delamnaton test. FEA, fnte element analyss. pure mode I delamnaton. When both forces are actng on the system, the mxed-mode delamnaton at the nterconnecton takes place. The results of the test are plotted n Fgure 11, showng the relatonshp between the load F 1 and the vertcal dsplacement at the left-hand sde of the beam usng two-noded mult-layer beam FEs. Smlar behavour as n the two prevous examples can be observed, consderng the shape of the dagram and the meshng nfluence on the results (nstabltes decrease wth an ncrease n number of the FEs). The results of the present mult-layer beam model agree very well wth the numercal results from [29], where 2D plane-stran FEs and two damage crtera (lnear D 2 and ellptcal D 4) for the mxed-mode delamnaton parameter (3) were used. In the present mult-layer beam model, only the lnear crteron has been used, and consderng that the numercal results n [29] are obtaned for an orthotropc materal model, the agreement of the results s more than satsfactory. The analytcal results for delamnaton [29] also show excellent agreement wth the numercal results. Note that the vertcal dsplacement v./ D 4 mm presented n Fgure 11, wth respect to the length of the specmen 2L D 1 mm, sgnfcantly exceeds the lmt of small dsplacements and rotatons. Thus, a geometrcally non-lnear model would be more approprate n ths example, whch would probably produce results dfferent form those presented n Fgure Dual-nterconnecton mxed-mode delamnaton. Based on an expermental nvestgaton of a mult-layered specmen manufactured from HTA913 carbon epoxy materal, numercal models are presented by Robnson et al. [3] and later by Alfano and Crsfeld [1]. The specmen s supported only at the bottom of the left-hand sde wth a vertcal force F actng at the top of the left-hand sde causng a crack to propagate frst as a contnuaton of the upper ntal crack only, but

17 MIXED-MODE DELAMINATION IN 2D LAYERED BEAM FINITE ELEMENTS 783 later along both ntal cracks smultaneously (Fgure 12). The HTA913 specmen s made of 18 layers of equal thckness, where the ntal cracks are located between the 1th and the 11th layer and between the 12th and the 13th layer (counted from the bottom). Because t s assumed that all the other nterlayer connectons are rgd, a mult-layer beam model s specfed as havng three layers and two nterconnectons. The geometry of the specmen s shown n Fgure 12, where a D 2 mm, H D 3:18 mm and wdth B D 2 mm. Orthotropc materal propertes for HTA913 gven n [3] are adapted for the beam model as E D 115: GPa, G D 4:5 GPa,. D 1; 2; 3/, whereas for the nterconnecton, accordng to [1], materal propertes G c1; D :8 N/mm, G c2; D :33 N/mm,! 1; D 7: MPa,! 2; D 3:3 MPa and d j; =d cj; D 1 6,where D 1; 2 and j D 1; 2 are used. It can be further derved that d c1; D 2G c1; =! 1; D :23 mm, d c2; D 2G c2; =! 2; D :2 mm, d 1; D 2:3 1 7 mm, d 2; D mm, S 1; D t 1; =! 1; D 3:6 1 7 N/mm 3 and S 2; D t 2; =! 2; D 1: N/mm 3. In Fgure 13, a comparson between the mult-layer beam model and the model proposed by Alfano and Crsfeld [1] s shown. The vertcal dsplacement of the top layer at the left-hand sde of the beam s plotted aganst the appled load F. Alfano and Crsfeld have used a mesh wth 3 36 quadrlateral mxed four-node plane-stran elements wth enhanced strans (QM4) and 68 INT4 elements wth a total of 4322 degrees of freedom and the tangent predctor for the soluton (see [1] for more detals). On the other hand, for the mult-layer beam model, two meshes wth two-node beam FEs and INT4 nterface elements are adopted. A 3-pont Newton Cotes ntegraton s used for the INT4 nterface elements. The frst mesh wth 18 FEs uses 1629 degrees of freedom, whle the second mesh wth 36 FEs uses 3249 degrees of freedom. It can be noted that the results obtaned usng the mult-layer beam model show excellent agreement wth the converged soluton usng 2D othotropc plane-stran FEs and use sgnfcantly less degrees of freedom. Fgure 12. Specmen for the double mxed-mode delamnaton. Fgure 13. Comparson of the results for the double mxed-mode delamnaton example. FE, fnte element.

18 784 L. ŠKEC, G. JELENIĆ AND N. LUSTIG The results shown n Fgure 13 are obtaned usng the model where reference axes of all three layer concde wth ther centrodal axes (a D h =2, D 1; 2; 3). However, changng the poston of reference axes of the layers has an nfluence on the results, whch s shown n Fgure 14. The frst poston s as mentoned, whle n the second poston, the reference axs of the mddle layer s postoned n the mddle of the beam (.e. on top of ths layer, a 2 D h 2 ), and the two remanng reference axes are postoned symmetrcally wth respect to the mddle axs wth a 1 D a 3 D H=4. The thrd poston s the same as the second, only that the reference axs of the second layer now concdes wth the reference axs of the thrd layer and a 2 D.h 2 h 1 /=2. Other postons, such as a poston where all reference axes lay on the centrodal lne, cause numercal problems, and no acceptable solutons can be obtaned. At ths pont of nvestgaton, we only recommend to keep the reference axes of the layers always n lne wth ther centrodal axes n order to obtan the desred results. However, more nvestgaton on the nfluence of the reference axes poston on the results s planned n order to propose an optmal poston of the layers reference axes. In the work whch follows Reference [1], Alfano and Crsfeld have examned the nfluence of varaton of some nterface materal parameters, whle keepng the energy release rate constant, on the results of the same double mxed-mode delamnaton problem (see [23] for detals). Three sets of materal propertes used n ths comparson are presented n Table II. It can be observed that all of the three sets have dentcal energy release rates G cj; and penalty stffness parameters S j;, where j D 1; 2, but other quanttes relevant for the nterconnecton sgnfcantly dffer. Ths means that the area under the relatve dsplacement-contact tracton dagram, as well as the slope of the lnearelastc branch remans the same n all three cases, but the peak contact tracton! j; and maxmum relatve dsplacement d cj; are dfferent. In ths example, two dfferent non-unform fnte element meshes are used. The horzontal doman s dvded n fve zones. The frst zone, accordng to Fgure 12, s near the left-hand sde of the beam wth the ntal crack along the upper nterface. D 2/, whle the second zone covers the Fgure 14. Influence of the poston of the layers reference axes on the results. Table II. Three sets of materal propertes for the nterconnecton for the double mxed-mode delamnaton example. Case G c1; G c1; d j; =d cj;! 1;! 2; d j; d cj; S 1; S 2; [N/mm] [N/mm] [MPa] [MPa] [mm] [mm] [N/mm 3 ] [N/mm 3 ] A B C

19 MIXED-MODE DELAMINATION IN 2D LAYERED BEAM FINITE ELEMENTS 785 Table III. Two fnte element meshes for the double mxed-mode delamnaton example wth dfferent materal propertes for the nterconnecton. Zone 1 Zone 2 Zone 3 Zone 4 Zone 5 Total Length [mm] Intal crack D 2 None D 1 None None Mesh 1/No. of FE Mesh 2/No. of FE FE, fnte element. ntally uncracked part of the beam between the two ntal cracks, and the thrd zone covers the regon of the second ntal crack, whch has been placed along the lower nterface. D 1/. The ntally undamaged rght-hand part of the beam s dvded n two zones, zone 4, where delamnaton s expected, and zone 5, where no delamnaton s expected. In contrast to the prevous example (Fgure 12), ths part of the beam n ths example s 7 mm long. However, ths dfference does not nfluence the results as long as the cracks reman suffcently dstant from the rght-hand end of the specmen. The lengths of the meshng zones, along wth the number of FEs for both meshes, are gven n Table III. In ths example, n contrast to the one prevously presented, nterface elements wth no contact tractons, except n compresson n mode I, are used at the notches. Ths feature gves more realstc behavour of the model (penetraton between layers at the notches s almost completely prevented) so the second peak n Fgure 13 (v 3./ 9 mm and F 35 N), whch s obvously a non-physcal artefact of the model dsappears. Compresson tractons n ths example can occur only at the bottom notch n whch they grow as the upper crack approaches t, but as soon as the upper crack reaches the horzontal poston of the start of the bottom notch, delamnaton and crack propagaton n the bottom notch start, too. After that, the upper crack propagaton s slowed down, whle the bottom crack propagates more rapdly. In [23], no nterface elements are nserted between the mddle and the bottom layer n meshng zones 1 and 2, that s, the left-hand sde of the bottom nterconnecton s assumed to be absolutely rgd, whch s justfed, because the results show that the bottom crack propagates only to the rght. In the present work, nterface elements wth 1 tmes hgher values of penalty stffness parameters are used at the left-hand sde of the bottom nterface. In Fgure 15, a comparson between the mult-layer beam model and the model proposed by Alfano and Crsfeld [23], whch uses 2D plane-stran FEs wth nterface elements, s shown. As n [23], mesh 1 s used for case A, whle mesh 2 s used for cases B and C accordng to Table III. Three-node beam FEs and INT6 are used n the mult-layer beam model. The 3-pont Newton Cotes ntegraton s used for the nterface elements. Usng 97 quadrlateral plane-stran eght-node (QPSN8) fnte elements per layer wth INT6 nterface elements at the nterconnectons, as dd Alfano and Crsfeld [23], results n 2938 degrees of freedom for case A, whle usng 189 QPSN8 fnte elements per layer wth INT6 nterface elements at the nterconnectons results n 551 degrees of freedom for cases B and C. On the other hand, usng the presented mult-layer beam model accordng to Table III gves 1755 degrees of freedom for mesh 1 (case A) and 3411 degrees of freedom for mesh 2 (cases B and C). Even though the total number of degrees of freedom s consderably reduced, the mult-layer beam model stll gves results that, for the most crtcal part of the load-dsplacement dagram, show an excellent agreement wth the results presented by Alfano and Crsfeld [23] (Fgure 15). It can be also noted that cases B and C show better agreement wth the expermental results [3]. Oscllatons, typcal for fnte element analyss of delamnaton problems, however, can stll be noted n the dagram, especally n the fnal softenng branch. Usng mesh 1 for case A, a very small amount of oscllatons occurs, whle usng a fner mesh 2 for the cases B and especally C, oscllatons become more pronounced. Thus, t can be concluded that the nfluence of materal parameters of the nterconnecton has even bgger nfluence on the oscllatons of the results than the meshng tself. Along the fnal softenng branch of the dagrams shown n Fgure 15, delamnaton occurs n both nterconnectons smultaneously. It s notced that oscllatons have dfferent ampltudes.

20 786 L. ŠKEC, G. JELENIĆ AND N. LUSTIG Fgure 15. Results for dfferent cases of materal propertes of the nterconnecton: (a) plane-stran fnte element model from lterature and (b) mult-layer beam model. Smaller ampltudes correspond to a total damage at an ntegraton pont of one nterconnecton, whle the larger oscllatons occur when the total damage s reached at ntegraton ponts n both layers n the same load step. Compared wth the plane-stran FE formulaton [23], the present model shows larger oscllatons, whch can be reduced usng numercal ntegraton of hgher order (more ntegraton ponts). In Fgure 16, a comparson of the results obtaned usng dfferent order of Newton Cotes numercal ntegraton s shown for the last part of the dagram. It can be noted that usng 4 ntegraton ponts nstead of 3, the oscllatons reduce consderably and the results are much closer to those proposed by Alfano and Crsfeld [23]. On the other hand, further ncrease n number of ntegraton ponts does not lead to any sgnfcant mprovement of the results. It has to be emphassed that results n Fgure 16 correspond to the FE mesh wth 2 elements (nstead of 4) n the second zone accordng to Table III, whch, however, does not have any sgnfcant nfluence n ths part of dagram. It seems obvous that further ncrease n the number of ntegraton ponts s unlkely to lead to any sgnfcant reducton of oscllatons. Note, also, that ncreasng the number of ntegraton ponts further s not recommended as the soluton procedure s then expected to requre smaller ncrements [1].

21 MIXED-MODE DELAMINATION IN 2D LAYERED BEAM FINITE ELEMENTS 787 Fgure 16. Comparson of the results for dfferent order of numercal ntegraton. 6. CONCLUSIONS AND FUTURE WORK In the present work, a mult-layer beam wth nterconnecton allowng for delamnaton between layers has been presented. The bulk materal s modelled usng beam FEs, whle the cohesve-zone model ncorporated nto the nterface elements proposed by Alfano and Crsfeld [1] s used for the nterconnecton. Modellng the bulk materal structure as beams, n comparson wth commonly used 2D plane-stran fnte elements, gves comparable accuracy usng a reduced total number of degrees of freedom. The results of the numercal examples presented agree very well wth the results form the lterature, whch use 2D FEs for the bulk materal. It was notced that the mesh-dependent oscllatons caused by the numercal ntegraton reduce wth the mesh refnement or addton of the ntegraton ponts. In case of slender layers, the beam elements have an advantage over the sold elements because of the lmt n the aspect rato of sold elements. Wth an ncrease n the beam elements lengths, however, t has been shown that the spurous mesh-dependent oscllatons also ncrease, that s, for ths reason beam elements may stll need to be lmted n length. In future, we plan to extend ths model to a geometrcally non-lnear analyss. The man ssue, whch n ths case remans to be resolved, s how to defne the matrx ƒ from (27) approprately. Such a model would allow to analyse problems wth large deformatons and rotatons, such as the peel test. ACKNOWLEDGEMENTS These results were obtaned wthn the research project No IP (confguraton-dependent approxmaton n non-lnear fnte-element analyss of structures) fnancally supported by the Croatan Scence Foundaton. We also acknowledge the Unversty of Rjeka s fnancal support for the ongong research No (testng of slender spatal beam structures wth emphass on model valdaton). REFERENCES 1. Daves GAO, Htchngs D, Ankersen J. Predctng delamnaton and debondng n modern aerospace composte structures 26; 66(6): Barenblatt GI. The formaton of equlbrum cracks durng brttle fracture - general deas and hypothess, axally symmetrc cracks. Journal of Appled Mathematcs and Mechancs 1959; 23(3): Grffth AA. The phenomena of rupture and flow n solds. Phlosophcal Transactons of the Royal Socety of London 1921; A 221: Bažant Z, Cedoln L. Stablty of Structures. Dover: New York, USA, Hllerborg A, Modéer M, Petersson PE. Analyss of crack formaton and crack growth n concrete by means of fracture mechancs and fnte elements. Cement and Concrete Research (Sec. 12.2) 1976; 6: de Borst R. Numercal aspects of cohesve-zone models. Engneerng Fracture Mechancs 23; 7(14):