FINITE ELEMENT IMPLEMENTATION OF INTRINSIC FIELD TENSORS: AN EXAMINATION OF FREE-EDGE SINGULARITIES IN COMPOSITE LAMINATES

Transcription

1 21 st Internatonal Conference on Composte Materals X an, th August 2017 FINITE ELEMENT IMPLEMENTATION OF INTRINSIC FIELD TENSORS: AN EXAMINATION OF FREE-EDGE SINGULARITIES IN COMPOSITE LAMINATES Danel J. O Shea 1, Davd C. Kellermann 2 and Maro M. Attard 3 1 School of Cvl and Envronmental Engneerng, UNSW Australa d.oshea@unsw.edu.au 2 School of Mechancal and Manufacturng Engneerng, UNSW Australa d.kellermann@unsw.edu.au 3 School of Cvl and Envronmental Engneerng, UNSW Australa m.attard@unsw.edu.au Keywords: contnuum mechancs, fnte element, free-edge sngularty, asymmetrc stran tensor ABSTRACT Ths paper presents the mplementaton of a novel orthotropc-based elastcty theory nto fnte element applcatons; employng t to the analyss of nterlamnar stress behavour of mult-layered fbre-renforced compostes. At the scale of nfntesmal deformatons, classcal contnuum mechancs enforces an equlbrum condton based on the geometrc deformatons of the body (assumng an averagng of shear strans), thus utlses symmetrc lnearzed stran and stress tensors. The new approach mparted here resolves materal rotatons and equlbrum as a functon of the ntrnsc materal propertes of the contnuum n addton to geometrc deformatons; replacng the classcal stran tensor wth an orthotropc-based Intrnsc Feld Tensor. Ths has sgnfcant advantages when modellng strongly orthotropc meda, generatng an asymmetrc lnear stran tensor capable of descrbng the ntrcate drectonal nature and response of the contnuum. In ths paper, a novel eghtnode lnear hexahedral fnte element s derved, capturng the new elastcty theory through the ncorporaton of rotatonal degrees of freedom and a modfed consttutve law. The effects of the new theory on studyng nterlamnar behavour of fbre-renforced compostes are readly apparent when observng the well-researched and longstandng free-edge stress sngularty problem, and comparng to the classcal mechancal approach. 1 INTRODUCTION Substantal defcences reman n accurately determnng the stress feld n mult-layered compostes due to ther orthotropc nature; lmtng predctve capabltes n engneerng analyss. Classcal methods of modellng transversely sotropc structures, such as a fbre-renforced lamnate, gve rse to non-physcal stresses at the ntersecton of an nterface plane between any two dssmlar layers and the free-edge. Ths s commonly referred to as the free-edge sngularty problem n composte lamnates [1-4]. Notably, ths s a typcal ste where delamnaton ntates, and we are therefore currently restrcted n predctons of ths crtcal falure mode [5]. Ths paper ams to assess the nfluence that opposng ansotropc prncple materal drectons n adjacent bonded ples has on the development of these sngulartes. A novel approach s presented whch apples an orthotropc elastcty theory, based on Intrnsc-Feld Tensors (IFTs), to conventonal fnte element methods. Ths mproves the modellng of complex drectonal-behavour and deformaton modes, n addton to descrbng the mpact of mcropolar couplngs that occur n lamnated fbre-renforced compostes. Ths s acheved by ncorporatng addtonal rotatonal degrees of freedom to fnte element analyses

2 Danel J. O Shea, Davd C. Kellermann and Maro M. Attard though not requrng a length scale as n Cosserat theory [6] whch allows us to capture novel asymmetrc lnear stran and stress tensors. These asymmetres are drven by the mcropolar couplng that occurs between adjacent drectonal layers of the lamnate. The new orthotropc elastc consttutve law presented by Kellermann [7] s ntroduced to conventonal fnte element methods such that equlbrum s mantaned on both an elemental and global level sgnfcantly wthout any addtonal constrants or specal treatment at the nterface requred. Prelmnary results show a dstnct dfference n stress predctons at the ntersecton of the nterface plane between layers and the freeedge n composte lamnates under unaxal tenson, n comparson to classcal mechancal technques. In classcal mechancs, equlbrum s enforced as a result of an mposed stran symmetry condton. The new approach removes the requste assumpton of symmetry nstead equlbrum s now resolved on both an elemental and global level as a functon of the ntrnsc materal propertes of the system tself. In a composte lamnate, two adjacent layers of dssmlar orthotropc materal wll possess dstnct asymmetrc stran tensors as a result of the unque drectonal response of each. The IFT approach allows for ths to occur, whlst also mantanng compatblty of the stran feld across the ply nterface wthout ntroducng addtonal constrants or requrng hgher order elements. Ths method s mplemented n ths paper usng a novel lnear 6DOF hexahedron element, where the three addtonal rotatonal degrees of freedom capture the asymmetry of shear strans n each respectve plane. At the nterface of two dssmlar materals, a couplng moment s measured n ths degree of freedom whch consequently enforces stran contnuty across the nterface; producng an asymmetrc nfntesmal Cauchy stress tensor. In the case of a fbre-renforced materal under unaxal tenson, ths couplng moment s necessary to physcally model the constrant of rotatons of the mcrostructural fbres n each ply toward the drecton of an appled load due to the bondng of the two dssmlar materals. Ths addtonal mcro-torsonal energy captured n the model s reflected n the results produced, and s a mechansm that cannot be captured usng classcal contnuum mechancs approaches. A commonly used experment for analysng free-edge effects s a symmetrc, fxed wdth, four-ply lamnate under unaxal extenson. Usng classcal methods, ncompatblty of dsplacement gradent n adjacent ples gve rse to large nterlamnar stresses, wth through-thckness normal stress exhbtng sngular behavour at the free-edge [2, 3]. Past attempts to address these non-physcal stress predctons have ncluded addtonal constrants or specal treatment of the nterface, or the use of hgher-order elements [1, 5, 8]. Specalsed solutons have been proposed whch combne modfed lamnaton theores to solve specfc mposed boundary condtons [9]. The am of ths paper s to develop a generalsed approach, whch can address factors contrbutng to free-edge sngulartes n fbre-renforced composte models, n order to predct falure mechansms such as delamnaton. The next secton wll outlne the dervaton of the orthotropc lnear stran tensor and hghlght ts dfferences wth classcal contnuum mechancs, before detalng ts novel mplementaton nto the fnte element method. Followng ths, a numercal experment nvolvng a balanced angle-ply fbrerenforced composte lamnate under symmetrc unaxal loadng wll be used to assess the usablty and physcal sgnfcance of the IFT approach. 2 IMPLEMENTATION OF INTRINSIC FIELD TENSORS TO THE FINITE ELEMENT METHOD 2.1 The classcal mechancal approach The deformaton gradent tensor F of a contnuum can be expressed n tensor form as a functon of the dsplacement gradent tensor u such that:

3 21 st Internatonal Conference on Composte Materals X an, th August 2017 T F u I (1) Here, refers to the forward gradent operator. On the scale of small deformatons, the stretch U and rotaton R tensors are defned as a result of an addtve decomposton of the deformaton gradent tensor nto symmetrc and skew-symmetrc parts such that: 1 T 1 T F U R F F F F (2) 2 2 It then follows that a lnearzed stran tensor ε can be approxmated from the above defnton of stretch; expressed n both tensoral and explct component forms below ε U I 1 T 2 u u (3) 1 j u, j uj, (4) 2 The symmetrcal nature of the nfntesmal stran tensor shown n Equaton (4) s a result of the defnton of a symmetrc stretch U n the decomposton shown n Equaton (2) (5) j To dstngush wth the novel stran tensor presented n the next secton, Kellermann [7] denotes ths symmetrc lnear stran tensor as the Isotropc Lnear Stran Tensor (ILST); and ts use hereon wll be referred to as the classcal contnuum mechancs approach. The classcal lnearzed rgd body rotaton tensor ω s defned as the skew-symmetrc part of the dsplacement gradent tensor wrtten n component form below j 1 j uj, u, j k 2 (6) Here, are components of the thrd-order permutaton tensor, used to defne the postve sgn conventon of rotatons on a gven plane as a result of the cross product. It s mportant to note that n the classcal contnuum mechancs approach ths rotaton tensor s only a functon of the geometrc deformatons of the body (the dsplacement gradent tensor). Note that we can express the followng general defnton relatng stran and rgd body rotaton as an addtve decomposton of the dsplacement gradent u εω (7) u (8) j j, k The nfntesmal Cauchy stress tensor s obtaned through a mappng of the ILST wth the fourthorder materal tensor, contanng components C l, to form the consttutve equaton: σ : ε (9)

4 Danel J. O Shea, Davd C. Kellermann and Maro M. Attard As a result of the symmetry of the ILST, n order to satsfy equlbrum of stresses on an element absent of appled body moments, classcal mechancs requres only a sngle materal constant (shear modulus) n order to entrely descrbe the shear response of a gven plane. Therefore, n addton to major symmetry, the fourth-order materal tensor possesses the property that tensoral components descrbng behavour on a partcular shear plane are equal C jj C (10) jj For numercal calculatons, n a three dmensonal problem ths condton allows for the amount of stress and stran components studed to be reduced from the full nne to the typcally used reduced sx. 2.2 The Intrnsc Feld Tensor approach The ILST derved n the classcal contnuum mechancs approach s based on an assumpton of sotropc materal behavour; and therefore cannot correctly be automatcally extended to the orthotropc case [7]. In other words, a lnear stran tensor expressed only as a functon of the geometrc dsplacement response of a contnuum s nsuffcent to descrbe the mechancal response of an orthotropc materal due to ts ntrnsc drectonal materal behavour. The orthotropc-based elastcty theory s therefore prmarly drven by the novel noton that that shear planes of an orthotropc meda have dstnct horzontal and vertcal behavour and thus n the consttutve law, a gven plane requres two materal constants to descrbe shear response C jj C (11) jj To now defne an Orthotropc Lnear Stran Tensor (OLST), we begn by decomposng the dsplacement gradent of a contnuum nto adapted lnearzed stran and rotaton tensors; analogous to Equatons (7) and (8) u œ φ (12) œ u (13) j j, k For the new elastcty theory, rather than the classcal decomposton of the dsplacement gradent nto smple symmetrc and skew-symmetrc parts, the novel OLST tensor s determned as a functon of the orthotropc propertes of the contnuum. Hence, equlbrum s no longer solely enforced as a result of the geometrc deformaton of a contnuum, but rather also dependent on the ntrnsc materal behavour. We can defne a set of general stress equlbrum equatons on the contnuum such that on a gven plane mk j j (14) Once more, components of the permutaton tensor are used to denote the postve sgn conventon of the couplng moment m for each plane. In prncple materal drectons, by usng k Equatons (8) and (9), the above equaton can then be expressed n terms of dsplacement gradent components and components of the fourth-order materal tensor mk Cjju, j Cjju j, ( Cjj Cjj ) k (15)

5 21 st Internatonal Conference on Composte Materals X an, th August 2017 In a homogenous contnuum, absent of any appled body moments, equlbrum states that all must be zero. The materal rotaton tensor φ can then be explctly determned n terms of both dsplacement gradent components and materal constants where components are gven by: m k C u C u k C C jj j, jj, j jj jj (16) Unlke the classcal form, whch represented a geometrc rgd body rotaton, ths new tensor gves a change n materal prncple drecton [7]. From Equatons (13) and (16), the closed form soluton of the OLST s therefore also materal property dependent, and s no longer assumed symmetrc: C œ u u jj j, j j, Cjj Cjj (17) œ œ (18) j j By applyng the classcal sotropc materal constrant on the consttutve law gven n Equaton (10) t becomes readly apparent ths new defnton reduces to form the ILST and correspondng nfntesmal rgd body rotaton tensor ω defned earler removng any dependence on the ntrnsc materal characterstcs of the contnuum: 1 k u, j uj, k (19) 2 1 œj u j, u, j u j, j (20) 2 Sgnfcantly, the OLST shown here s an encompassng model whch retans the ablty to accurately model the sotropc case. Ths s opposed to the current approach of smply forcng the extenson of the classcal ILST dervaton to the orthotropc case. The nfntesmal Cauchy stress tensor s formed as a result of the mappng of the OLST wth a modfed fourth-order consttutve law σ : œ (21) In the orthotropc case, due to the asymmetrc nature of the stran tensor, numercal methods must explctly determne all nne dstnct components of stress and stran on a three-dmensonal element. It wll be shown n the numercal experment to follow that at the nterface between two dssmlar orthotropc materals n a contnuum, the new method allows for an asymmetrc Cauchy stress tensor, whch has sgnfcant physcal meanng n the analyss of fbre-renforced compostes. 2.3 Applcaton to Fnte Element Method: The new orthotropc-based elastcty theory presented n Secton 2.2 can be ncorporated nto smple lnear 2D and 3D elements smply through manpulatng the calculaton of the typcal stffness matrx relatng forces and dsplacements. In ths paper, an 8-noded soparametrc lnear hexahedral element wll be consdered. In order to mplement the classc ILST nto the fnte element method, we are requred to capture only 3DOF per node (3 translatonal dsplacements). Therefore the classcal element can be attrbuted the followng dsplacement vector

6 Danel J. O Shea, Davd C. Kellermann and Maro M. Attard Here, element dsplacements each node, (where T ILST 1, 2, 3 q u u u (22) u are a lnear nterpolaton of the respectve dsplacements from N are lnear shape functons) (23) u N u u N u u N u 1 1( ) 2 2( ) 3 3( ) However, each node of the proposed orthotropc hexahedral element s capable of capturng 6DOF (3 translatonal dsplacements, 3 rotatons). For the new orthotropc-based element, we therefore attrbute the followng extended dsplacement vector: T q u, u, u,,, (24) Once more, components of the above dsplacement vector are determned as a lnear nterpolaton of the value at each node usng smlar expressons for the rotatonal degrees of freedom to those shown n Equaton (23) (25) N N N 1 1( ) 2 2( ) 3 3( ) The asymmetry of the OLST requres all 9 dstnct components of stran and stress to be determned. In prncple drectons, these second-order tensors can be flattened to vector form usng Vogt notaton, such that σ [,,,,,,,, ] (26) { M} T œ [ œ, œ, œ, œ, œ, œ, œ, œ, œ ] (27) { M} T Here, { M } denotes the prncple materal coordnate system. From the orthotropc-based consttutve law presented n Equaton (21) the fourth order materal tensor matrx form such that can be flattened to { M} { M} { M} σ C. œ (28) { } C M Here, s the materal matrx for a gven element n prncple materal drectons. Ths matrx s symmetrc, nvertble and of sze [9 9]. The materal matrx can be transformed to the global coordnate system { G } through an angle usng the orthogonal rotaton matrx T such that { G} 1 { M} C T. C.T (29) For a transversely sotropc fbre-renforced materal, s typcally used to descrbe the angle of whch fbres are offset from the global Cartesan longtudnal axs. As dscussed n Secton 2.2, n the new approach equlbrum s to be determned as a result of the dstnct shear behavour n dfferent drectons on a gven plane. Hence, the equlbrum equatons gven n Equaton (15) are resolved as

7 21 st Internatonal Conference on Composte Materals X an, th August 2017 part of the fnte element process rather than through an appled symmetry constrant to the system. To acheve ths, the stran vector n Equaton (27) s wrtten n terms of the decomposton of the dsplacement gradent tensor shown n Equaton (13) where rotaton tensor components are resolved as varables n the model, captured through the addtonal degrees of freedom (30) { M} œ u1,1, u2,2, u3,3, u3,2, u1,3, u2,1, u2,3, u3,1, u1,2, 1, 2, 3 T Smlarly, we ntroduce the equlbrum equatons gven n Equaton (14) to the consttutve law by extendng the stress vector shown n Equaton (26) σ [,,,,,,,,, m, m, m ] (31) { M} T Smple manpulaton of the consttutve law n prncple drectons to extend the lnear system of equatons results n a modfed [12 12] matrx { } C M ; whch mportantly remans symmetrc and nvertble, and can be transformed to global coordnates usng an extended rotaton tensor analogous to Equaton (29): { G} 1 { M} C T. C. T T, (32) At a gven node, the dsplacement gradent components formng extended stran vector approxmated usng a matrx of dfferentals such that { } œ M are { G} œ [ ]. q (33) The components of the dfferental matrx, are numercally calculated by ntroducng the nterpolaton expressons (23) and (25) to extend the vector q, and rearrangng terms to form a matrx B such that { G} œ B. Q (34) Here, Q s a vector contanng all dsplacements at each node () n an element Q u, u, u,,,,, u, u, u,,, (35) The contents of matrx B are determned through dfferentaton of the lnear soparametrc shape functons, and convertng between natural and global coordnate systems usng the Jacoban matrx J : x y z N N N x y z 8 x y z N N N J x y z (36) 1 x y z N N N x y z T

8 Danel J. O Shea, Davd C. Kellermann and Maro M. Attard Through Gaussan ntegraton, the stffness matrx for the element () e s gven by p1 p2 p3 ( e) T { G} ( e) T { G} dv w wjwk ( e) V 1 j1 k1 K B C B. B. C. B det J (37) Here, w and p are the Gaussan weghts and ntegraton ponts respectvely, and e dv s the volume of the nfntesmal element Note that usng ths new 6DOF lnear fnte element and applyng the classcal constrant statng that the two materal constants defnng shear n a gven plane are equal wll produce dentcal results to the lnear 3DOF hexahedral element descrbed at the begnnng of ths secton. 3 CASE STUDY: FREE-EDGE STRESSES IN COMPOSITE LAMINATES 3.1 Detals of Numercal Experment Fgure 1 shows a symmetrc angle-ply lamnate of fbre-renforced materal (T700/VTM264) [10] under unaxal extenson along the longtudnal axs. The lamnate has a wdth of b 18 mm and has four ples each of thckness h 1 mm. The prncple orentaton of the fbres n a gven layer s offset from the longtudnal axs by an angle. The lay-up sequence of the composte lamnate studed n ths paper s a balanced angle-ply of the form: [ / ] s Fgure 1 - Detals of Numercal Experment In the fnte element model, usng dsplacement control, the coupon s extended such that an arbtrarly chosen constant axal stran s mposed ( xx ). We are nterested n the nterlamnar stress response between dssmlar layers, partcularly along the transverse axs y as we approach the nterface corner (marked pont A n Fgure 1). At the free-edge, classcal methods predct sngulartes n the zz and zx components of the stress tensor [2]. The composte s constructed of four layers of fbre-renforced composte, each ply dealsed as homogenous and elastc usng the materal propertes (n prmary materal coordnates) provded n Table 1

9 σ zx (MPa) 21 st Internatonal Conference on Composte Materals X an, th August 2017 Materal Property T700/VTM264 E 1 [MPa] E, E [MPa] G, G [MPa] G [MPa] , Table 1 - Materal propertes of fbre-renforced composte used n numercal experment [10] In order to assess any sngular behavour n stress components, the balanced angle-ply lamnate has been dscretsed usng four meshes of ncreasng refnement, wth more elements favoured towards the free-edge of the specmen n each case: Coarse, Medum, Fne and Very Fne (800, 6400, and elements respectvely) 3.2 Dscusson of Results Fgure 2 shows the calculated transverse shear stress at the pont A ndcated n Fgure 1 for symmetrc [ / ] s angle-ply lamnates rangng from 0 to 90. For ths shear component, the peak value occurs at the free-edge, whlst the component vanshes towards the centre of the specmen. Sngular behavour n the stress component can be observed for both the classcal and IFT approaches, wth the new model predctng slghtly lower values. The classcal approach shows good agreement wth results presented by both Becker [11] and Herakovch [12], whereby the maxmal nterlamnar shear occurs for the lay-up angle 15. Ths therefore provdes a sold bass to assess the IFT approach Classc - Coarse Classc - Medum Classc - Fne Classc - Very Fne IFT - Coarse IFT - Medum IFT - Fne IFT - Very Fne θ (degrees) Fgure 2 Free-edge nterlamnar shear stress for a range of [ / ] s lamnates zz

10 σ zz (MPa) Danel J. O Shea, Davd C. Kellermann and Maro M. Attard The behavour of the through-thckness normal stress component along the nterface plane conssts of a steep stress gradent approachng the free-edge, wth the maxmum value occurrng at Pont A (Fgure 1). These maxmal values are shown below n Fgure 3 for the same range of lay-up angles Classc - Coarse Classc - Medum Classc - Fne Classc - Very Fne IFT - Coarse IFT - Medum IFT - Fne IFT - Very Fne θ (degrees) Fgure 3 Free-edge through-thckness normal stress for a range of [ / ] s lamnates Usng the symmetry-based ILST, classcal mechancs predcts a compressve sngularty for all layup angles. A maxmum occurs at a lay-up angle close to30, and these results replcate predctons offered n the lterature [2, 3, 8]. The novel IFT fnte element predcts sgnfcantly lower stresses and the sngular behavour s reduced for all lay-up angles. The orthotropc-based approach predcts convergng tensle behavour for 15and 50, whlst the md-range predcts a compressve sngularty. Free-edge sngulartes are a well-researched topc n contnuum mechancs. These non-physcal predctons can be sad to be mult-modal, wth several factors contrbutng to ther development at the nterface of dssmlar materals. Among others, these nclude the ntersecton of dstnct sotropc elastc propertes [13], the effect of dfferent Posson s ratos n adjacent ples, and the bondng of ansotropc materals wth opposng prncple materal drectons. It has been shown, that by addressng the latter of these, for balanced angle-ply lamnates under symmetrc unaxal tenson, the approach presented n ths paper has had mnmal effect on transverse shear stresses, though greatly reduced the free-edge sngularty found n the component. In future studes t would be desrable zz to solate the extent of whch certan other factors contrbute to ths sngularty n order to completely assess the sgnfcance of ths work. Accurate predctons of the through-thckness normal stran component are requste n understandng the falure mechansms of composte lamnates. It s known that there are three dstnct falure modes n compostes, wth smaller angle lay-ups falng n delamnaton and the larger angle ples undergong lamna falure [14]. For the prelmnary results of the IFT fnte element approach shown n Fgure 3, the three dstnct ranges of stress behavour outlned earler tend to occur n ranges that match those dfferent falure mechansms. Ths s more useful than classcal mechancs whch zz

11 Depth, z (mm) Depth, z (mm) Depth, z (mm) Depth, z (mm) 21 st Internatonal Conference on Composte Materals X an, th August 2017 predcts a compressve sngularty for all lamnates, regardless of ther known dfferent methods of falure. As outlned n the dervaton, a sgnfcant property of the new model s ts encompassment of sotropc behavour. For a homogenous materal, the novel orthotropc fnte element wll reduce to calculate stress results matchng those of classcal mechancs. Ths can be observed n the results of the 0 and90 lay-up angles n both Fgures 2 and 3. The IFT model provdes an mproved physcal understandng of the deformaton mechansms occurrng n a fbre-renforced lamnate under tenson E E E E E-04 Rotaton, φ z (-) E E E-08 Rotaton, φ z (-) Classc - Coarse Classc - Medum Classc - Fne Classc - Very Fne IFT - Coarse IFT - Medum IFT - Fne IFT - Very Fne Couple moment, m z (MPa) E-09 4E-09 6E-09 8E-09 1E-08 Couple moment, m z (MPa) Fgure 4 - In-plane rotatons (top) and couple moments (bottom) versus lamnate depth for [ 30 / 30] s and [ 0 / 0 ] s lamnates (left and rght respectvely) The n-plane materal rotaton around the z axs, as defned n Equaton (13), s shown n the top row of Fgure 4. Recall that ths s a measure of the asymmetry of stran n a gven plane. For a lay-up angle 0, as expected we observe the new formulaton reduces to match the classcal response. For the 30 case, we observe that the new formulaton ndcates that the 30and 30 layers rotate relatve to each other as fbres attempt to algn n the drecton of the appled load. As the materal s assumed perfectly bonded, the rotaton at the nterface correctly s zero, and stran contnuty s ensured. To enforce ths, a couple moment, defned n Equaton (14), s captured at the nterface whch physcally

12 Danel J. O Shea, Davd C. Kellermann and Maro M. Attard represents the resstance to load caused by bondng two ansotropc materals wth opposng prncple fbre drectons. Smlar results are observed for all other lay-up angles. Classcal mechancs s unable to capture ths addtonal torsonal energy; nstead addng t to the hgh stress gradent at the free-edge. Ths explans the reducton n sngular behavour we observe n Fgure 3 by usng the new orthotropc-based theory to addresses ths ssue. 4 CONCLUSIONS A novel lnear 3D fnte element s derved n ths paper whch removes a symmetry assumpton nherent n classcal mechancs through mplementaton of an orthotropc-based elastcty theory dependent on the ntrnsc materal propertes of the system. Ths new approach s a generalsed model whch can be appled to any geometry or loadng condtons wthout the requrement for specal treatment of nterfaces or lamnaton theores. The model s shown to have physcal sgnfcance n both ts theores and applcaton. Numercal nvestgaton of a transversely sotropc balanced angleply composte lamnate shows a reducton n the sngularty as a result of capturng the addtonal torsonal energy resultng from the bondng of adjacent layers wth dstnct ansotropc prncple materal drectons. When symmetrc sotropc shear materal behavour s rentroduced, the model collapses to classcal contnuum mechancs theory, and equvalent results to past lterature are replcated. Future expermental work s needed to consoldate these prelmnary fndngs, and detect the sutablty of the new approach n predctng falure mechansms of fbre-renforced compostes. REFERENCES [1] Km, H.S., J. Lee, and M. Cho, Free-edge nterlamnar stress analyss of composte lamnates usng nterface modelng. Journal of Engneerng Mechancs, [2] Pagano, N., Free edge stress felds n composte lamnates. Mechancs of Composte Materals, 1994: p [3] Schellekens, J. and R. De Borst, Free edge delamnaton n carbon-epoxy lamnates: a novel numercal/expermental approach. Composte structures, (4): p [4] Salamon, N.J., An assessment of the nterlamnar stress problem n lamnated compostes. Journal of composte materals, (1): p [5] Schellekens, J. and R. De Borst, A non-lnear fnte element approach for the analyss of mode-i free edge delamnaton n compostes. Internatonal Journal of Solds and Structures, (9): p [6] Cosserat, E. and F. Cosserat, Theory of Deformable Bodes, Scentfc Lbrary, A. Hermann and Sons, Pars, [7] Kellermann, D., T. Furukawa, and D. Kelly, Strongly orthotropc contnuum mechancs and fnte element treatment. Internatonal journal for numercal methods n engneerng, (12): p [8] Raju, I. and J.H. Crews Jr, Interlamnar stress sngulartes at a straght free edge n composte lamnates. Computers & Structures, (1): p [9] Ppes, R.B. and N. Pagano, Interlamnar stresses n composte lamnates under unform axal extenson, n Mechancs of Composte Materals. 1994, Sprnger. p [10] Ahamed, J., et al., Ply-nterleavng technque for jonng hybrd carbon/glass fbre composte materals. Compostes Part A: Appled Scence and Manufacturng, : p [11] Becker, W., Free-edge stress concentraton n angle-ply lamnates. Archve of Appled Mechancs, (1): p [12] Herakovch, C., Free edge effects n lamnated compostes [13] Whtcomb, J., I. Raju, and J. Goree, Relablty of the fnte element method for calculatng free edge stresses n composte lamnates. Computers & Structures, (1): p [14] Rotem, A. and Z. Hashn, Falure modes of angle ply lamnates. Journal of Composte Materals, (2): p