Influence of the 90/0 ply thickness ratio on the stresses associated with the edge effect phenomenon

In luence o he 90/0 ply hickness a io on he s esses associa ed wi h he edge e ec phenomenon S. S´ anchez-Ca mona * , A. Ba oso , E. Co ea G upo de Elas icidad y Resis encia de Ma e iales, Depa amen o de Mec´ anica de Medios Con inuos y Teo ía de Es uc u as, Escuela T´ ecnica Supe io de Ingenie ía, Uni e sidad de Se illa 41092 Se illa, Spain ARTICLE INFO Keywo ds: Scale e ec F ee edge Ca bon and glass ib es C oss-ply lamina e ABSTRACT The scale e ec phenomenon has shed ligh on he he mo-mechanical beha iou o ul a- hin plies. These laminas in c oss-ply lamina es (unde ce ain 90/0 hickness a ios) ha e gi en ise o a ele an edge e ec phenomenon, causing signi ican h ough- he- hickness s ess in he weakes ply along i s ee edge unde bo h he mal and (0-longi udinal ension) mechanical loading. Thus, a biaxial s ess s a e mus be conside ed o analyse wha happens along he ee edges om expe imen al samples o s uc u al componen s, such as d ill holes. A pa ame ic nume ical analysis is pe o med aking h ee di e en ea u es in o accoun : he hickness o he ply blocks, he c oss-ply s acking sequences and he ype o ib e, ei he ca bon o glass. The nume ical p edic ions a e in acco dance wi h expe imen al esul s, which a e ob ained unde a he mal cooldown. The biaxial s ess s a e p edic ions could be used in u u e nume ical p ocedu es including he p esence o compo- nen s’ ee edges. 1. In oduc ion The possibili y o manu ac u ing ul a- hin plies indus ially leads o pe o ming se e al esea ch wo ks o imp o e he knowledge conce n- ing he phenomenon known as he scale e ec , which has been s udied since he appea ance o composi e ma e ials by Pa izi e al [1], Flaggs and Ku al [2] and Pagano e al [3], among o he s. The in e es o he scien i ic communi y in ul a- hin plies in he las decades has shed ligh on hei beha iou unde di e en he mal and mechanical condi ions [4,5,6,7]. Some yea s ago, Pa ís e al [8] ga e a physically based explana ion o he scale e ec phenomenon. In his way, hese au ho s s a ed o s udy hese ul a- hin plies unde di e en loading condi ions, inding some non-con en ional damages on he ul a- hin 90◦ply blocks o c oss-ply lamina es which appea ed longi udinal o he ib es o he 0◦ply blocks along he samples’ ee edges, e en jus a e he cooling o he cu ing p ocess [9]. The p esence o ee edges, om expe imen al specimens ( o cha - ac e isa ion and esea ch p og ammes) o s uc u al componen s (holes, edge inishing o wo kpieces and join s, e c.), gi es impo ance o he edge e ec phenomenon, since an accu a e knowledge o he s ess s a e is equi ed. Due o his ac , se e al s udies ha e been pe o med con- ce ning his phenomenon, such as Kassapoglou and Lagace [10], Becke [11], Pagano e al[3], Lo io e al [12] and Mi els ed and cowo ke s ([13,14]), among o he s. I is wo h highligh ing ha Hajikazemi and Van Paepegem [15] showed he impo ance o he h ough- he- hickness s esses ha appea when hin plies a e conside ed. Taking his idea in o accoun , S´ anchez-Ca mona e al [16] p o- oundly s udied he edge e ec phenomenon o he case o c oss-ply lamina es, which akes special ele ance when ul a- hin plies a e in ol ed in he 90◦ply block. The high cons ain s o he su ounding 0◦plies o he 90◦ul a- hin ones led o impo an in alamina h ough- he- hickness s esses in he 90◦ply block along he pa s’ ee edges. The ele ance o his s ess componen implies ha a biaxial s ess s a e appea s in he 90◦ply block jus a e he cu ing p ocess (due o he he mal dec emen associa ed wi h he cooling p ocess) and when es ed unde uniaxial ension, a ac ecen ly s udied by he au ho s (S´ anchez- Ca mona e al [17]). In his way, his wo k aims o es ablish he edge-e ec consequences associa ed wi h he use o di e en 90 / 0 a ios in c oss-ply lamina es o analyse om hick o ul a- hin ply blocks. A pa ame ic nume ical analysis is pe o med based on di e en hicknesses o 0◦and 90◦ply blocks, di e en c oss-ply con igu a ions, and di e en ib e sys ems. Once his wide ange o possibili ies is calcula ed, an accu a e p edic ion o he biaxial s ess s a e along he 90◦ply block’s ee edge is ob ained, * Co esponding au ho . E-mail add ess: [email p o ec ed] (S. S´ anchez-Ca mona). Con en s lis s a ailable a ScienceDi ec Composi es Pa A jou nal homepage: www.else ie .com/loca e/composi esa h ps://doi.o g/10.1016/j.composi esa.2025.108961 Recei ed 6 Feb ua y 2025; Recei ed in e ised o m 16 Ap il 2025; Accep ed 17 Ap il 2025 Composi es: Pa A 195 (2025) 108961 A ailable online 20 Ap il 2025 1359-835X/© 2025 The Au ho s. Published by Else ie L d. This is an open access a icle unde he CC BY-NC license ( h p://c ea i ecommons.o g/licenses/by- nc/4.0/ ). p o iding a a be e app oxima ion o he ac ual s ess s a e ha can be calcula ed by simply using he Classical Lamina e Theo y. These nu- me ical p edic ions a e co obo a ed expe imen ally o he he mal case (jus a e he cu ing p ocess) o bo h ib e sys ems. Finally, hese es ima ions may be used o p edic he biaxial s ess s a e ha occu along componen s’ ee edges o u u e nume ical s udies pe o med by he scien i ic communi y. 2. De ini ion o he case s udies The lamina e scheme unde s udy is de ined in Fig. 1, whe e he 0◦ ib e ply block is aligned o he x-axis and he 90◦ ib e ply block is aligned o he y-axis. The ee edges a e hose ma ked in g een which co espond o he planes y=0 and y=w. The edge e ec phenomenon in he c oss-ply con igu a ion is asso- cia ed wi h wo p ope y misma ches, he di e en he mal expansion coe icien s and elas ic p ope ies, which can be s udied sepa a ely. On he one hand, he cu ing p ocess en ails ha he c oss-ply lamina e su e s a he mal dec emen du ing he cooling s age, which p o okes ee-edge s esses due o he di e en coe icien s o he mal expansion (CTE) o he 90◦and 0◦ply blocks. On he o he hand, a mechanical load leads o a nominal σ xx in he lamina e and also gene a es ee-edge s esses due o he di e ences in he elas ic modulus and he Poisson a ios be ween he 0◦and 90◦ply blocks. Bo h p oblems p o oke h ough- he- hickness s esses, σ zz , in he 90◦ply block. Hence, a biaxial s ess s a e is p esen along he lamina e’s ee-edges (S´ anchez-Ca mona e al [17]), he analysis o which is he main mo i a ion o his wo k. As men ioned abo e, he p e ious wo k o hese au ho s [16] showed he ele ance o he edge e ec phenomenon in a c oss-ply lamina e when an ul a- hin 90◦ply block is used oge he wi h hick 0◦ply blocks. A mo e in-dep h s udy abou he di e en ply cons ain s be- ween 0◦and 90◦ply blocks is pe o med in his wo k. The case s udies which a e going o be analysed a e based on a pa ame ic analysis o conside he e ec o h ee pa ame e s, summa ised in Table 1: he ply hicknesses 0◦and 90◦, he ply loca ions gi ing ise o di e en lay-ups, and he ype o ma e ial used, a ying he ype o ib e while keeping he same he mose ma ix. Fi s , a ange o di e en hicknesses, om ul a- hin o ul a- hick, a e used o each ply block, 0◦and 90◦. The selec ed hicknesses a e 50, 65, 100, 170, 340, 510 and 680 µm. In Table 1, a pa ame e , n, a ying om 1 o 7, is employed o ep esen each hickness in se s o s acking sequences wi h he same hickness o he 0◦ply blocks. Second, wo di e en c oss-ply lamina es a e selec ed as he ex eme cases, [0/90/0] and [0/90/0/90/0] s . The i s lamina e has all he 0◦and 90◦plies concen a ed in one ply block on one side wi h espec o he cen al plane o he lamina e, named as he egula c oss-ply, whe eas he second c oss-ply con igu a ion has he 0◦and 90◦plies al e na i ely dis ibu ed. Thi d, he in luence o wo di e en ib es on he edge e ec phe- nomenon is assessed. T700S Ca bon and 2026 E-glass ib es a e conside ed using he same epoxy esin, TP-402. Any combina ion o hese h ee di e en cha ac e is ics is e alua ed o p o ide he biaxial s ess s a e ha occu s along he 90-ply ee edges o hese ma e ial sys ems. 3. Nume ical modelling The nume ical model is de ined as indica ed in Fig. 2 using a 2D geome y (elemen ype PLANE 182) unde he assump ion o Gene al- ized Plane S ain, ui=ui(y,z) o i=x,y,z, in ANSYS® so wa e. The esul s ob ained om he Fini e Elemen Model (FEM) we e p ocessed using he MATLAB® so wa e. Each ply block has a pa icula hickness, 0 o he 0◦and 90 o he 90◦ply blocks, leading o a o al o 49 combina ions o 50, 65, 100, 170, 340, 510 and 680 µm (as shown in Table 1) o each speci ic case o s acking sequence and ib e sys em, gi ing ise o a o al o 196 cases s udied. The model is implemen ed as a qua e o he 2D geome y wi h a wid h (w) o 5 mm, conside ing symme y along he middle plane o he lamina e h ough he hickness and he wid h, as indica ed by he ed dash lines in Fig. 2. The smalles elemen s (10 -6 mm leng h) a e loca ed in he 90◦ply block close o he 0/90 in e ace. As de ailed in Sec ion 2, he model is sol ed o he he mal and mechanical p oblems independen ly; hence, a o al o 392 nume ical cases a e pe o med in his pa ame ic analysis. The he mal case is simula ed using a he mal dec emen o −110 ◦C, ep esen ing he cooling down p ocess om cu e empe a u e (135 ◦C) o oom em- pe a u e (25 ◦C). The mechanical p oblem is associa ed wi h a longi- udinal displacemen (u x ) applied in he elas ic cen e o he p oblem (whe e ensile o ce and bending momen s a e uncoupled) as i does no coincide wi h he geome ical cen e because only he uppe hal pa o he p oblem is modelled. This p esc ibed displacemen , u x , gi es ise o a longi udinal s ain ( ε x ) ha gene a es in he 90◦ply block a nominal σ xx =38 MPa o he ca bon/epoxy lamina e and σ xx =46 MPa o he glass/ epoxy one, which coincides wi h he expe imen ally measu ed ans- e se s eng h (Y T ) o each ma e ial sys em. This alue, p edic ed by he Classical Lamina e Theo y, has been e i ied in a eas a away om he ee edge in he nume ical model. I is wo h highligh ing ha a s ess singula i y exis s due o he p esence o he bima e ial co ne a he 0/90 ee-edge in e ace. The p esen au ho s pe o med a nume ical analysis o emo e he in luence o he weak s ess singula i y h ough a local geome ical modi ica ion o he co ne geome y, gi ing ise o a leng h a ec ed by he singula s ess ield below 10 -2 mm [16] unlike he ex en o he edge e ec phenomenon, which is o he o de o 1 o 2 hicknesses ac oss he wid h w. As a las commen , le ’s ake in o accoun ha his dis ance is o he same o de o magni ude as he ib e diame e i sel , and he homoge- nised calcula ed s esses may no ully cap u e he eal he e ogeneous na u e o he ma e ial. Hence, he ee-edge consequences shown below a e exclusi ely due o he edge e ec phenomenon wi hou being a ec ed by he s ess singula i y a he 0/90 in e aces. Taking he ca bon/epoxy [0/90/0/90/0] s lamina e ( 0 =680 µm and 90 =50 µm) in o conside a ion, Fig. 3 shows he σ zz con ou plo s along he 90◦ply blocks, showing he h ough- he-wid h dep h o he edge e ec phenomenon (pu ple ame) and ema king wo di e en de ails a di e en scales; i s , he bi-ma e ial co ne a he 0/90 in e ace whe e he singula s ess ield a ises (magen a ame); and, second, he zone Fig. 1. Scheme o he ma e ial sys em unde s udy. S. S´ anchez-Ca mona e al. Composi es Pa A 195 (2025) 108961 2 conside ed o he calcula ion o he a e age alue o he edge-e ec s esses p o oked is inside an o ange ame. The he mo-mechanical p ope ies o bo h p ep eg g ammages needed o hese models a e indica ed in Table 2, aken om S´ anchez- Ca mona e al [18] o he ca bon/epoxy ma e ial and measu ed expe imen ally o he glass/epoxy ape. The ou -o -plane p ope ies (z- axis) a e es ima ed om he in-plane p ope ies. E 33 and α 3 a e aken equal o E 22 and α 2 , espec i ely. Rega ding ν 23 , an inc easing pe cen - age o i s co esponding ν 12 is assumed ollowing he li e a u e ([19,20]). A pa icula conside a ion has been made ega ding G 23 due o he di icul ies in ob aining his elas ic p ope y expe imen ally. These au ho s a e expe ienced in he complexi ies o cha ac e ising shea p ope ies [21]; hence, i s assump ion is pe o med om he li e a u e. Some publica ions ob ain his alue by assuming he ans e sely iso opic beha iou whe eas o he s di ec ly ake a pe cen age o he alue o G 23 wi h espec o G 12 , e en no mally aking a sligh educ ion ([20,22]). To a oid any in luence o G 23 in his nume ical analysis, wo ex eme alues a e chosen in his pa ame ic model o he glass ib e case. The lowes alue selec ed is G 23 =3200 MPa, ep esen ing a minimum educ ion o G 12 , and he highes alue used is G 23 =4181 MPa, calcula ed unde he assump ion o ans e sely iso opic beha - iou . Fig. 4 shows he negligible in luence o his elas ic p ope y on he a e age mechanical solu ion. I is wo h no ing ha he ex eme alue due o he s ess singula i y inc eases o he i s case conside ed al hough he a e age alues a e qui e simila . Bo h esul s can be conside ed equal as he esul s a ec ed by he s ess singula i y will be emo ed o jus analyse he e ec o he nominal edge e ec phenome- non. The o he h ee s ess componen s analysed show he same ac . Thus, G 23 is selec ed as a sligh educ ion o G 12 o bo h ib es cases in his wo k (al hough only he glass ib e is shown he e), aking simila pe cen ages o G 23 wi h espec o G 12 han hose used in Kaddou e al [20]. The nex wo sec ions p esen he hickness-based pa ame ic anal- ysis o he p esen ed p oblem, showing he dimensionless in-plane ( σ xx ) and ou -o -plane ( σ zz ) s esses, i.e. each s ess is di ided by Y T (due o he biaxial s ess s a e associa ed wi h he edge-e ec consequences, on a e age along he 90◦ply block hickness) e sus he hicknesses o he 0◦and 90◦ply blocks ( 0 , 90 ) o bo h he mal and mechanical p oblems ( om Fig. 8 o Fig. 11). These ou 3D cha s show he hicknesses o each ply block, 0 and 90 , in he ho izon al axes, and he dimensionless s ess analysed in he e ical axis. Fu he mo e, a g ey cons an plane o dimensionless s ess equal o 1 is d awn in each plo o help he isu- aliza ion o he dimensionless s esses ha a e abo e he Y T alue. The i s sec ion shows he dependence on he s acking sequence o each ib e ype and he second sec ion de ails he e ec o he ib e ype on each s acking sequence. Rega ding he a e age edge-e ec s ess ob ained om each s udy case analysed, which is de ined as he selec ed s ess e sus he ma e ial ans e se s eng h, some conside a ions a e ma ked: •These a e age alues a e calcula ed by disca ding he alues asso- cia ed wi h he s ess singula i y ield, i.e. emo ing hose alues ha a e wi hin 10 -2 mm om each 0/90 ee-edge in e ace acco ding o S´ anchez-Ca mona e al [16]. •Mos cu es a e qui e la along he 90◦ hickness conside ed; ne e heless, he e a e some cases whose dis ibu ions equi e a cla i ica ion. In pa icula , and wi h espec o he [0/90/0] con igu a ion, he edge e ec ends a e no so uni o m o some speci ic 90/0 hickness a ios. Fig. 5 shows he he mal dimensionless in-plane s ess, σ xx , e sus he 90◦ply block hickness o he 0 =340 µ m, a ying 90 om 50 o 680 µ m (CP-50 up o CP-680, espec i ely). A mono- onically inc easing end o he edge-e ec s ess wi h espec o he he mal σ xx componen is obse ed especially o CP-680 (bo om cu e). The minimum, a e age and maximum alues om his end a e ma ked, aken as he maximum alue ha emains a e emo ing hose alues associa ed wi h he s ess singula i y ield. Al hough his a e age alue, 0.596, is no as ep esen a i e as can be he one selec ed o ins ance o CP-50, i is an in e media e alue ha can be seen as a e e ence. Howe e , all he cases ha p esen his speci ici y ha e a dimensionless s ess less han 1; hence, hose alues a e he esidual he mal elaxa ion conside ed show a lowe ele ance. This change om he la end occu s because he esin o hick 90◦ply blocks ends o dec ease hese blocks olume wi h a mino e ec om he cons ain o he 0◦ply blocks. Table 1 Pa ame ic analysis ma ix, a ying he 0◦and 90◦ply block hicknesses om 50 o 680 µm (only he ex eme hicknesses case o he 0◦ply block, n =1 and 7, a e shown). Ca bon ib e – Epoxy esin Glass ib e – Epoxy esin   [0/90/ 0] [0/90/0/ 90/0] s [0/90/ 0] [0/90/0/ 90/0] s Combina ion o hicknesses (in µm) o 0◦and 90◦ply blocks n =1 (50 µm) [50/ 50/50] [50/50/ 50/50/ 50] s [50/ 50/50] [50/50/ 50/50/ 50] s [50/ 65/50] [50/65/ 50/65/ 50] s [50/ 65/50] [50/65/ 50/65/ 50] s [50/ 100/ 50] [50/100/ 50/100/ 50] s [50/ 100/ 50] [50/100/ 50/100/ 50] s [50/ 170/ 50] [50/170/ 50/170/ 50] s [50/ 170/ 50] [50/170/ 50/170/ 50] s [50/ 340/ 50] [50/340/ 50/340/ 50] s [50/ 340/ 50] [50/340/ 50/340/ 50] s [50/ 510/ 50] [50/510/ 50/510/ 50] s [50/ 510/ 50] [50/510/ 50/510/ 50] s [50/ 680/ 50] [50/680/ 50/680/ 50] s [50/ 680/ 50] [50/680/ 50/680/ 50] s …F om n =2 (65 µm) o n =6 (510 µm) … n =7 (680 µm) [680/ 50/ 680] [680/50/ 680/50/ 680] s [680/ 50/ 680] [680/50/ 680/50/ 680] s [680/ 65/ 680] [680/65/ 680/65/ 680] s [680/ 65/ 680] [680/65/ 680/65/ 680] s [680/ 100/ 680] [680/ 100/ 680/ 100/ 680] s [680/ 100/ 680] [680/ 100/ 680/ 100/ 680] s [680/ 170/ 680] [680/ 170/ 680/ 170/ 680] s [680/ 170/ 680] [680/ 170/ 680/ 170/ 680] s [680/ 340/ 680] [680/ 340/ 680/ 340/ 680] s [680/ 340/ 680] [680/ 340/ 680/ 340/ 680] s [680/ 510/ 680] [680/ 510/ 680/ 510/ 680] s [680/ 510/ 680] [680/ 510/ 680/ 510/ 680] s [680/ 680/ 680] [680/ 680/ 680/ 680/ 680] s [680/ 680/ 680] [680/ 680/ 680/ 680/ 680] s S. S´ anchez-Ca mona e al. Composi es Pa A 195 (2025) 108961 3 •The cons ain o hick 0◦ply blocks leads o edge-e ec s esses ha a e no uni o mly dis ibu ed along he 90◦ply block hickness o he dis ibu ed c oss-ply con igu a ion, as shown in Fig. 6. In he dis ibu ed s acking sequence, he inne 90◦ply block is mo e es ained by he cen al 0◦ply block (loca ed in he plane o sym- me y) han by he in e media e 0◦ply block, which is also con- s aining he ou e 90◦ply block. In o he wo ds, he 0/90 in e aces close o he in e media e 0◦ply block a e less es ained because his 0◦ply block is su ounded by wo di e en 90◦ply blocks. This can be illus a ed g aphically by employing he scheme o Fig. 7, whe e he dimensionless s esses o he CP-680 om Fig. 6 a e shown in each 90◦ply block. Fig. 2. FEM mesoscopic model o he ma e ial sys ems unde s udy. Fig. 3. σ zz con ou plo s (MPa) o he ca bon/epoxy [0/90/0/90/0] s lamina e ( 0 =680 µm and 90 =50 µm), highligh ing he singula s ess ield a he 0/90 in e ace and he 90◦-ply zone aken o he calcula ion o he edge-e ec s ess a e age alue. Table 2 The momechanical p ope ies o unidi ec ional ca bon/epoxy and glass/epoxy ape p ep eg. Ca bon/Epoxy Glass/Epoxy E 11 (MPa) 113,910 43,210 E 22 (MPa) 7810 11,540 E 33 (MPa) 7810 11,540 ν 12 0.321 0.269 ν 13 0.321 0.269 ν 23 0.4 0.38 G 12 (MPa) 3254 3380 G 13 (MPa) 3254 3380 G 23 (MPa) 2350 3200 α 1 (◦C −1 ) 6.16e-6 1.13e-5 α 2 (◦C −1 ) 3.03e-5 3.54e-5 α 3 (◦C −1 ) 3.03e-5 3.54e-5 S. S´ anchez-Ca mona e al. Composi es Pa A 195 (2025) 108961 4 3.1. On he s acking sequence All he esul s ob ained o he compa ison o he wo c oss-ply con igu a ions conside ed a e shown below. Fig. 8 shows he esul s associa ed wi h he 90◦ca bon ib e blocks (bo h egula c oss-ply and dis ibu ed) whe eas Fig. 9 p esen s he same collec ion o esul s o he E-glass ib e case. A simila end occu s o each dimensionless s ess analysed in Fig. 8 and Fig. 9. The 90◦ply block in he egula c oss-ply con igu a ion is mo e cons ained by he su ounding 0◦ply blocks han hose 90◦ply blocks in he dis ibu ed lay-up. Thus, he egula c oss-ply con igu a- ion shows g ea e edge-e ec s esses, excep o he he mal in-plane s ess. This pa icula case, shown in Fig. 8.(b) o ca bon ib e and Fig. 9.(b) o glass ib e, has a lowe edge-e ec s ess han he dis ib- u ed con igu a ion, excep o he ex eme cons ain cases o hick 0◦ply blocks agains hin 90◦ply blocks. This di e en beha iou comes om he non-uni o m edge e ec consequences expounded in he i s pa o sec ion 3, speci ically in Fig. 8. Ne e heless, he g ea e alues ob ained in each cha o Fig. 8 and Fig. 9 a e no so di e en o he wo c oss-ply con igu a ions, excep o he he mal h ough- he- hickness s ess ha has alues a ound 20 % highe o all he hicknesses a ios. I is impo an o ema k ha he in-plane mechanical s esses σ xx a e always g ea e han hei co esponding Y T alue because he ex e nal mechanical load applied o he lamina e is Y T ; he σ xx di e ence wi h Y T is he e o e he edge e ec componen . 3.2. On he ib e ype The esul s ob ained compa ing he wo ib e ypes a e shown in his sec ion. Fig. 10 shows he esul s associa ed wi h [0/90/0] egula lamina es whe eas Fig. 11 p esen s he esul s o he dis ibu ed c oss- ply con igu a ion. Al hough he edge-e ec s esses a e lowe o he dis ibu ed c oss- ply lamina es, he ends o he edge-e ec consequences o he wo ypes o ib e a e almos iden ical o he wo lamina es s udied, as deduce om he compa ison o Fig. 10 and Fig. 11. Fig. 10.(a) and Fig. 11.(a) show a ele an edge e ec phenomenon when he e a e hick 0◦ply blocks agains ul a- hin 90◦ply blocks o he ca bon/epoxy case. In con as , his phenomenon is no so p o- nounced o he glass/epoxy ma e ial, which p esen s a mode a e in- c ease o hick 0◦ply blocks e sus ul a- hin 90◦ply blocks. Since a ce ain elaxa ion o he esidual he mal s esses has o be aken in o accoun a e he cooling p ocess, as expe imen ally con i med by he au ho s [9], he 90◦ply block would inally bea s esses lowe han Y T o glass ib e whe eas hese s esses may be equal o o highe han Y T o ca bon ib e. Conce ning he dimensionless he mal in-plane s ess ( σ xx / Y T ), Fig. 10.(b) shows almos iden ical su aces o bo h ypes o ib e whe eas some di e ences can be obse ed in Fig. 11.(b). In he case o he ca bon/epoxy ma e ial, he su ace endency changes o hick 90◦ ply blocks agains ul a- hin 0◦ply blocks, being g ea e han o he glass case. I should be no ed ha he edge-e ec s esses a e so low in hese cons ain cases ha he di e ence be ween he wo ma e ials may be conside ed negligible. These he mal di e ences a e ela ed o he di e en coe icien s o he mal expansion (CTE) expe ienced by hese ma e ials in he Fig. 4. F ee-edge consequences conce ning mechanical ou -o -plane s esses o he glass/epoxy ma e ial sys em: (a) G 23 =3200 MPa, and (b) G 23 =4181 MPa. Fig. 5. F ee-edge consequences conce ning he he mal in-plane s esses ( σ xx ) o 0 =340 µm ( a ying 90 ), highligh ing he minimum, he a e age and he maximum alues o he hickes 90◦ply block hickness ( 90 =680 µm, CP-680). S. S´ anchez-Ca mona e al. Composi es Pa A 195 (2025) 108961 5 longi udinal di ec ion o he ib e ( α 1 ), while α 2 and α 3 alues a e almos iden ical o bo h bi-ma e ial sys ems. Mechanical σ zz / Y T , Fig. 10.(c) and Fig. 11.(c), show he same end o each ib e ype used. Whe eas he ou -o -plane s ess is qui e simila o all 90/0 hicknesses a ios in glass ib e, he ca bon ib e shows impo an inc emen s. The lowe he 90 / 0 a io is, he highe he edge- e ec s ess is. The e o e, he ex eme case o a hick 0◦ply block and an ul a- hin 90◦ply block leads o ele an mechanical s esses in he hickness di ec ion ( σ zz ) in he 90◦ply block (~80 %Y T ). A simila end can be seen o he in-plane s ess, Fig. 10.(d) and Fig. 11.(d), al hough he inc ease is less p onounced o lowe 90 / 0 a ios. As commen ed in he i s pa o Sec ion 3, all esul s a e abo e Y T because a displacemen is p esc ibed in he FEM model ha is asso- cia ed wi h an in-plane ans e se no mal s ess in he 90◦ply block o Y T . In he ex eme lowes 90 / 0 case, he e is an inc ease o ~ 40 % along he ee edges o he 90◦ply block. These mechanical ends can be explained by he ac ha he E 11 /E 22 a io is much highe o ca bon ib e (=14.6) han o glass ib e (=3.7). 3.3. Polynomial i o he edge-e ec s esses The esul s p esen ed in he p e ious sec ions (Fig. 8 o Fig. 11) may be use ul o u u e s ess p edic ions needed by designe s and e- sea che s. In o de o acili a e he use o hese esul s, a 4 h-o de polynomial unc ion, eq. (1), is ob ained using he leas squa es me hod and is p oposed o each s ess analysed ( σ ii, i =x,z) o gi e ise o he s esses ha appea in he weakes lamina, he 90◦ply block, along he c oss-ply lamina es’ ee edges o hese wo ma e ial sys ems. These p edic ions could be ob ained o any bi-ma e ial case. σ ii YT ( 0, 90) =p40 04+p31 03 90 +p22 02 902+p13 0 903+p04 904 +p30 03+p21 02 90 +p12 0 902+p03 903+p20 02 +p11 0 90 +p02 902+p10 0+p01 90 +p00 (1) The alues o he polynomial coe icien s equi ed o eq. (1) o each di e en con igu a ion (24 in o al) and hei i s a is ics a e de ailed in Annex 1. An example o one o he mos non-plana su aces is shown in Fig. 12, speci ically he ca bon [0/90/0] case. I is wo h poin ing ou ha he polynomial ha he polynomial i o his case is in pe ec ag eemen wi h he edge-e ec s esses p edic ed by FEM, as shown by i s i s a is ics, he sum o e o squa es and i s R-squa ed (see Annex 1). 4. Edge-e ec s esses as a unc ion o he 90/0 hickness a io The esul s ob ained so a ema k he in luence o he hicknesses o he 0◦and 90◦ply blocks on he edge-e ec s esses. In an a emp o unde ake a deepe analysis, he s ess dis ibu ions a e plo ed di ec ly e sus he 90/0 hickness a io o each c oss-ply con igu a ion, Fig. 13. F om hese esul s, i can be seen ha he e is a clea and s aigh o wa d ela ionship be ween he edge e ec s esses and he hickness a io. Speci ically, 90 / 0 a io is calcula ed as 1/2 o he egula con igu a ion and 2/3 o he dis ibu ed case. Once each dimensionless s ess is cha ed e sus he 90/0 hickness a io, i can be seen ha he e is a a ional unc ion ha ollows he exp ession: Fig. 6. F ee-edge consequences conce ning he mal ou -o -plane s esses ( σ zz ) o dis ibu ed c oss-ply s acking sequence using 0 =680 µm and a ying he hickness o he 90◦ply block om 50 o 680 µm: (a) inne 90◦ply, and (b) ou e 90◦ply. Fig. 7. Scheme o he dis ibu ion o he ee-edge consequences conce ning he mal ou -o -plane s esses ( σ zz ) o he dis ibu ed c oss-ply con igu a ion wi h 0 = 90 =680 µm. S. S´ anchez-Ca mona e al. Composi es Pa A 195 (2025) 108961 6 Fig. 8. Compa ison o he hickness e ec on he a e age edge-e ec s esses (dimensionless alues) o he ca bon/epoxy c oss-ply con igu a ions: (a) he mal σ zz , (b) he mal σ xx , (c) mechanical σ zz , and (d) mechanical σ xx . Fig. 9. Compa ison o he hickness e ec on he a e age edge-e ec s esses (dimensionless alues) o he glass/epoxy c oss-ply con igu a ions: (a) he mal σ zz , (b) he mal σ xx , (c) mechanical σ zz , and (d) mechanical σ xx . S. S´ anchez-Ca mona e al. Composi es Pa A 195 (2025) 108961 7 Fig. 10. Compa ison o he hickness e ec on he a e age edge-e ec s esses (dimensionless alues) o he egula c oss-ply con igu a ion: (a) he mal σ zz , (b) he mal σ xx , (c) mechanical σ zz , and (d) mechanical σ xx .Fig. 11. Compa ison o he hickness e ec on he a e age edge-e ec s esses (dimensionless alues) o he dis ibu ed c oss-ply con igu a ion: (a) he mal σ zz , (b) he mal σ xx , (c) mechanical σ zz , and (d) mechanical σ xx . S. S´ anchez-Ca mona e al. Composi es Pa A 195 (2025) 108961 8 σ ii YT ( 90/ 0) = k a+ 90 0 +b(2) whe e a, b and k a e he i ing pa ame e s adjus ed om he FEM alues ob ained. The pa ame e s alues o each case a e de ailed in Annex 2. Eq. (2) sheds ligh on he in luence o he hickness’s ela ion be- ween 0◦and 90◦plies, whe eas eq. (1) allows esea che s o ha e a good p edic ion o he edge e ec s esses as he 3D s ess su aces ( om Fig. 8 o Fig. 11) a e i ed. I is wo h poin ing ou ha only he s ess alues o he ou e 90◦ ply block a e shown in Fig. 13, as hose o he inne ply block a e qui e simila . Fig. 12. Compa ison o he a e age he mal σ zz along he 90◦ply block o he ca bon/epoxy egula c oss-ply lamina es’ ee edges be ween he FEM p e- dic ion and he polynomial i . Fig. 13. σ ii/Y T (i =x,z) s. 90 / 0 (p edic ed −FEM, and adjus ed −Fi ) o each case (ca bon/glass ib e, egula /dis ibu ed c oss-ply con igu a ion) o : (a) he mal and (b) mechanical ou -o -plane s ess, σ zz , and (c) he mal and (d) mechanical in-plane s ess, σ xx . Fig. 14. The mal ou -o -plane s ess, σ zz , (di ided by Y T ) along he ee edge o a [0/90/0] lamina e made o ca bon/epoxy ma e ial, o 0 =680 µm and 90 = 50, 65, 100 and 510 µm. S. S´ anchez-Ca mona e al. Composi es Pa A 195 (2025) 108961 9