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
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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
