J. Appl. Compu . Mech., 11(2) (2025) 439-450
DOI: 10.22055/jacm.2024.47328.4695
ISSN: 2383-4536
jacm.scu.ac.i
Published online: Augus 28 2024
Shahid Cham an
Uni e si y o Ah az
Jou nal
o
Applied
and
Compu a ional
Mechanics
Resea ch Pape
Nume ical Cha ac e iza ion o he In-Plane Shea Beha iou o
Non-C imp Fab ic Composi es
L.M. Fe ei a
1,2
, E. G aciani
2
, F. Pa ís
2
1 G upo de Elas icidad y Resis encia de Ma e iales. Escuela Técnica Supe io de Ingenie ía, Uni e sidad de Se illa. Camino Descub imien os,
S/N 41092 Se illa, España, Email: lma [email protected]
2 Escuela Poli écnica Supe io , Uni e sidad de Se illa. C/ Vi gen de Á ica, 7, Se illa, 41011, España, Email: eg [email protected] (E.G.); pa [email protected] (F.P.)
Recei ed July 02 2024; Re ised Augus 13 2024; Accep ed o publica ion Augus 13 2024.
Co esponding au ho : L.M. Fe ei a (lma [email protected])
© 2024 Published by Shahid Cham an Uni e si y o Ah az
Abs ac . Expe imen al o -axis ensile es s aimed a cha ac e ising he in-plane shea beha iou o biaxial non-c imp ab ic (NCF)
composi e lamina es ob ained om di e en di ec ions wi hin he same panel (wa p and we ) e eal no ewo hy a ia ions. The
in-plane shea modulus, shea s eng h, and shea s ain a ailu e a e signi ican ly highe in he we di ec ion compa ed o he
wa p di ec ion. To gain insigh s in o he unde lying easons o hese disc epancies, a pa ame ic s udy is pe o med. This nume ical
s udy u ilises mesoscopic 3D ini e elemen (FE) models, ep esen ing he uni cell o a [+45,-45]
2S
NCF lamina e. The analysis
indica es ha nei he he p esence o s i ching ya ns no he ou -o -plane ib e c imp induced by he ya ns accoun o he
subs an ial di e ences obse ed in he expe imen al indings. The maximum ini ial angen in-plane shea modulus (G
xy
), due o
ou -o -plane ib e c imp is only 2%. Howe e , his di e ence inc eases o app oxima ely 6.1% when s i ching ya ns a e also included
in he FE model. Mo eo e , i is ound ha ib e olume ac ion o he ows and he non-linea beha iou negligibly impac he in-
plane shea pe o mance o he NCF lamina e in bo h di ec ions. In con as , he s udy e eals ha he p ima y ac o con ibu ing
o he di e ences be ween he wa p and we di ec ions is ela ed o misalignmen be ween he nominal 45° ows wi hin he NCF
composi e lamina e panels. The misalignmen obse ed in he expe imen ally es ed specimens is app oxima ely 6°, leading o an
8.5% dec ease in he G
xy
o SP lamina e (wa p di ec ion) compa ed o ST lamina e (we di ec ion).
Keywo ds: In-plane shea beha iou ; Composi es; Non-c imp ab ic; Fini e elemen analysis; Fib e c imp.
1. In oduc ion
Non-c imp ab ic (NCF) composi es ep esen a signi ican al e na i e wi hin he ealm o high-pe o mance composi e
applica ions. The de ining ea u e o NCF composi es esides in he a angemen o unidi ec ional ib e ows, which a e no wo en
bu ins ead posi ioned side by side and held in place by s i ching ya ns [1, 2]. The alignmen o ib es pe mi s he cus omisa ion o
mechanical pe o mance, enabling he op imisa ion o s eng h and s i ness in mul iple di ec ions, hus accommoda ing speci ic
load-bea ing equi emen s [3]. Mo eo e , NCF composi es o e se e al ad an ages o e adi ional wo en ab ics, p ima ily due o
hei lowe c imp, which leads o enhanced mechanical p ope ies. The alignmen o ib es wi hou he in e lacing ypically seen
in wo en ab ics educes s ess concen a ions and po en ial weak poin s wi hin he composi e ma e ial. This esul s in be e load
dis ibu ion and highe ensile s eng h, making NCF composi es pa icula ly sui able o applica ions equi ing high pe o mance
unde complex loading condi ions. In he ae ospace and au omo i e indus y, o example, he use o NCF composi es allows o
he c ea ion o ligh weigh componen s, con ibu ing o uel e iciency and o e all pe o mance [4-7]. Simila ly, in ma ine
applica ions, he esis ance o a igue and en i onmen al ac o s. The manu ac u ing p ocess o NCF composi es is ano he c i ical
aspec ha se s hem apa . The use o s i ching ya ns o hold he ib es in place be o e esin in usion allows o be e con ol o e
ib e o ien a ion and placemen , leading o mo e consis en mechanical p ope ies h oughou he ma e ial. This p ocess also
acili a es he p oduc ion o la ge, complex shapes. Despi e hei ad an ages, he manu ac u ing p ocess o NCF composi es
in oduces ce ain challenges, such as he po en ial o in-plane and ou -o -plane ib e c imp and he c ea ion o esin- ich a eas,
all o which can impac he o e all pe o mance o he composi e. The ib e c imp o he ows is in luenced by se e al ac o s,
including he ension in he s i ching ya ns, and he nes ing o he ows du ing layup [8-13]. Addi ionally, esin pocke s o m
be ween he ows du ing manu ac u ing, u he complica ing he in e nal s uc u e [14-16]. Collec i ely, hese mesoscale geome ic
ea u es con ibu e o a complex h ee-dimensional (3D) in e nal a chi ec u e ha signi ican ly impac s he mechanical
pe o mance o NCF composi es.
The goal o his wo k is o add ess he challenges posed by his complex in e nal a chi ec u e in accu a ely modelling NCF
composi es using Fini e Elemen (FE) me hods. Some ea lie ep esen a i e s udies employed a 2D modelling app oach [11, 17-21].
Howe e , he mesoscopic a chi ec u e o hese ma e ials is no ully cap u ed by a 2D FE model [22], as i exhibi s conside able
440
L.M. Fe ei a e al., Vol. 11, No. 2, 2025
Jou nal o Applied and Compu a ional Mechanics, Vol. 11, No. 2, (2025), 439-450
i egula i y along all ma e ial di ec ions, leading o inaccu acies in s ess and s ain p edic ions. Consequen ly, he li e a u e has
seen he de elopmen o se e al 3D FE models designed o eplica e he in ica e in e nal s uc u e o NCF lamina es [14, 23-27].
The modelling s a egy adop ed in his s udy builds upon he app oach in oduced by [24] and u he de eloped in subsequen
wo ks [25, 26, 28], which ocused on p edic ing he comp essi e esponse o NCF composi es h ough nume ical analysis. This
app oach in ol es modelling he ows using a geome ically s aigh 3D FE mesh, wi h ib e c imp accoun ed o by o a ing he
elemen s’ coo dina e sys em o align wi h he ac ual ib e di ec ion. This me hod has been no ed o i s simplici y in modelling he
ow geome y and i s pa ame iza ion [22]. Howe e , i also highligh s he ongoing ension be ween model ideli y and
compu a ional e iciency, a ecu ing heme in he li e a u e [24, 29-31].
This s udy builds on he expe imen al cha ac e isa ion o NCF composi es ha was ca ied ou in [32, 33], whe e di e ences in
he in-plane shea beha iou o [+45,-45]
2S
lamina es we e obse ed du ing o -axis ensile es s in di e en loading di ec ions (wa p
and we ). To unde s and he unde lying causes o hese expe imen ally obse ed di e ences, a pa ame ic s udy was pe o med
using a mesoscopic scale 3D FE model o a [+45,-45]
2S
NCF composi e lamina e. The FE model speci ically conside ed ou -o -plane
ib e c imp induced by non-s uc u al s i ching ya ns, a po en ial ac o in luencing he mechanical esponse o he lamina es [34,
35]. Pa ame e s analysed included he e ec o ib e c imp due o he s i ching, a ia ions in ib e olume ac ion, he non-linea
beha iou o bo h ows and esin, and he o ien a ion o he ows. The s udy sys ema ically explo ed how hese ac o s con ibu e
o he di e ences in mechanical pe o mance be ween lamina es cu in he wa p and we di ec ions. The indings we e hen
compa ed and co ela ed wi h he expe imen al esul s epo ed in [32, 33] o deepen he unde s anding o he s uc u al beha iou
o NCF composi es and o iden i y c i ical ac o s in luencing hei in-plane shea esponse. In his way, he no el y o his s udy
lies in i s ex ended pa ame ic analysis using a mesoscopic-scale 3D ini e elemen model o explo e he impac o se e al di e en
pa ame e s, on he in-plane shea beha iou o NCF composi es, p o iding new insigh s in o he causes o expe imen ally obse ed
di ec ional di e ences in mechanical pe o mance. Addi ionally, his wo k is mo i a ed by he need o b idge he gap be ween he
cu en modelling app oaches and he complex eali y o NCF composi es, aiming o de elop accu a e and compu a ionally e icien
FE models. In his con ex , ou app oach aligns wi h ecen ends in he li e a u e ha seek o enhance model accu acy wi hou
p ohibi i e inc eases in compu a ional cos [31, 36, 37].
The documen is s uc u ed as ollows: Sec ion 2 showcases he expe imen al indings, emphasising he dispa i ies obse ed
in lamina es cu in wo di e en di ec ions. Sec ion 3 de ails he de eloped nume ical models, co e ing he geome ic pa ame e s,
ma e ial p ope ies, bounda y condi ions, and elemen ypes. Sec ion 4 p esen s he esul s om he pa ame ic s udy, examining
he in luence o ac o s such as ib e c imp, non-s uc u al s i ching, ib e olume ac ion o he ows, nonlinea ma e ial beha iou
o he ows, and ows misalignmen . Subsequen ly, Sec ion 5 jux aposes he nume ical esul s wi h he expe imen al e idence.
Finally, he s udy’s main indings a e p esen ed in Sec ion 6.
2. Expe imen al E idence
To cha ac e ise he in-plane shea p ope ies o NCF composi es, expe imen al ensile o -axis es s we e ca ied ou acco ding
o he EN6031 s anda d [38] on [+45,-45]
2S
NCF lamina es as pa o he “Failu e, pe o mance and p ocessing p edic ion o enhanced
design wi h non-c imp ab ic composi es p ojec (FALCOM)” [39]. The layou o he 8-ply s anda d es ed specimens is shown in Fig.
1(a), ea u ing a leng h o 250 mm, a wid h o 25 mm and a hickness o 3 mm, as de ailed in Table 1. As i is possible o obse e,
longi udinal s ain was measu ed using an ex ensome e a ached o he specimen, while ans e se s ain was measu ed wi h a
s ain gauge glued o he specimen.
The specimens we e cu in wo dis inc o ien a ions: one aligned wi h he s i ching ya ns, deno ed as “SP” (wa p di ec ion), and
he o he pe pendicula o he s i ching ya ns, e e ed o as “ST” (we di ec ion). A ep esen a ion o SP and ST specimens is
depic ed in Fig. 1(b), p o iding an illus a ion o he di e ences be ween he wo o ien a ions.
(a) (b)
Fig. 1. (a) Layou o he s anda d es ed specimen, (b) di e ences be ween SP and ST specimens.
Table 1. Speci ica ions o he es ed specimen.
Layup Leng h [mm] Wid h [mm] Thickness [mm] Fib e olume ac ion
𝑉
𝑓
𝑙
[+45,-45]2S 250 25 3 60%
Nume ical Cha ac e iza ion o he In-Plane Shea Beha iou o Non-C imp Fab ic Composi es
441
Jou nal o Applied and Compu a ional Mechanics, Vol. 11, No. 2, (2025), 439-450
Table 2. Expe imen al esul s o Gxy and Sxy wi h s anda d de ia ions, ob ained o di e en NCF lamina e con igu a ions [31, 34].
Layup Specimen Gxy [GPa] Sxy [MPa]
[+45,-45]2S SP 4.95±0.29 62.27±1.25
ST 5.43±0.29 80.54±1.83
This s udy u ilised NCF lamina es manu ac u ed om biaxial Tenax
®
HTS ca bon ib e 5632 12k ab ic, wi h a weigh o 534 g/m
2
.
Resin ilm in usion (RFI) using he HexFlow
®
RTM6 esin sys em was employed [33, 40]. Nons uc u al Sin e ama Ze bion
®
50 d ex
polyes e s i ching ya ns we e added o main ain he ab ic’s s uc u al in eg i y. The NCF lamina es exhibi ed a ib e olume
ac ion o app oxima ely 𝑉
𝑓
𝑙
= 60%.
Gi en ha he specimens we e ex ac ed om a common panel, simila esul s we e expec ed o bo h di ec ions. Howe e , he
ST specimens demons a ed supe io shea modulus, shea s eng h, and shea s ain a ailu e compa ed o hei SP coun e pa s.
Cha ac e is ic shea s ess-s ain cu es ob ained o he o -axis ensile es s in bo h o ien a ions a e p esen ed in Fig. 2 along wi h
he uppe and lowe limi s o he expe imen al da a. Resul s pe aining o ini ial angen in-plane shea modulus G
xy
and in-plane
shea s eng h S
xy
a e ou lined in Table 2. No ice ha he x-axis and y-axis co espond o he ans e se and longi udinal di ec ions
o he specimens, espec i ely, as shown in Fig. 1.
The longi udinal and ans e se s ains we e measu ed in he specimens using s ain gauges posi ioned a 0° (longi udinal) and
90° ( ans e se) di ec ions. In his way, 𝜎
𝑥𝑦
was calcula ed using Eq. (1):
𝜎
𝑥𝑦
=
𝑃
2
𝑤𝑡
(1)
whe e, P ep esen s he applied ensile load, and w and deno e he wid h and hickness o he specimen, espec i ely. S
xy
was
calcula ed conside ing he highes ensile load P
max
sus ained by he specimens du ing es ing. Rega ding G
xy
, i was de e mined by
applying a s anda d linea i ing p ocedu e o he s ess-s ain cu e wi hin he ange o 0.1% o 0.4% shea s ain.
I is possible o obse e ha in he case o G
xy
, he di e ences be ween specimens ST and SP can each a ound 9%. In he case
o S
xy
, he di e ences a e mo e p onounced, eaching app oxima ely 25%. No ewo hy, his mechanical beha iou o he biaxial NCF
specimens, when cu in wo di e en o ien a ions wi hin he same panel, was also epo ed by González e al. in [41].
3. Nume ical Model
A mesoscopic scale 3D FE model was de eloped o he ep esen a i e uni cell (RUC) o a [+45,-45]
s
NCF lamina e using ANSYS
FE code [42]. The RUC comp ises o ou laminas s acked wi h co esponding +45° and -45° o ien a ions, and each lamina ea u es
wo hal - ec angula c oss-sec ion ows wi h esin ich a eas be ween hem, as shown in Fig. 3. Wi h e e ence o he local
coo dina e sys em, he RUC’s plane co esponds o he 12-plane, wi h di ec ions 1 and 2 co esponding o he ib e di ec ion and
in-plane di ec ion no mal o he ib e. Di ec ion 3 ep esen s he h ough- hickness di ec ion o he lamina e. Non-s uc u al
s i ching ya ns, along wi h he ou -o -plane ib e c imp hey induce ac oss he leng h o he ows, we e also inco po a ed in o he
nume ical model. Since a s aigh 3D FE mesh was employed, he ou -o -plane ib e c imp was modelled ollowing he app oach
ou lined in [24] and which has p o en success ul in p e ious s udies [25, 26, 28].
Each colou used in he RUC depic ed in Fig. 3 ep esen s a dis inc ib e o ien a ion angle, deno ed as α. In his s udy, a linea
a ia ion in ib e c imp angle is assumed. Consequen ly, he heo e ical and he app oxima e o a ions a e de e mined based on
he y-coo dina e o he poin . Figu e 3(c) illus a es he a ia ion o he ib e o ien a ion angle α conce ning he maximum c imp
angle β o each column o elemen s along he leng h o he RUC (in he y-di ec ion). I ’s impo an o no e ha he o a ion o he
elemen ’s axis is execu ed abou he x-axis. The modelling app oach used enables he de ini ion o bo h cons an and a iable ou -
o -plane ib e c imp angles, as well as dis inc c imp o ien a ions o each lamina (ei he cu ed upwa ds o downwa ds). Howe e ,
o he pu pose o maximising he impac o ib e c imp, all laminas we e uni o mly cu ed o he same side, cha ac e ised by a
consis en h ough- hickness angle β.
As men ioned be o e, he e a e wo p ima y ac o s con ibu ing o ib e c imp: he p esence o esin pocke s, which may lead
o he nes ing o ows, and he exis ence o s i ching ya ns wi h a de e mina e s i ching ension. I ’s wo h no ing ha p e ious
s udies p edominan ly ocused on ou -o -plane ib e c imp esul ing om he p esence o esin pocke s [24–26, 28]. Howe e , in his
s udy, his ib e c imp is o e looked, and he ocus is on examining he “s i ching ya ns” ib e c imp.
Fig. 2. Typical shea s ess-s ain cu es ob ained om he o -axis ensile es s [31, 34].
442
L.M. Fe ei a e al., Vol. 11, No. 2, 2025
Jou nal o Applied and Compu a ional Mechanics, Vol. 11, No. 2, (2025), 439-450
Fig. 3. 3D FE model o he RUC o a [+45,-45]s NCF lamina e: (a) Shape o he ows, (b) comple e RUC, including s i ching and ou -o -plane ib e c imp,
(c) geome ic pa ame e s o he comple e RUC illus a ed h ough de ailed d awing iews.
The geome y and mechanical p ope ies o he cons i uen s o he laminas we e pa ame ically de ined, enabling he s udy o
he e ec o dimensions and mechanical p ope ies on he in-plane shea beha iou o he NCF lamina e. Figu e 3(c) depic s he
geome ical pa ame e s employed, wi h a ep esen ing he leng h o he RUC, deno ing he hickness o each lamina, and g
indica ing he wid h o he gap be ween wo adjacen ows o each lamina. In his s udy, he nume ical alues o he geome ic
pa ame e s we e de e mined o a 𝑉
𝑓
𝑙
= 60%, and hey we e es ima ed based on he in e nal geome y and ib e con en o he
ma e ials analysed in [33]. Consequen ly, he speci ic pa ame e alues we e ob ained: a = 3.67 mm, = 0.24 mm and g = 0.26 mm.
3.1. Ma e ial P ope ies
The ows we e modelled as a homogeneous ans e sely iso opic elas ic ma e ial wi h bilinea shea beha iou . This non-
linea i y mani es s in he longi udinal and ans e se h ough- hickness planes, speci ically he 12-plane and 13-plane, whe e he
esponse o he ows o shea loads is in luenced by he non-linea beha iou o he esin [11,17]. To inco po a e his non-linea i y
in o he nume ical model, a bilinea 𝜎
12
/𝛾
12
ela ionship was in oduced. The s ess-s ain cu e by Di che [43] was used as a
e e ence. The ansi ion om he ini ial slope o he second slope occu s a 𝜎
12
∗
= 𝜎
13
∗
= 60 MPa and 𝛾
12
∗
= 1.4%. Acco dingly, he
ollowing alues we e assigned o he ini ial slope 𝐺
12
𝑖
= 𝐺
13
𝑖
= 4.03 GPa and second slope 𝐺
12
𝑖𝑖
= 𝐺
13
𝑖𝑖
= 0.75 GPa o he shea moduli.
Con e sely, he esin- ich a eas we e ea ed as a homogeneous iso opic ma e ial. The elas ic cons an s o he esin- ich a eas (𝐸
𝑟
,
𝜈
𝑟
and 𝐺
𝑟
) and o he s i ching ya ns (𝐸
𝑠
, 𝜈
𝑠
and 𝐺
𝑠
) we e ob ained om [44, 45]. The mechanical p ope ies employed in he
pa ame ic a e summa ised in Table 3, along wi h he co esponding sou ce e e ences.
3.2. Bounda y Condi ions and Elemen Types
The applied bounda y condi ions on he aces o he RUC aim o eplica e he ensile es in bo h he SP and ST di ec ions. Fo
he case in which he load is applied in he SP di ec ion, a symme y condi ion was imposed on one o he aces pa allel o he yz-
plane (𝑢
𝑥
= 0, and 𝜎
𝑥𝑦
= 𝜎
𝑥𝑧
= 0), and a pu e longi udinal ensile s ain was applied along he x-axis on he opposi e ace (𝜀
𝑥
= 3%,
and 𝜎
𝑥𝑦
= 𝜎
𝑥𝑧
= 0). When he load is applied in he ST di ec ion, a symme y condi ion was assumed on one o he aces pa allel o
he xz-plane (𝑢
𝑦
= 0, and 𝜎
𝑥𝑦
= 𝜎
𝑥𝑧
= 0), and a pu e longi udinal ensile s ain (along he y-axis) was applied on he opposi e ace (𝜀
𝑦
= 3%, and 𝜎
𝑥𝑦
= 𝜎
𝑥𝑧
= 0). In bo h cases coupling bounda y condi ions we e applied in he emaining aces and poin suppo s we e
used o a oid igid body mo ion. In his way, he in-plane shea s ain 𝛾
𝑥𝑦
o he SP and ST lamina es was calcula ed as he
di e ence be ween he longi udinal s ain and he ans e se s ain.
Table 3. Mechanical p ope ies assigned o he ib e ows, esin- ich a eas and s i ching ya ns.
Mechanical p ope y Uni s Value
Fib e ows [24, 26]
𝐸
11
𝑡
GPa 167.6
𝐸
22
𝑡
=
𝐸
33
𝑡
GPa 11.44
𝜈
12
𝑡
=
𝜈
13
𝑡
- 0.3
𝜈
23
𝑡
- 0.42
𝜎
12
∗
=
𝜎
13
∗
MPa 60
𝐺
12
𝑖
=
𝐺
13
𝑖
GPa 4.03
𝐺
12
𝑖𝑖
=
𝐺
13
𝑖𝑖
GPa 0.75
𝐺
23
𝑡
GPa 4.03
Resin- ich a eas [44, 45]
𝐸
𝑟
GPa 3.5
𝜈
𝑟
- 0.42
𝐺
𝑟
GPa 1.23
S i ching ya ns [44, 45]
𝐸
𝑠
GPa 61
𝜈
𝑠
- 0.29
𝐺
𝑠
GPa 2.9
Nume ical Cha ac e iza ion o he In-Plane Shea Beha iou o Non-C imp Fab ic Composi es
443
Jou nal o Applied and Compu a ional Mechanics, Vol. 11, No. 2, (2025), 439-450
The ows and he esin ich a eas we e modelled u ilising linea solid elemen s (SOLID45) wi h eigh nodes and h ee deg ees
o eedom a each node ( ansla ions in he di ec ions 1, 2 and 3). The non-s uc u al s i ching was implemen ed using 3D spa
elemen s (LINK180). This elemen wo ks as a uniaxial ension-comp ession elemen , p o iding h ee deg ees o eedom a each
node: ansla ion in he nodal x, y, and z di ec ions. This app oach was employed in a ious s udies using s i ched composi es [46-
49]. The con igu a ion o each NCF lamina, deno ed as [+45,-45], equi es he duplica ion o s i ching ya ns in he middle o he
lamina e hickness. As a esul , he c oss-sec ional a ea o he spa elemen s be ween wo consecu i e -45° laminas is wice ha o
he emaining elemen s. I is no ewo hy ha a ious s i ching pa e ns we e es ed, bu no signi ican changes in he esul s we e
obse ed. As indica ion, he comple e nume ical model is composed o 65,600 elemen s and 68,901 nodes.
4. Resul s o he Pa ame ic S udy
A pa ame ic s udy was conduc ed o examine he impac o a ious pa ame e s on he in-plane shea beha iou o a [+45,
-45]
n
NCF lamina e. Pa ame e s unde analysis encompassed ou -o -plane ib e c imp, non-s uc u al s i ching, he non-linea
beha iou o he ows, lamina ib e olume ac ion, and ows misalignmen . The goal was o iden i y po en ial ac o s con ibu ing
o he obse ed di e ences in in-plane shea pe o mance be ween he SP and ST lamina es, as indica ed in he expe imen al
e idence. Fo his pu pose, he nume ically p edic ed alues o he in-plane shea s ess 𝜎
𝑥𝑦
a e plo ed agains he in-plane shea
s ain 𝛾
𝑥𝑦
. Conside ing he geome ic pa ame e s de ined in he nume ical model depic ed in Fig. 3, 𝜎
𝑥𝑦
was compu ed using Eq. (2).
No ice ha he di e ence be ween Eqs. (1) and (2) a ises because he o al hickness in he nume ical model is de ined as 4 , as
shown in Fig. 3(c), whe eas in he expe imen al samples, i co esponds o he o al lamina e hickness.
𝜎
𝑥𝑦
=
𝑃
8
𝑎𝑡
(2)
4.1. E ec o he Ou -o -Plane Fib e C imp
The modelling app oach employed o he ou -o -plane ib e c imp enables he speci ica ion o ei he cons an o a iable ou -
o -plane angles, while also allowing dis inc c imp o ien a ions o each lamina, as illus a ed schema ically in Fig. 4. Ne e heless,
i was ound ha hese a ia ions exe ed a negligible in luence on he nume ical p edic ions. In his con ex , he same ib e c imp
o ien a ion and angle β was assumed o all laminas.
To assess he e ec o β on esul s in bo h SP and ST lamina es, a ious c imp angles ( anging om 0° o 45°) we e conside ed.
Fo cla i y, only he esul s ob ained wi hou s i ching ya ns, and wi h ib e c imp angles o β = 0°, 15°, and 45° a e p esen ed.
Al hough β = 45° may be conside ed a high c imp alue, i was chosen as a e e ence o clea ly disce n i s e ec on esul s be ween
SP and ST lamina es.
Figu e 5 depic s he in-plane shea s ess 𝜎
𝑥𝑦
plo ed agains he in-plane shea s ain 𝛾
𝑥𝑦
. The indings e eal ha , unless
un ealis ic high ib e c imp angles a e conside ed, almos no di e ences can be app ecia ed in he in-plane shea pe o mance
be ween SP and ST lamina es. This is e iden in Fig. 5, whe e he esul s ob ained wi h β = 15° closely esemble hose wi h pe ec ly
s aigh ows, and some di e ences only eme ge be ween SP and ST o β = 45°. The esul s indica e ha his dispa i y inc eases
upon eaching he second pa o he bilinea shea cons i u i e equa ion o he ows. Addi ionally, i is obse ed ha bo h
o ien a ions demons a e almos iden ical ini ial angen in-plane shea modulus 𝐺
𝑥𝑦
. Speci ically, o β = 45°, he 𝐺
𝑥𝑦
alue o ST
lamina es is app oxima ely 2% highe compa ed o SP lamina es.
While he nume ical p edic ions sugges enhanced in-plane shea pe o mance in ST lamina es, aligning wi h expe imen al
obse a ions, i can be concluded ha inco po a ing ou -o -plane ib e c imp has a ma ginal e ec on he o e all in-plane shea
pe o mance o bo h SP and ST lamina es. No iceably, Yin e al. [14] simila ly obse ed he negligible in luence o ib e c imp on he
in-plane shea s eng h o NCF composi es. Thei analysis employed 3D mesoscale FE models inco po a ing ow wa iness angles
anging om 1° o 3°.
4.2. E ec o he Non-S uc u al S i ching
The analysis in his sec ion aims o assess he e ec o non-s uc u al s i ching ya ns on he obse ed expe imen al di e ences
be ween SP and ST lamina es. Fo his pu pose, assuming β = 45°, he in-plane shea s ess-s ain cu es ob ained wi h and wi hou
s i ching ya ns a e compa ed in Fig. 6. In e ms o modelling, he inco po a ion o non-s uc u al s i ching ya ns is accomplished
h ough he u ilisa ion o 3D spa elemen s, speci ically he LINK180, as de ailed in sec ion 3.2. Consequen ly, he FE model wi hou
s i ching ya ns does no include hese elemen s. I is no ewo hy ha despi e conside ing la ge s i ching ya n a eas and di e en
s i ching pa e ns, no signi ican impac was obse ed on he nume ical p edic ions.
The in oduc ion o s i ching ya ns con ibu es o a sligh inc ease in he s i ness o he NCF lamina e, pa icula ly o ST.
Consequen ly, he ini ial angen in-plane shea modulus o ST wi h s i ching ya ns is abou 6.5% highe han wi hou s i ching
ya ns, while o SP, his di e ence is app oxima ely 2.2%. Mo eo e , he p esence o s i ching ya ns con ibu es o an inc ease in
he di e ences ound in 𝐺
𝑥𝑦
be ween SP and ST. Speci ically, wi h he inclusion o s i ching ya ns, he maximum di e ence obse ed
is abou 6.1%, whe eas wi hou s i ching ya ns is 2%. These indings sugges ha he inco po a ion o non-s uc u al s i ching in
he nume ical models, coupled wi h ou -o -plane ib e c imp, in ensi ies he obse ed di e ences in in-plane shea pe o mance
be ween SP and ST lamina es.
Fig. 4. Schema ic ep esen a ion o he ou -o -plane c imp o ien a ions conside ed o he NCF laminas: (a) cons an ou -o -plane ib e c imp wi h
iden ical o ien a ion, (b) cons an ou -o -plane ib e c imp wi h dis inc o ien a ions, (c) a iable ou -o -plane ib e c imp wi h dis inc o ien a ions.
444
L.M. Fe ei a e al., Vol. 11, No. 2, 2025
Jou nal o Applied and Compu a ional Mechanics, Vol. 11, No. 2, (2025), 439-450
Fig. 5. In-plane shea s ess 𝜎𝑥𝑦 s. he in-plane shea s ain 𝛾𝑥𝑦 o SP and ST wi h a maximum c imp angle β = 45°, 15° and 0°.
Fig. 6. In-plane shea s ess 𝜎𝑥𝑦 s. he in-plane shea s ain 𝛾𝑥𝑦 o SP and ST wi h and wi hou s i ching ya ns and conside ing β = 45°.
Fig. 7. In-plane shea s ess 𝜎𝑥𝑦 s. he in-plane shea s ain 𝛾𝑥𝑦 o SP and ST wi h 𝑉𝑓
𝑙 = 60%, 65% and 70%, conside ing β = 45° and s i ching ya ns.
4.3. E ec o he Fib e Volume F ac ion
Lamina ib e olume ac ions o 𝑉
𝑓
𝑙
= 60%, 65% and 70% we e conside ed o assess hei impac on he in-plane shea beha iou
o SP and ST lamina es. Fo his pu pose, he RUC wi h a ib e c imp angle β = 45° and inco po a ing s i ching ya ns was u ilised.
In Fig. 7, he in-plane shea s ess 𝜎
𝑥𝑦
is plo ed e sus he in-plane shea s ain 𝛾
𝑥𝑦
o he di e en lamina ib e olume
ac ions. As could be expec ed, inc easing he lamina ib e olume ac ion 𝑉
𝑓
𝑙
enhances he ini ial angen inplane shea modulus
𝐺
𝑥𝑦
o bo h SP and ST lamina es. Fo example, aising 𝑉
𝑓
𝑙
om 60% o 65% esul s in a 16% inc ease in 𝐺
𝑥𝑦
. Howe e , al e ing he
ib e olume ac ion o he lamina does no accoun o he expe imen al disc epancies be ween he wo lamina es. In ac , an
inc ease in 𝑉
𝑓
𝑙
educes he di e ences ound be ween ST and SP. Fo example, inc easing 𝑉
𝑓
𝑙
o 65% and 70% esul s in di e ences
o app oxima ely 5.3% and 4.6%, espec i ely.
Nume ical Cha ac e iza ion o he In-Plane Shea Beha iou o Non-C imp Fab ic Composi es
445
Jou nal o Applied and Compu a ional Mechanics, Vol. 11, No. 2, (2025), 439-450
Table 4. Pa ame e s used o analyse he e ec o he in-plane shea moduli 𝐺12
𝑖𝑖 = 𝐺13
𝑖𝑖 .
𝜎
12
∗
=
𝜎
13
∗
(MPa) Fi s Slope
𝐺
12
𝑖
=
𝐺
13
𝑖
(GPa)
Second Slope
𝐺
12
𝑖𝑖
=
𝐺
13
𝑖𝑖
(GPa)
60 4.03
0.10
0.75
1
Table 5. Pa ame e s used o analyse he e ec o he in-plane shea s esses 𝜎12
∗= 𝜎13
∗.
𝜎
12
∗
=
𝜎
13
∗
(MPa) Fi s Slope
𝐺
12
𝑖
=
𝐺
13
𝑖
(GPa)
Second Slope
𝐺
12
𝑖𝑖
=
𝐺
13
𝑖𝑖
(GPa)
40
4.03 0.75 60
80
4.4. E ec o he Non-Linea Beha iou o he Tows
To assess he e ec o he non-linea shea beha iou o he ows, h ee di e en alues o he second slope o he in-plane
shea moduli 𝐺
12
𝑖𝑖
= 𝐺
13
𝑖𝑖
we e analysed. Addi ionally, h ee dis inc alues o 𝜎
12
∗
= 𝜎
13
∗
, co esponding o he in-plane shea s ess
alues a which he bilinea ma e ial cu e ansi ions o i s second slope.
I is impo an o no e ha , o all con igu a ions, he
ini ial slope o he in-plane shea moduli was kep cons an . The nume ical models also accoun ed o he p esence o he s i ching
ya ns and a ib e c imp o β = 45°. The pa ame e s used in his s udy a e de ailed in Tables 4 and 5.
The e ec o a ying he second slope o he in-plane shea moduli on he on he in-plane shea s ess-s ain cu es is
illus a ed in Fig. 8(a). As p e iously no ed, he ini ial slope emains cons an ac oss all con igu a ions. Mo eo e , he pe cen age
di e ence be ween he cu es o ST and SP lamina es emains la gely unchanged, wi h a consis en alue a ound 6%.
The in luence o he s ess alue a which he cu e ansi ions o he second slope is shown in Fig. 8(b). The esul s clea ly
demons a e ha choosing di e en s ess alues a ec he lamina es esponse, bu do no con ibu e o he obse ed di e ences
in he in-plane shea pe o mance o SP and ST lamina es. The di e ence be ween he in-plane shea moduli o ST and SP lamina es
emains consis en h oughou he en i e cu e, wi h ST alues consis en ly abou 6% highe han hose o SP. O e all, he
pa ame ic s udy e eals ha al e ing he non-linea beha iou o he ows has no signi ican impac on he in-plane shea
pe o mance o SP and ST lamina es.
4.5. E ec o he Misalignmen o he Tows
Va ious phases in he manu ac u ing p ocess o NCF panels can impac he o ien a ion o he ib e ows, leading o a ce ain
deg ee o o a ion o misalignmen conce ning he p ojec ed o ien a ion. This phenomenon is e iden in he C-Scan images o he
NCF panels es ed in [50], as illus a ed in Fig. 9.
Upon isual inspec ion o he panels, a misalignmen be ween he nominal di ec ion o he ows was obse ed, as shown in Fig.
10. The p e ailing misalignmen was a ound 6° and displayed a nea -symme y wi h espec o he di ec ion o he s i ching ya ns.
This misalignmen in oduced a de ia ion om heo e ical uni o mi y in he lamina es. The smalles angle be ween he ows
consis en ly appea ed in he loading di ec ion o he ST lamina es, esul ing in he highes angle in he loading di ec ion o he SP
lamina es, as ep esen ed in Fig. 11.
By applying he Classical Lamina e Theo y (CLT) [51, 52], i becomes possible o assess he impac o misalignmen o he ows
on 𝐺
𝑥𝑦
. Fo ins ance, i he misalignmen o 6° is aken in o accoun , co esponding o a SP lamina e wi h a s acking sequence [+42,
-42]
S
and a ST lamina e wi h [+48, -48]
S
as shown in Fig. 11, 𝐺
𝑥𝑦
inc eases by 12% o ST and dec eases by 7% o SP when compa ed
o pe ec ly aligned ows, i.e., SP = ST = [+45, -45]
S
. The esul s om he CLT unde sco e he impac o misalignmen on he in-plane
shea modulus. Ne e heless, i is essen ial o no e ha hese esul s do no conside he ou -o -plane ib e c imp o he p esence
o s i ching ya ns, bo h o which, as p e iously demons a ed, can also sligh ly con ibu e o he obse ed di e ences be ween he
wo lamina es.
(a) (b)
Fig. 8. In-plane shea s ess
e sus he in-plane shea s ain
o SP and ST conside ing β = 45° and s i ching ya ns:
(a) 𝐺12
𝑖𝑖 = 𝐺13
𝑖𝑖 = 0.1 GPa, 0.75 GPa and 1 GPa, (b) 𝜎12
∗ = 𝜎13
∗ = 40 MPa, 60 MPa and 80 MPa.
446
L.M. Fe ei a e al., Vol. 11, No. 2, 2025
Jou nal o Applied and Compu a ional Mechanics, Vol. 11, No. 2, (2025), 439-450
Fig. 9. C-Scan images o NCF lamina e panels [50].
Fig. 10. Misalignmen o ows ound in SP and ST lamina es.
Fig. 11. Schema ic ep esen a ion o a misalignmen o 6° be ween he ows in SP and ST lamina es.
To p edic how he misalignmen angle o he ows in luences in-plane shea pe o mance, new 3D FE models o he RUC we e
gene a ed wi h misalignmen angles o 2°, 4°, and 6°. As a esul , he FE models we e modi ied o adop a ec angula shape o
ep esen he angle o misalignmen . The e o e, he wid h a used in Eq. (2) o calcula e 𝜎
𝑥𝑦
was adjus ed o i he ec angula shape
o he RUC. Figu e 12 illus a es he new 3D FE model, inco po a ing he ele an geome ic pa ame e s, wi h a misalignmen o 6°.
This con igu a ion co esponds o an SP lamina e wi h a s acking sequence o [+42, -42]
S
and an ST lamina e wi h [+48, -48]
S
, as
shown in Fig. 11.
To acili a e he unde s anding o he nume ically p edic ed in-plane s ess s ain cu es, Fig. 13 only showcases he cu es
co esponding o 2° and 6°. The esul s unde sco e he impac o ows misalignmen on he in-plane shea pe o mance o bo h ST
and SP lamina es. Fo example, when an angle o 2° is conside ed, he in-plane shea modulus (𝐺
𝑥𝑦
) o SP expe iences an 8.5%
dec ease ela i e o ST. This pe cen age change becomes mo e p onounced wi h inc easing angles, eaching app oxima ely 11.8%
and 15.1% o 4° and 6°, espec i ely. I is impo an o highligh ha hese esul s we e de i ed unde he condi ions o an ou -o -
plane ib e c imp angle o 45° and he inclusion o s i ching ya ns. No ably, when hese ac o s we e no conside ed, he p edic ions
exhibi ed less p onounced e ec s.
Nume ical Cha ac e iza ion o he In-Plane Shea Beha iou o Non-C imp Fab ic Composi es
447
Jou nal o Applied and Compu a ional Mechanics, Vol. 11, No. 2, (2025), 439-450
Fig. 12. Geome ic pa ame e s o he 3D FE model o he RUC gene a ed wi h a misalignmen angle o 6° be ween he ows, including ou -o -plane
ib e c imp and s i ching ya ns.
Fig. 13. In-plane shea s ess 𝜎𝑥𝑦 e sus he in-plane shea s ain 𝛾𝑥𝑦 o SP and ST wi h misalignmen angles o 2° and 6°, conside ing β = 45° and
s i ching ya ns.
Fig. 14. Nume ical-expe imen al co ela ion o he In-plane shea s ess 𝜎𝑥𝑦 e sus he in-plane shea s ain 𝛾𝑥𝑦.
Table 6. Pa ame e s used o adjus he nume ical p edic ions o he expe imen al e idence.
𝜎
12
∗
=
𝜎
13
∗
(MPa) Fi s Slope
𝐺
12
𝑖
=
𝐺
13
𝑖
(GPa)
Second Slope
𝐺
12
𝑖𝑖
=
𝐺
13
𝑖𝑖
(GPa)
55 5.57 0.45
