THE ON SET OF MIXED-MODE INTRALAMINAR CRACK ING

IN A CROSS-PLY COMPOSITE LAM I NATE

J. Andersons,* E. Spârniòð,*

and R. Joffe**

Keywords: poly mer-ma trix com pos ites, lam i nates, intralaminar crack ing, frac ture tough ness, mixed-mode

load ing

The intralaminar frac ture tough ness of a unidirectionally re in forced glass/ep oxy com pos ite is de ter mined ex -

per i men tally at sev eral mode I and mode II load ing ra tios. The crack prop a ga tion cri te rion, ex pressed as a

qua dratic form in terms of sin gle-mode stress in ten sity fac tors (al ter na tively, lin ear in terms of en ergy re lease

rates), ap prox i mates the test re sults rea son ably well. The mixed-mode crack ing cri te rion ob tained is used to

pre dict the intralaminar crack on set in a cross-ply glass/ep oxy com pos ite un der off-axis ten sile load ing.

1. In tro duc tion

The fail ure pro cess of con tin u ous fi ber-re in forced com pos ite lam i nates in ten sion usu ally com mences with the for ma -

tion of intralaminar cracks. The cracks run along the fi bers of the plies sub jected to trans verse ten sile stresses. A par tic u lar case

of crack ing of trans verse plies in a cross-ply lam i nate has al ready been stud ied in de tail, see, e.g., the re cent re views [1-3]. The

com pos ite lay-ups typ i cally con sid ered are of the type [0n/90m]s, in which, when loaded in ten sion along, e.g., the re in force -

ment di rec tion of the outer plies, cracks nor mal to the load ing di rec tion de velop in the in ner trans verse plies. The ini ti a tion of

the trans verse cracks is gov erned by the ply stress, whereas their prop a ga tion is de ter mined by the mode I en ergy re lease rate

(ERR) in the plies [4, 5]. There fore, a strength cri te rion of fail ure is ap plied to thick trans verse plies, where the crack ing is ini ti -

a tion-con trolled, and the crit i cal ERR cri te rion is used for thin trans verse plies, where crack prop a ga tion con trols the crack ing

[6-10].

In com pos ite lam i nates with more com plex lay-ups and/or un der com bined load ing, intralaminar cracks may de velop

in plies with dif fer ent re in force ment di rec tions. A re view of ex per i men tal and mod el ing ac tiv i ties re lated to the off-axis ply

dam age in lam i nates is pre sented in [11]. Mod els for the stiff ness re duc tion of a lam i nate with a com plex crack sys tem have re -

cently been de rived [12-19] and ver i fied [20-23]. The de vel op ment of intralaminar crack ing of a unidirectionally re in forced

(UD) ply un der a com plex stress state, com pris ing the in-plane ten sile and shear stresses, in a com pos ite lam i nate is con sid ered

in [12, 15, 16, 24-27]. An in dis pens able part of mixed-mode crack ing mod els is the cri te rion of crack ing, usu ally for mu lated in

terms of mode I and II ERRs, GI and GII , and the cor re spond ing crit i cal ERRs, GIc and GIIc . A num ber of such em pir i cal cri te -

5490191-5665/08/4406-0549 © 2008 Springer Sci ence+Busi ness Me dia, Inc.

*In sti tute of Poly mer Me chan ics, Uni ver sity of Lat via, 23 Aizkraukles St., LV-1006, Riga, Lat via. **Di vi sion of

Poly mer En gi neer ing, Luleå Uni ver sity of Tech nol ogy, SE-971 87 Luleå, Swe den. Trans lated from Mekhanika Kompozitnykh

Materialov, Vol. 44, No. 6, pp. 785-794, No vem ber-De cem ber, 2008. Orig i nal ar ti cle sub mit ted April 29, 2008.

Me chanics of Com pos ite Ma te rials, Vol. 44, No. 6, 2008

ria have been pro posed, for both the inter- and the intralaminar cracking, see, e.g., [28] and the references therein. Specifically,

the criterion

G

G

G

G

N MI

Ic

II

IIc

æ

èçç

ö

ø÷÷ +

æ

èçç

ö

ø÷÷ = 1 (1)

with N = 1 and M = 2 has been ap plied in [25-27] to pre dict the crack on set in [ ]0 q s lam i nates. Us ing typ i cal crit i cal ERR val -

ues, a good agree ment with the test re sults re ported in [29] was ob tained for the ply ori en ta tion an gles 75° £ £q 90°, while for

45° £ £q 75° the model pre dic tion mark edly un der es ti mated the crack on set strain (COS).

In the cur rent study, the frac ture tough ness of a UD glass-ep oxy com pos ite is de ter mined ex per i men tally for mode I

and mixed-mode load ing con di tions, and a mixed-mode frac ture cri te rion is es tab lished. The lat ter is used to pre dict the COS of

a cross-ply lam i nate un der off-axis ten sion, the test re sults for which have been reported in [30].

2. Ex per i men tal

The ma te rial tested was an E-glass fiber/ep oxy ma trix com pos ite. UD plates were pre pared by hand lay-up from a UD

prepreg VICOTEX NVE 913/28%/192/EC9756. The cur ing cy cle com prised 30 min at 90°C fol lowed by 60 min at 120°C un -

der a 3- to 7-bar pres sure. Rect an gu lar spec i mens of length 220-240 mm and width 22-26 mm were cut from the plates un der

dif fer ent an gles to the ply re in force ment di rec tion. Thus, spec i mens with lay-ups [208], [308], [458], and [908] were ob tained.

The spec i men thick ness was about 1 mm, and they were pro vided with GFRP end tabs, leav ing a 120 to 140-mm-long test

section.

A precrack was made in each spec i men ac cord ing to the pro ce dure de scribed be low. First, a small-di am e ter hole was

drilled in the mid dle of the spec i men by us ing a 0.6-mm drill. Then, a min ia ture saw blade 0.2 mm thick was in serted in the

hole, and a saw cut was made along the fi bers in both di rec tions from the hole for about 2.5 mm. As a re sult, a precrack of about

5 mm length, ori ented in the fi ber di rec tion, was ob tained in the cen tral part of each spec i men. A sche matic pic ture of the

specimens is shown in Fig. 1.

The spec i mens were sub jected to ten sile load ing by us ing a Zwick/Roell Z2.5 test ing ma chine. The load was mea sured

by a 2.5-kN load cell. All the spec i mens were tested up to fail ure, with stroke con trol, at a 2-mm/min rate. In all cases, the fail -

ure oc curred due to the un sta ble prop a ga tion of the precrack along fibers.

550

2H

2L

1

2

Fig. 1. Sche matic of a UD spec i men with a precrack par al lel to the fi ber di rec tion (1) and end

tabs (2).

3. Frac ture Tough ness un der Mixed-Mode Load ing

In or der to eval u ate the frac ture tough ness of a UD com pos ite, the ap plied stress at the on set of crack prop a ga tion has

to be re lated to the mode I and mode II stress in ten sity fac tors (SIFs) of the crack. It has been shown in [31] that, for prob lems

in volv ing self-equilibrating loads, the SIFs for anisotropic ma te ri als are iden ti cal to those of iso tro pic ones. Hence, the SIFs for

a crack of length 2a in an in fi nite anisotropic plate sub jected to ten sion by a stress s are

K aI = s p qsin 2 , K aII = s p q qsin cos , (2)

where q is the an gle be tween the crack line and the load ing axis. The ef fect of the fi nite size of test spec i mens ne ces si tates cor -

rec tions to the SIF val ues pro vided by Eq. (2). Such cor rec tions have been ob tained by a nu mer i cal anal y sis in [32] for UD

glass/ep oxy and car bon/ep oxy com pos ites hav ing a crack par al lel to the re in force ment. They are pre sented as func tions of the

re in force ment an gle q, the crack length to spec i men width ra tio a W, and the spec i men length to width ra tio H W, both for

mode I and mode II SIFs. The lim ited spec i men size leads to higher SIFs com pared with those for an in fi nite plate.

The geo met ri cal fac tors in the pres ent case amount to H W » 5 and a W » 0.2. The re spec tive cor rec tions do not ex -

ceed a few per cent for q in the range of 30 to 90° con sid ered in [32]. In the fol low ing, we will ne glect the size ef fect, as its in flu -

ence is small com pared with the scat ter of ex per i men tal data, and eval u ate the SIFs by Eq. (2). The SIF val ues at the on set of

crack growth are plot ted in Fig. 2. They ap pear to com ply with the qua dratic criterion for crack propagation

K

K

K

KI

Ic

II

IIc

æ

èçç

ö

ø÷÷ +

æ

èçç

ö

ø÷÷ =

2 2

1. (3)

[908] com pos ite tests yielded K Ic = 2.1 ± 0.2 MPa ×m1 2 . Ap prox i mat ing the data in Fig. 2 by Eq. (3), we ob tained K IIc = 3.6

MPa ×m1 2 .

For a crack ly ing in the sym me try plane of an orthotropic ma te rial, the ba sic modes are in de pend ent [31]. Hence, the

mode I and mode II ERRs de pend only on the re spec tive SIFs:

G Ka a a

a

a a

aI I

=æ

èç

ö

ø÷

æ

èçç

ö

ø÷÷ +

+2 11 221 2

22

11

1 212 66

112

2

2

é

ë

êê

ù

û

úú

1 2

,

(4)

G Ka a

a

a a

aII II

=æ

èçç

ö

ø÷÷ +

+é

ë

êê

ù

û

úú

2 11 22

11

1 212 66

11

1

2

2

2

2

,

where aij are the com pli ance ma trix el e ments of the UD com pos ite (with axis 1 along the crack, i.e., in the fi ber di rec tion, see

Fig. 1). Crack prop a ga tion cri te rion Eq. (3), ex pressed in terms of ERRs, takes the form

G

G

G

GI

Ic

II

IIc

+ = 1, (5)

and the crit i cal ERRs amount to GIc = 300 ± 56 J/m2 and GIIc = 500 J/m2.

551

4. Crack On set in Cross-Ply Lam i nates un der Off-Axis Load ings

Hav ing es tab lished the mixed-mode crack ing cri te rion for the UD glass/ep oxy com pos ite, we pro ceed by ap ply ing it

to the crack on set pre dic tion in a cross-ply glass/ep oxy lam i nate un der uni ax ial ten sile load ings at dif fer ent an gles to the ma te -

rial orthotropy axis. To this end, we ex press the mode I and II steady-state ERRs via the cor re spond ing crack face dis place -

ments and the far-field ply stresses and re late the stresses to the ap plied load by us ing the classical laminate theory.

The intralaminar crack ing of [02/902]s, [152/–752]s, [302/–602]s, and [452/–452]s lam i nates un der uni ax ial ten sion is

re ported in [30]. The off-axis lay-ups cor re spond to the [02/902]s cross-ply lam i nate ro tated by 15, 30, and 45°, re spec tively.

The cracks ap pear in the in ner plies of the lam i nates with the first three lay-ups and in all plies of the [452/–452]s lam i nate. The

cracks in the plies run along the orthotropy axis of the lam i nate, there fore, the load ing modes are in de pend ent. Know ing the

crack open ing dis place ment at a given stress, the ERR can be eas ily eval u ated via the work of crack closure (see, e.g., [33]).

Spe cif i cally, the mode I ERR for a trans verse crack in a cross-ply lam i nate is de ter mined as

Ghu

EI =

2 22

2

s, (6)

where h is the thick ness of the trans verse ply, s2 des ig nates the un per turbed stress in the ply nor mal to the crack plane, E2 is

the trans verse Young’s modulus of the UD ply, and u2 is the nor mal ized av er age crack open ing dis place ment, as de fined in

[17]. The nondimensional dis place ment u2 is eval u ated by an ap prox i mate re la tion based on a se ries of FEM cal cu la tions and

has the form [17]

u A BE

E

n

2 2 22

1

2

= +æ

èçç

ö

ø÷÷ . (7)

For a crack in the in ner trans verse layer of a sym met ric cross-ply com pos ite with equal thick nesses of the in ner and

(to tal) outer plies (i.e., of the lay-up [0n/90n]s), the pa ram e ters of Eq. (7) are as fol lows: A2 = 0.52, B2 = 0.3075, and n2 =

0.671767. If an outer ply is crack ing, the pa ram e ters of Eq. (7) are A2 = 1.2, B2 = 0.5942, and n2 = 0.57057.

Sim i larly, the mode II ERR for a trans verse crack in a cross-ply lam i nate is eval u ated as

Ghu

GII =

1 122

12

s, (8)

552

1 2 3 40

3

2

1

1/2.K , MP a mI

1/2.K , MPa mII

Fig. 2. Frac ture tough ness of UD glass/ep oxy com pos ite un der mixed-mode load ing: (l) — ex -

per i ment and ( ) — ac cord ing to Eq. (3).

where s12 stands for the un per turbed in-plane shear stress in the ply, G12 is the in-plane shear modulus of the UD ply, and u1 is

the nor mal ized av er age slid ing dis place ment of the crack face. An ap prox i mate re la tion for u1, sim i lar to Eq. (7), is de rived in

[18]. For cross-ply com pos ites with equal to tal thick nesses of the in ner and outer plies, it re duces to a con stant: u1 = 0.42 for a

crack in the in ner ply and u1 = 0.839 for a crack in an outer ply.

Since the lam i nates are treated as lin ear elas tic, the ply stresses are lin ear func tions of the ap plied ten sile strain e:

s s e2 2 2= +r k , s e12 12= k , (9)

where s2r is the re sid ual stress, and the fac tors k2 and k12 are eas ily ob tained by the clas si cal lam i nate the ory for each lay-up

and ply con sid ered. Com bin ing Eqs. (5), (6), (8), and (9), the crack on set cri te rion can be put in the form

hu k

G E

hu k

G Gr2 2 2

2

2

1 12

12

1( (s e) e)2+

+ =Ic IIc

. (10)

Solv ing Eq. (10) for e, the ap plied strain at crack on set in the cor re spond ing ply is ob tained. The ex per i men tal data,

bor rowed from [30], are shown in Fig. 3, where the crack den sity as a func tion of ap plied strain for the [02/902]s, [152/–752]s,

[302/–602]s, and [452/–452]s lam i nates is pre sented along with the pre dicted crack on set strain ac cord ing to Eq. (10). Note that

553

0.5 1.0 1.5 2.0 2.50

1.5

1.0

0.5

-1r, mm à

e, %

[0 /90 ]2 2 s

b

0.5 1.0 1.50

0.2

0.1

-1r, mm

e, %

[15 /-75 ]2 2 s

c

0.5 1.0 1.5 2.0 2.50

0.15

0.10

0.05

-1r, mm

e, %

[30 /-60 ]2 2 s

d

1 2 3 40

0.3

0.2

0.1

-1r, mm

e, %

[45 /-45 ]2 2 s o45-

o45

Fig. 3. Evo lu tion of the crack den sity r [30] and the pre dicted on set strain [Eq. (10)] for the

[02/902]s (a), [152/–752]s (b), [302/–602]s (c), and [452/–452]s (d) lam i nates. (- - -) — the crack

on set strain of the in ner plies; ( ) — the crack on set strain of the outer plies (for the

[452/–452]s lam i nate).

the the o ret i cal COSs of the in ner and outer plies of the [452/–452]s lam i nate are al most equal, be cause the larger nor mal ized

crack face dis place ments of the outer plies are off set by the smaller ply thickness.

It is seen that the the o ret i cal es ti mate agrees rea son ably well with the test re sults for the [02/902]s and [152/–752]s lam -

i nates, whereas the COS of [302/–602]s and [452/–452]s lam i nates is un der es ti mated. The dis crep ancy is likely to be caused by

the de vi a tion from lin ear ity in the lam i nate re sponse, ap par ent in stress–strain di a grams [30], which pre cedes the crack ing on -

set in these lam i nates. The nonlinearity is caused, in part, by the non lin ear shear stress–strain re sponse typ i cal of UD poly mer

ma trix com pos ites [34]. Thus, the lin ear lam i nate the ory may over es ti mate the shear stresses in the plies of [302/–602]s and

[452/–452]s lam i nates, which even tu ally could lead to a con ser va tive COS pre dic tion. A non lin ear lam i nate anal y sis, as in, e.g.,

[35-38] and cor re spond ing cor rec tions to the ERR estimates would be needed to improve the model accuracy.

5. Con clu sions

The frac ture tough ness of UD glass/ep oxy com pos ite un der mode I and mixed-mode load ings has been de ter mined

ex per i men tally. The crack prop a ga tion cri te rion is found to be lin ear in terms of sin gle-mode en ergy re lease rates and the crit i -

cal ERRs eval u ated at GIc = 300 J/m2 and GIIc = 500 J/m2. The mixed-mode crack ing cri te rion was em ployed to pre dict the

intralaminar crack on set in a [02/902]s cross-ply glass/ep oxy com pos ite un der off-axis ten sile load ing. The COS pre dicted

agrees with test re sults at small load ing an gles (on-axis and 15° off-axis load ings) and un der es ti mates the COS at larger (30 and

45°) load ing an gles. The dis crep ancy is likely to be caused by the de vi a tion from lin ear ity in the lam i nate re sponse be fore

crack ing on set in these lam i nates, re lated to the nonlinear shear characteristics of the UD plies.

Ac knowl edge ments. E. Spârniòð grate fully ac knowl edges the sup port by ESF.

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