Stiffness degradation in cross-ply laminates damaged by transverse

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Stiffness degradation in cross-ply laminates damaged by transversecracking and splitting

M. Kashtalyan, C. Soutis*

Department of Aeronautics, Imperial College of Science, Technology and Medicine, Prince Consort Road, London SW7 2BY, UK

Received 15 March 1999; received in revised form 30 July 1999; accepted 1 September 1999

Abstract

In contrast to the few existing theoretical models (Highsmith and Reifsnider, ASTM STP 1986;907:233– 251; Hashin, Trans ASME J Appl

Mech 1987;54:872–879; Daniel and Tsai, Comp Eng 1991;1(6):355–362; Tsai and Daniel, Int J Solid Structures 1992;29(24)3251–3267;Henaff-Gardin et al., Comp Structures 1996;36:113–130; 1996;36:131–140), based on the consideration of a repeated laminate element

defined by the intersecting pairs of transverse cracks and splits, the new approach for evaluating the stiffness degradation in [0m /90n]slaminates due to matrix cracking both in the 90 (transverse cracking) and 0 (splitting) plies employs the Equivalent Constraint Model

(Fan and Zhang, Composites Science and Technology 1993;47:291–298). It also uses an improved 2-D shear lag analysis (Zhang et al.,

Composites 1992;23(5):291–298; 1992;23(5):299–304) for determination of stress field in the cracked or split lamina and In-situ Damage

Effective Functions for description of stiffness degradation. Reduced stiffness properties of the damaged lamina are found to depend

explicitly upon the crack density of that lamina and implicitly upon the crack density of the neighbouring lamina. Theoretical predictions

for carbon and glass fibre reinforced plastic cross-ply laminates with matrix cracking in the 90 ply revealed significant reduction in the

Poisson’s ratio and shear modulus due to additional damage (splitting) in the 0 ply. 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Cross-ply composite laminates

1. Introduction

Matrix cracking has long been recognised as the first

damage mode observed in composite laminates under static

and fatigue tensile loading. It does not necessarily result in

the immediate catastrophic failure of the laminate and there-

fore can be tolerated. However, its presence causes stiffness

reduction and can be detrimental to the strength of the

laminate. It also triggers the development of other harmful

resin-dominated damage modes, such as edge and local

delaminations, which can cause fibre-breakage in the

primary load-bearing plies.Since the early 1970s transverse cracking in composite

laminates has been the subject of extensive research, both

theoretical and experimental. A number of theories have

appeared in an attempt to predict initiation of transverse

cracking and describe its effect on the stiffness properties

of the laminate, among them theories based on the self-

consistent method [1], variational principles [2,3],

continuum damage mechanics [4,5], shear lag [6–11],

approximate elasticity theory solutions [12] and stress trans-

fer mechanics [13,14]. Most of the models developed were

confined, however, to cross-ply laminates under uniaxial

tensile loading. More recently, the focus of investigation

into stiffness reduction due to matrix cracking has shifted

towards transverse cracking in unbalanced m 90ns

laminates under general in-plane loading [15–18], matrix

cracking in the off-axis plies [19], multilayer matrix

cracking of angle-ply and quasi-isotropic laminates

[20–22], and transverse cracking interacting with edge

and local delaminations [23–27].Shear-lag-based models remain the most commonly used

ones for calculating the reduced stiffness properties of trans-

versally cracked composites. They are being modified and

generalised to enable better description of wider classes of

laminates. Thus, the modified 1-D shear-lag approach [28],

suitable for cross-ply laminates of various stacking

sequences, is based on the assumption that longitudinal

displacement is independent of the 0 ply thickness and

width and is a power function of the thickness co-ordinate

and indeterminate function of the length co-ordinate in the

90 piles. While in most of existing shear-lag models

Composites: Part A 31 (2000) 335–351

JCOMA 631

1359-835X/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved.

PII: S1359-835X(99)00077-9

www.elsevier.com/locate/compositesa

* Corresponding author. Tel.: 44-0171-594-5070; fax: 44-0171-584-

8120.

E-mail address: [email protected] (C. Soutis).



longitudinal displacement of the 90 piles in a linear or

parabolic function of the thickness co-ordinate, in Ref.

[28] the exponent in power function was taken equal to

1.1, the value thought to be reasonable for many laminates.

The generalised shear-lag approach [29], aimed at better

description of laminates with thick 0 layers, assumes an

indeterminate variation of the longitudinal displacement

across the thickness of 0 layer, which is then chosenempirically, by best fitting finite element results, and proved

to be an exponential function of cracking density in the 90

ply. The modified shear-lag model [30] assumes existence

of a thin interlaminar adhesive layer between neighbouring

layers, able to transfer not only interlaminar shear stress but

also interlaminar normal stress.

When a cross-ply laminate is subjected to biaxial tensile

loading (Fig. 1), matrix cracking may occur both in the

90 (transverse cracking) and in the 0 piles (splitting).

Transverse cracking and splitting in CFRP cross-ply lami-

nates may also be observed under uniaxial tension [31].

Interestingly, multilayer matrix cracking of cross-ply lami-nates (i.e. transverse cracking combined with splitting) has

been the subject of a very small number of studies. The

reason for this lies perhaps in the problem itself, which is

mathematically more complex than that with just one

cracked ply, due to complicated interaction between two

damage modes. Highsmith and Reifsnider [32] were appar-

ently the first who examined cross-ply laminates damaged

by matrix cracking and splitting. Their study was concerned

with evaluation of stresses and involved extensive numer-

ical analysis. Hashin [2] treated the problem of stiffness

reduction and stress analysis of orthogonally cracked

cross-ply laminates under uniaxial tension by the variational

method based on the principle of minimum complementaryenergy. He obtained a strict lower bound for the Young’s

modulus and an approximate value of Poisson’s ratio.

Hashin’s analysis revealed an increase in Poisson’s ratio

with increasing cracking and splitting density in orthogon-

ally cracked laminates with thin 90 layer. Reduction of the

Young’s modulus proved to occur mainly due to transverse

cracking, with splitting having an insignificant effect.

Experimental data and theoretical predictions of axial stiff-

ness reduction due to cracks in both layers were obtained by

Daniel and Tsai [33], who later also predicted and verified

experimentally the reduction of shear modulus due to

transverse cracking and splitting [34]. A finite difference

iteration method was used to solve a system of governing

equations that involves only in-plane displacements in both

layers. Although the interlaminar-shear-analysis, from

which a system of governing equations were derived

based on equilibrium, continuity and boundary conditions,

did not start with a usual assumption of the classical shear

lag approach that the interlaminar shear stresses are propor-

tional to the displacement different between two layers, it

has shown agreement with this assumption. The initial value

for the iterative procedure, provided by superposition of

solutions for a single set of cracks, was thought to be

more than adequate for calculation of shear modulus reac-

tion, with the difference between initial and final value less

than 1%. Experimental results for graphite/epoxy cross-ply

laminates appear to be in a good agreement with the analy-

tical predictions. Henaff-Gardin et al. [35,36] examined

cross-ply composite laminates, damaged by doubly periodic

matrix cracking (i.e. transverse and longitudinal, or split-

ting), both under general in-plane and thermal biaxial load-ing. They assumed that the in-plane displacements in each

lamina vary parabolically through the lamina thickness in

the direction normal to the crack plane and are constant in

the other direction. While evaluating elastic constants of the

damaged laminate, it was further assumed that the elastic

constants of a cracked lamina depend only on the crack

spacing in that lamina, but not on the crack density in the

adjacent piles. Stresses in the direction perpendicular to

the crack plane were found to be almost independent of

the other in-plane co-ordinate. This observation was made

earlier by Highsmith and Reifsnider [32] in their experimen-

tal work and used as a basic assumption by Hashin [2].It appears therefore that at present there are few theore-

tical models that can predict reduction of all in-plane elastic

properties ( E, G, v) of cross-ply laminates due to transverse

cracking and splitting under general in-plane loading. None

of the existing models seems to be simple enough to allow

any feasible generalisation, with a purpose to describe, for

instance, delaminations growing from the crack/split tips.

The objective of the present investigation is to evaluate

efficiently the reduced stiffness properties of [0m /90n]scross-ply laminates damaged by transverse cracking and

splitting.

2. Theoretical modelling

In the current model, transverse cracks and splits in a

[0m /90n]s laminate are assumed to be spaced uniformly and

to span the full thickness and width of the 90 and 0 plies.

The assumption of the uniform spacing, crucial for theore-

tical modelling as it allows to solve the problem via analysis

of a representative element, was shown by many researches

to be justified from an engineering point of view. A

schematic of the cross-ply laminate containing bi-direc-

tional cracks is shown in Fig. 1. Spacings between splits

M. Kashtalyan, C. Soutis / Composites: Part A 31 (2000) 335– 351336

Fig. 1. Cross-ply laminate damaged by transverse cracking and splitting.



and transverse crack are denoted, respectively, 2s1 and 2s2.

A global set of Cartesian co-ordinates with the origin in the

centre of the laminate is introduced, with the x1-axis coin-

ciding with the fibre direction in the 90 lamina (ply group),

the x2-axis parallel to the fibre direction in the 0 lamina and

the x3-axis is directed through the laminate thickness. The

laminate is subjected to biaxial tension ( 11 and 12) andshear loading 12

2.1. Application of the Equivalent Constraint Model

In contrast to the existing analytical models [32–36],

based on consideration of a repeated laminate element

defined by the intersecting pairs of transverse cracks and

splits, the present model uses a different approach and

employs the Equivalent Constraint Model (ECM) of the

damaged lamina [37]. It is now increasingly accepted that

a crack lamina behaves within a laminate in a different

manner compared to an infinite effective medium containing

many cracks. In Ref. [1], while evaluating stiffness and

compliance changes of a cracked laminate, it was suggested

to replace a cracked lamina with an effective medium

containing many cracks, following an assumption that inter-

action between the cracked lamina and the neighbouringlayers is limited to the vicinity of the crack tip and, there-

fore, has minor effect of the lamina stiffness. It was revealed

in Ref. [12] that the laminate-independent approach [1]

resulted in significant overestimation of the changes in

damaged lamina compliances. The stiffness of the damaged

lamina was clearly shown to be strongly influenced by the

laminate in which it is contained. In order to take into

account the effect of the in-situ constraint on the stiffness

of a particular cracked lamina, in Ref. [37] the ECM of the

damaged lamina was introduced. In the ECM (Fig. 2), all the

laminae below and above the damaged lamina under consid-

eration are replaced with homogeneous layers (I and II)having the equivalent constraining effect. The in-plane stiff-

ness properties of the equivalent constraining layers can be

calculated from the laminated plate theory, provided stres-

ses and strains in them are known. Although theoretically

the ECM does not impose any restrictions onto the laminate

lay-up, it may be less efficient if the equivalently constrain-

ing layers are anisotropic. Besides that, in its present form

the model does not reflect hierarchical constraining mechan-

isms, which may be acting in a multi-directional composite

laminate.

Thus, instead of considering the damaged laminate

configuration shown on Fig. 1, the following two equivalent

constraint models (ECMs) will be analysed. In ECM1 (Fig.3(a)), the 0 lamina (layer 1) contains damage explicitly,

while the 90 lamina (layer 2), damaged by transverse

cracking, is replaced with the homogeneous layer with

reduced stiffness properties. Likewise, in ECM2 (Fig.

3(b)), the 90 lamina (layer 2) is damaged explicitly,

while the split 0 lamina is replaced with the homogeneous

layer with reduced stiffness. ECM1 and ECM2 represent

two particular cases of the general model (i.e. ECMk , 1 k

N where 2 N is the number of piles in the laminate) shown

on Fig. 2(b), namely when the k th layer is either central or

outer ply. All the quantities associated with the 0 lamina

M. Kashtalyan, C. Soutis / Composites: Part A 31 (2000) 335– 351 337

Fig. 2. Equivalent constraint model (ECM) of a damaged lamina: (a) initial laminate; (b) ECMk .

Fig. 3. Representative segments of the two equivalent constraint models of

a laminate damaged by transverse cracking and splitting: (a) ECM1, (b)

ECM2.



(layer 1) will be henceforth denoted by a sub- or superscript

(1) and those associated with the 90 lamina (layer 2) with a

sub- or superscript (2).

The purpose of the analysis of the ECM (i.e. ECM1 if

1 and ECM2 if 2) to determine the reduced stiff-

ness properties of the th layer damaged by transverse

cracking or splitting. Since the reduced elastic properties

of the equivalently constraining layer , used in

the analysis of the ECM , are determined from the analysis

of the ECM , the problems for ECM1 and ECM2 are inter-

related. In the absence of splitting, the analysis of ECM2

coincides with that performed earlier in Ref. [6].

2.2. Stress analysis of the ECM

Due to periodicity of the damage configuration in the

ECM , only their representative segments (Fig. 3), contain-

ing either two pairs of splits or a single pair of transverse

cracks, need to be considered. As the representative

segments are symmetric with respect to the mid-plane andtheir material and geometry are noteworthy uniform in

direction perpendicular to the x0 x3 plane, the analysis

can be further restricted to one-quarter of the representative

segments.

The three-dimensional stress field in the ECM is

assumed to comply with the equilibrium equations

x j

k ij 0 i 1 2 3 k 1 2 1 2 1

where k ij denotes stress components in the k th layer of

the ECM . It is assumed that the equivalent constraining

lamina(e) in the ECM are homogeneous orthotropic, and

the constitutive equations for both layers can be written as

{ } C

{ } { } C

{ }

1 2 2

where C

denotes the stiffness matrix of the explicitly

damaged th layer, (a circumflex (^) is used for represent-

ing the elastic properties of the undamaged material), and

C

denotes the stiffness matrix of the homoge-

neous orthotropic material of the equivalent constraining

th layer. The equilibrium equations, Eq. (1), and constitu-

tive equations, Eq. (2), can be averaged across the layer

thickness and the width of ECM as indicated below

f k

1

2whk

h

w

f k d x3 d x 3

in order to make a transition to the in-plane microstresses

and microstrains. The equilibrium equations in terms of

microstresses become

d

d x

k j 1

k

j

hk

0

1 2 j 1 2 k 1 2

4

where k ij are the in-plane microstress in the k th layer of

the ECM and

j are the interface shear stresses at the

(0/90) interface of the EMC in the jth direction. The

in-plane microstresses are related to the total stresses ij

applied to the laminate by the following equilibrium

equations:

1ij

2ij 1 ij i j 1 2 h1 h2 5

The constitutive equations in terms of microstresses and

microstrains are

1

2

6

Q

11Q

12 0

Q12

Q22 0

0 0 Q

66

1

2

6

1

2

6

Q 11

Q 12 0

Q 12

Q 22 0

0 0 Q 66

1

2

6

1 2

6

where the components of the in-plane stiffness matrix are

related to the elastic moduli of the orthotropic material as

Q

ij C

ij

C

i3C

j3

C

33

Q ij C

ij

C i3 C

j3

C 33

7

In doing so, it is assumed that k 3 0 k 1 2 The in-

plane constitutive equations can also be written in the inverse

form as

1

2

6

S

11S

12 0

S 12

S 22 0

0 0 S

66

1

2

6

1

2

6

S 11 S

12 0

S 12 S

22 0

0 0 S 66

1

2

6

1

2

8

The boundary conditions on the stress-free crack/split

surfaces are

ij 0 i j 1 2 9

To determine the in-plane microstresses from the equili-

brium equations, Eq. (4), the interface shear stresses

j

have to be expressed in terms of the in-plane displacements

uk

j j 1 2 This can be done by averaging the out-of-

plane constitutive equations across the lamina thickness and

making an assumption about the variation of either the out-

of-plane shear stresses or the in-plane displacements. Here,




it is assumed that the out-of-plane shear stresses k

j3 j

1 2 vary linearly with x3, which corresponds to a parabolic

variation of the in-plane displacements. Besides that, it is

assumed that in the 0-lamina linear variation of out-of-

plane shear stresses 1

j3 j 1 2 is restricted to the region

of the shear layer. The thickness of the shear layer depends

on the cracked layer thickness and must be roughly propor-tional to it. For laminates with thick 0-layer this appears to

offer a more reasonable description of the cracked laminate

behaviour. For such laminates it was shown [29], by means

of a finite element (FE) analysis, that the assumption of

parabolic variation of the in-plane displacement across the

thickness of the whole 0-layer provides a very poor approx-

imation to the distribution of the longitudinal stress across

the laminate thickness. This approximation becomes even

poorer as the transverse crack density increases. Thus, the

out-of-plane shear stresses are assumed to vary as follows

(Fig. 4):

1

j3

j

hs

h2 hs x3 h2 x3 h2 hs

2

j3

j

h2

x3 x3 h2 j 1 2

10

After some mathematical calculations and equation rearran-

gements (see Appendix A.1), the interface shear stresses are

obtained as

j K

j u1

j u2

j 11

where the shear lag parameters K j are functions of ply

properties

K j 3 G

1 j3

G2

j3

h2 G

1 j3 1 1 2 h1

G2

j3

hs h1 hs mst j 1 2

12

Here, Gk j3 k 1 2 are the out-of-plane shear moduli of the

k th layer, hs is the thickness of the shear layer, ms is the

number of plies in the shear layer ms hs t and t is the

ply thickness, Fig. 4. The presence of aligned microcracks

does not affect the value of the out-of-plane shear moduli

(this fact is emphasised by marking them with a circumflex

(^)), therefore, they are the same for ECM1 and ECM2.

When the thickness of the shear layer is equal to the outer

layer thickness, i.e. when hs h1 or 1 Eq. (12) is

reduced to the expression derived in Ref. [12].

The equilibrium equations, Eq. (4), along with expres-

sions for the interface shear stresses, Eq. (10), the laminate

equilibrium equations, Eq. (6), and constitutive equations,

Eq. (8), provide a full set of equations, which are required

for determining the in-plane microstresses

j j 1 2

in the representative segment of the EMC . For instance,

11

11 can be found from the following set of eight equations

with respect to eight variables

d 11

11

d x1

11

h1

0 13a

11 K 1 u

111 u

121 13b

1 11

11 2 12

11 11 13c

1 11

22 2 12

22 22 13d

d u111

d x1

112

S 111

S 112

S 112

S 122

1111

11

22

13e f

d u121

d x1

122

S 211 S

212

S 212 S

222

1211

1222 13g h

After some rearrangement, this and other similar sets of

equations can be reduced to the single differential equations

d2

d x L

1

11 11

22 22 0

1 2

14a

d2

12

d x L

2

12

12 12 0 14b

where L1 L

1

11

22 and

12 are the laminate

constants depending on the layer compliances S

ij

S ij shear lag parameters K j and the layer thick-

ness ratio h1 h2 In detail, they are presented in

Appendix A.2. Given the boundary conditions, Eq. (9),

at the crack/split surfaces, solutions to Eq. (14) are

1

L

1

1 cosh

L

1

x

cosh

L

1

s

11 11

22 22

15a


Fig. 4. Variation of out-of-plane stresses in the improved 2-D shear lag

analysis.



12 1

L

2

1 cosh

L

2

x

cosh

L

2

s

12 12 15b

Once the in-plane microstresses, Eq. (15), in the expli-

citly damaged th layer of the ECM are known, the

laminate macrostresses can be found as

j 1

2s

su

s

j d x 16

2.3. Reduced stiffness properties of a cracked/split layer

The reduced stiffness properties of the layer , damaged

by transverse cracking or splitting, can be determined by

applying the laminate plate theory to the ECM after repla-

cing the explicitly damaged layer with an equivalent homo-

geneous one. The constitutive equations for the

homogeneous layer equivalent to the explicitly damaged

th layer are

{ } Q

{ } 17

where the macrostrains are assumed to be

j

j j 1

2s

su

s

j d x

j 1 2 6

18

The in-plane reduced stiffness matrix [Q( )] of the homo-

geneous layer equivalent to the th layer of the ECM is

related to the in-plane stiffness matrix Q

of the unda-

maged layer via the In-situ Damage Effective Functions

(IDEFs)

22

66 as

Q

Q

R

11

22Q

12

22 0

Q12

22

R22

22 0

0 0 Q

66

66

19a

R111 Q

111 R

122

Q112

2

Q111

R211

Q212

2

Q222

R222 Q

222

19b

The concept of the IDEFs

22

66 was introduced in Ref.

[37] on the basis of results acquired by the general theory of inhomogeneous media [1,38]. It was proved [38], by means

of the self-consistent method, that the tensor of elastic

compliance {S } of a linearly elastic brittle anisotropic

solid containing microcracks can be represented as a sum

of two tensors: the tensor of elastic compliance tensor { S } of

an undamaged solid and the tensor of additional

compliances {W }, dependent on the configuration and

distribution of microcracks. For a unidirectional fibrous

composite regarded as an effective homogenous orthotropic

medium and aligned microcracks regarded as elliptic

cylindrical cavities, the tensor of additional compliances

{W } was found to have only three non-zero components

[1]. The number of non-zero additional compliances was

assumed to be further reduced to two if the cracked material

were loaded only in the plane parallel to the fibres [37]. In

the co-ordinate system chosen as shown in Fig. 1, these non-

zero additional compliances due to matrix cracking (in the

Voigt notations) will be W 1

11

0 W 1

66

0 for the layer 1

(fibres directed along the x2-axis) and W 222 0 W 2

66 0

for the layer 2 (fibres directed along the x1-axis). Inversion

of the in-plane compliance matrix S

S

W

and subsequent extraction of the stiffness matrix of the

undamaged material Q

yields the in-plane stiffness

Q

in the form given by Eq. (19), if the following

notations are introduced

22

W

Q

1 W

Q

66 W

66Q

66

1 W

66Q

66

1 2

20

From Eq. (19) and the constitutive equations for the th

layer of the ECM , Eq. (17), the IDEFs 22

66 can be

expressed as

122 1

11

1

Q111

111 Q

112

112

222 1

222

Q212 22

1 Q222 22

2

66 1

6

Q

66

6

21

On substituting macrostresses, calculated from Eqs. (15)

and (16), and macrostrains, calculated from Eqs. (8), (15)

and (18), into Eq. (21), the closed from expressions for

IDEFs are obtained. They represent IDEFs as functions of

damage parameters Dmc h S associated with trans-

verse cracking/splitting, the layer compliances S

ij S ij

shear lag parameters K j and the layer thickness

ratio , i.e.

qq

qq Dmc S

ij S

ij K j 22

In detail, the closed form expressions for the IDEFs for the

th layer of the ECM are

22 1

1 D

mc

1

tanh

1

Dmc

1 1

Dmc

1

tanh

1

Dmc

66 1

1 D

mc

2

tanh

2

Dmc

1

2

Dmc

2

tanh

2

Dmc

1 2

23





T a b l e 1

Y o u n g ’ s m o d u l u s r e d u c t i o n r a t i o f o r t r a n s v e r s a l l y c r a c k e d l a m i n a t e s

D a m a g e

p a r a m e t e r D m c 2

D m c 1

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G F R P [ 0 / 9 0 ] s

G F R P

[ 0 / 9 0

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C F R P [ 0 / 9 0 ] s

R e f . [ 2 ]

E C M / 2 - D

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[ 3 5 , 3

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[ 3 5 , 3

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R e f .

[ 2 ]

E C M / 2 - D

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[ 3 5

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7 3




T a b l e 2

Y o u n g ’ s m o d u l u s r e d u c t i o n r a t i o f o r t r a n s v e r s a l l y c r a c k e d a n d s p l i t l a m i n a t e s

D a m a g e

p a r a m e t e r

D m c 2

D m c 1

0

G F R P [ 0 / 9 0 ] s

G F R P [ 0 / 9 0

3 ] s

C F R P [ 0 / 9 0 ] s

R e f . [ 2 ]

E C M / 2 - D

s h e a r l a g

a p p r o a c h

R e f s .

[ 3 5 , 3

9 ]

R e f

. [ 2 ]

E C M / 2 - D

s h e a r l a g

a p p r o a c h

R e f s .

[ 3 5 , 3 9

]

R e f . [ 2 ]

E C M / 2 - D

s h e a r l a g

a p p r o a c h

R e f s .

[ 3 5 , 3

9 ]

0 . 0

2

0 . 9

9 0

0 . 9

9 2

0 . 9

9 9

0 . 9 8 0

0 . 9

8 5

0 . 9

9 9

0 . 9

9 9

0 . 9

9 9

0 . 9

9 9

0 . 0

5

0 . 9

7 5

0 . 9

8 0

0 . 9

9 3

0 . 9 5 1

0 . 9

6 2

0 . 9

8 1

0 . 9

9 7

0 . 9

9 7

0 . 9

9 9

0 . 1

0 . 9

5 1

0 . 9

6 1

0 . 9

8 0

0 . 9 0 6

0 . 9

2 7

0 . 9

5 2

0 . 9

9 4

0 . 9

9 5

0 . 9

9 7

0 . 2

0 . 9

0 7

0 . 9

2 5

0 . 9

5 7

0 . 8 2 9

0 . 8

6 3

0 . 8

9 7

0 . 9

8 8

0 . 9

9 0

0 . 9

9 4

0 . 3

3

0 . 8

5 3

0 . 8

8 2

0 . 9

2 8

0 . 7 4 3

0 . 7

9 2

0 . 8

3 5

0 . 9

8 2

0 . 9

8 4

0 . 9

9 0

0 . 5

0 . 8

0 4

0 . 8

4 1

0 . 8

9 4

0 . 6 5 8

0 . 7

1 8

0 . 7

6 6

0 . 9

8 1

0 . 9

7 9

0 . 9

8 6

1 . 0

0 . 7

6 2

0 . 7

8 7

0 . 8

3 2

0 . 5 4 2

0 . 6

0 2

0 . 6

3 9

0 . 9

7 4

0 . 9

7 2

0 . 9

7 7

2 . 0

0 . 7

5 7

0 . 7

6 5

0 . 7

9 5

0 . 5 1 6

0 . 5

4 1

0 . 5

5 9

0 . 9

6 8

0 . 9

6 9

0 . 9

7 1



where the constants

1

1 i 1 2 (Appendix A.3)

depend solely on the layer compliance S

ij S ij

shear lag parameters K j and the layer thickness ratio .

The modified compliances S ij of the equivalently

constraining th layer of the ECM are determined from

the analysis of the ECM and therefore are functions of the

IDEFs

22

66

Thus, the IDEFs for the th layer depend

implicitly on the damage parameters Dmc h s

associated with the layer .

The IDEFs for both layers form a system of simultaneous

nonlinear algebraic equations

1qq 1

qq Dmc1 S

1ij S

2ij D

mc2 S

2ij 2

qq

2qq 2

qq Dmc2 S

2ij S

1ij D

mc1 S

1ij 1

qq

q 2 6

24

This system is solved computationally by a direct iterative

procedure, carried out in such a way that the newly calcu-

lated IDEFs qq are used to evaluate the reduced stiffness

of the equivalently constrained th layer repeatedly until

the difference between two iterative steps meets the

prescribed accuracy. As a result, all four IDEFs k qq q

2 6 k 1 2 are determined as functions of damage para-

meters Dmc1 D

mc2 If interaction between damage modes in

different laminae are neglected, IDEFs associated with the

th layer will depend only on damaged parameters for that

layer.

3. Verification of the model

Before predicting the reduced elastic properties of cross-

ply laminates damaged by transverse cracking and splitting

using the new ECM/2-D shear lag approach, testing it

against other recent theoretical models and experimental

data seems to be worthwhile. As mentioned in Section 1,

there are few theoretical models, which can describe the

stiffness loss in cross-ply laminates due to matrix cracking

both in the 90 and 0 piles [32–36].

Tables 1 and 2 show Young’s modulus reduction inGFRP and CFRP laminates considered earlier in Ref. [2].

Table 1 contains data for laminates damaged by transverse

cracking without splitting, while Table 2 for laminates

damaged by transverse cracking and splitting. The material

properties used are given in Table 3. Since Hashin evaluates

Young’s modulus of a cracked laminate on the basis of the

principle of minimum complementary energy, his predic-

tions are supposed to provide the rigorous lower bound for

the reduced Young’s modulus value. It may be seen that the

present ECM/2-D shear lag approach delivers results, which

not only comply with this expectation, but also are closer to

the lower bound than the results [39] based on the model by

Henaff-Gardin et al. [35]. Predictions for CFRP system are

the closest ones, while for GFRP results for the [0/90]s,

laminate are closer than those for the lay-up with a thicker

90 layer, i.e. for [0/903]s.

However, the situation is quite different for the Poisson’s

ratio. For transverse cracking without splitting (Fig. 5(a)),

the ECM/2-D shear lag approach predicts much greater

reduction in Poisson’s ratio than Hashin’s calculations.

This prediction is similar to that in Ref. [39] based on the

model in Ref. [35], though for small values of damage

parameter the results of model [35] are close to those in

Ref. [2]. For transverse cracking and splitting (Fig.

5(b)), Hashin predicts an increase of Poisson’s ratio

for a [0/90]s lay-up and asymptotic decrease to some

non-zero value for a [0/903]s lay-up, while, according to

ECM/2-D shear lag approach predictions, the Poisson’s

ratio should decrease almost to zero. The model [35]

predicts decrease in the Poisson’s ratio, yet the asymp-

totic value appears to be dependent upon the lay-up: non-zerofor the [0/90]s laminate and zero for the [0/90]s laminate

[39].

The Poisson’s ratio reduction due to transverse cracking

predicted by the present model has been also compared with

some other, recently developed, theories [40,41]. The prop-

erties of unidirectional material (E-glass/epoxy [2]) used for

comparison were taken from Table 3, and the ply thickness

was 0.203 mm. The theory of Pagano and Schoeppner [41]

employs the variational theorem by Reissner [42] to predict

the stress fields in flat laminates. McCartney’s theory [40] is

based on the generalised plane strain model of stress trans-

fer, which has been shown to lead to the stationary values of the Reissner energy functional. It also uses an assumption

that the direct stresses in the 0 and 90 piles are independent

of the through-thickness co-ordinate [40]. The comparison

of results for [0/90]s GFRP laminate [41,43], which was

considered earlier in Refs. [2,32], reveals an excellent

agreement between all three models (Fig. 6). Together

with Fig. 5(a), it indicates that the source of discrepancy

between the present ECM/2-D shear lag approach and

Hashin’s model is presumably in the latter rather than in

the former one. Indeed, let us consider Eq. (60) in Ref. [2]

for the case of transverse cracking only. Then function is

equal to zero and the rest of the equation yields that the

reduction of the Poisson’s ratio is proportional to thereduction in the Young’s modulus, which is obviously

incorrect.

As far as reduction of shear modulus is concerned, the

present model can be compared to those of Tsai and Daniel

[34], who especially developed it for the description of the

cracked cross-ply laminates under shear loading, and

Henaff-Gardin et al. [35]. It is worth noting that the model

of Tsai and Daniel [34] and the present ECM/2-D shear lag

approach yield exactly the same analytical expression for

the shear modulus reduction ratio due to transverse crack-

ing, if the thickness of the shear layer in the ECM/2-D shear





Table 3

Properties of unidirectional materials

Material Source E A (GPa) E T (GPa) E A (GPa) E T (GPa) A T t (mm)

GFRP (E-glass/epoxy) Ref. [2] 41.7 13.0 3.40 4.58 0.300 0.420 –

CFRP Ref. [2] 208.3 6.5 1.65 2.30 0.255 0.413 –

CFRP (AS4/3501-6) Ref. [34] 145. 10.6 6.9 3.7 0.27 – –

GFRP (E-glass/epoxy) Ref. [44] 40. 10. 5. 3.52 0.3 – 0.155CFRP (XAS/914) Ref. [44] 145. 9.5 5.6 3.35 0.3 – 0.125

Fig. 5. Poisson’s ratio as a function of damage parameter Dmc2 (a) D

mc1 0 (transverse cracking without splitting); (b) D

mc2 D

mc1 (transverse cracking and

splitting).



lag approach is taken equal to that of the 0 lamina, i.e. if

hs h1 In notations of the present paper this expression is

G GA

GA

1 D

mc2

222

tanh 22

2

Dmc2

1

25

For transverse cracking combined with splitting, Tsai and

Daniel [34] suggested a semi-empirical expression for the

shear modulus reduction ratio based on the “superposition”

of solutions for a single set of cracks as

G 1 D

mc1

112

tanh 11

2

Dmc1

1

Dmc2

222

tanh 22

2

Dmc2

1

26

In fact, it was obtained from Eq. (25) by adding one

more term to the expression within the brackets. Then

Tsai and Daniel calculated the shear modulus reduction

ratio from the work done by the external shear loading.The shear stresses along the boundary of the block (i.e.

the laminate element between two transverse and two

longitudinal cracks) were obtained by the finite differ-

ence iteration procedure, used to solve the general

system of governing equations of the interlaminar-

shear–stress analysis. The value of the shear modulus

reduction ration obtained by the finite difference itera-

tion appeared to be within 1% of the value given by Eq.

(26). The present ECM/2-D shear lag model, if the

interaction between transverse cracks and splits is

neglected and the shear layer has the thickness of the

0 lamina, yields an expression

G

1

Dmc1

112

Dmc2

222

tanh11

2

Dmc1

tanh22

2

Dmc2

×

1

Dmc1

112

tanh11

2

Dmc1

1

Dmc2

222

tanh22

2

Dmc2

D

mc1

112

Dmc2

222

tanh

112

Dmc1 tanh

222

Dmc2

1

27

It may be seen from Eqs. (26) and (27) that the two

expressions differ by the underlined terms and G G

In absence of splitting Dmc1 0 they are both reduced

to Eq. (25).

Predictions, based on the semi-empirical expression,

Eq. (26), the ECM/2-D shear lag approach (with the shear

layer having the thickness of one ply and neglecting the

interaction between transverse cracks and splits), Eq. (27),

and Henaff-Gardin et al. [35,39] are presented in Fig. 7 for a

AS4/3506-1 [03 /903]s laminate. Elastic properties of the

unidirectional material are given in Table 3. When Dmc1 0(splitting without transverse cracking), the results differ due

to the fact that in the ECM/2-D shear lag model the shear

layer (Fig. 4) is assumed to be of one ply thickness. In most

cases, predictions by Tsai and Daniel are within 10% of

those by the ECM/2-D shear lag approach. However, in

some cases the error of the semi-empirical expression [34]

can be as big as 20%. Predictions based on Henaff-Gardin et

al. [35] model appear to be in better agreement with those of

the ECM/2-D shear lag approach. The same slope of the

corresponding curves is particularly noticeable. Limited

experimental data acquired by Tsai and Daniel [34] are in


Fig. 6. Poisson’s ratio reduction ratio as a function of transverse cracking density C 2 2s21 for transversally cracked [0/903]s GFRP laminate.



an acceptable agreement with all predictions. Yet, further

experimental work is needed to validate the analytical

models.

4. Prediction of stiffness degradation

All results on the loss of stiffness in graphite/epoxy and

glass/epoxy [0m /90n]s cross-ply laminates due to transversecracks and splitting in this section were obtained taking into

account the interaction between transverse cracks and splits.

Up to 12 iterations were required to solve a system of simul-

taneous non-linear algebraic equations, Eq. (24), with the

accuracy of 109. The number of iterations increased with

increasing crack/splitting density. Two lay-ups were

analysed: [0/90]s and [0/903]s. The properties of unidirec-

tional materials [44] used in the analysis are given in Table

1.

Fig. 8(a) shows reduction in the axial modulus, shear

modulus and Poisson’s ratio in a CFRP [0/90], laminate

with and without splitting as a function of transverse

crack density. Splitting density is taken as C 1 2s11 10 cracks/cm. It may be seen that the axial modulus of the

transversally cracked and split laminate is almost the same

as that of the laminate damaged by transverse cracking only.

As one would expect, splitting does not affect its value.

However, it causes further reduction in the shear modulus

and Poisson’s ratio, by 15–18% and 21–22%, respectively,

for a given splitting density. The effect of splitting, more

pronounced on the Poisson’s ratio than on the shear

modulus, slightly increases with an increase in transverse

cracking density. Fig. 8(b) shows analogous predictions

for a CFRP [0/903]s laminate. Splitting density is again

C 1 2s11

10 cracks/cm. Additional reduction in the

shear modulus and Poisson’s ratio due to splitting about 10

and 11%, for a given splitting density.

Predictions for GFRP [0/90]s and [0/903]s laminates are

shown in Fig. 9(a) and (b), respectively. Again, there is

almost no further reduction in the Young’s modulus value

due to splitting, although reduction due to transverse crack-

ing in GFRP laminates is greater that in CFRP ones. As far

as shear modulus and Poisson’s ratio are concerned, reduc-tion in their values due to splitting is approximately the

same, respectively, 17– 22% and 18–23% in [0/90]s and

10–15% and 11–14% in [0/903]s for a given splitting

density. The effect of splitting is also observed to increase

with an increase in the transverse cracking density.

Fig. 10(a) and (b) illustrate reduction in the axial, trans-

verse and shear moduli of CFRP and GFRP [0/90]s lami-

nates when the extent of damage, described by the damage

parameter Dmci hi si is the same in both layers. Such

situation appears when a laminate is subjected to biaxial

loading with the biaxiality ratio close to 1. Results for trans-

versally cracked and split Dmc2 D

mcl laminates are shown

in comparison with the data for laminates, which containdamage in the 90 layer only D

mc2 0 The maximum

value of damage parameter corresponds to crack density

of 37.8 cracks/cm in CFRP and 32.3 cracks/cm in GFRP

laminates.

5. Conclusions

While stiffness loss due to matrix cracking in the 90 plies

of cross-ply [0m /90n]s laminates has been the subject of

numerous studies in the literature, matrix cracking occurring


Fig. 7. Shear modulus reduction ratio as a function of damage parameter Dmc2 D

mc1 for transversally cracked and split [03 /903]s CFRP laminate.



both in the 90 (transverse cracking) and 0 (splitting) plies

has received considerably less attention. Existing theoretical

models [32–36], on the one hand, do not describe reductionof all stiffness components, and on the other hand, appear to

be complicated enough to allow any further extension or

generalisation, with the aim to involve into consideration

other damage modes, e.g. local delaminations that grow

from the crack/split tips. The new approach, developed to

overcome the above-mentioned limitations, is based on the

Equivalent Constraint Model (ECM) of the damaged

lamina. In the ECM, only one damaged layer contains

damage explicitly, while all neighbouring layers, damaged

or undamaged, are replaced with two (one) homogeneous

orthotropic layers with reduced stiffness properties. An

improved 2-D shear lag analysis is then performed to deter-

mine the in-plane microstresses in the explicitly damaged

layers of the two ECMs, which are considered instead of theoriginal one. Closed form expressions are obtained for the

IDEFs, which characterise reduced stiffness properties of

the damaged layers. IDEFs for a given layer are represented

as explicit functions of the damage parameters (relative

crack/split density) associated with that layer and implicit

functions of the damage parameters associated with the

neighbouring plies. Thus, interaction between all damage

modes is taken into account in the new approach.

The new ECM/2-D shear lag based approach is in an

excellent agreement with the models [2,35] as far as axial

stiffness reduction of the transversally cracked and split


Fig. 8. Elastic moduli reduction ratios as functions of transverse crack density C 2 2s21 (cm1) for transversally cracked CFRP laminates with (solid lines)

and without (hatched lines) splitting: (a) [0/90]s; (b) [0/903]s. Splitting density 10 cm1.



laminate is concerned. For the Poisson’s ratio, significant

qualitative and quantitative scattering of predictions isobserved, the source of which appears to lie in the Hashin’s

approach. Comparison with estimations [34,35] of the shear

modulus reduction in transversally cracked and split lami-

nates shows that all three models predict the same trend,

although some quantitative discrepancy (10–20%) between

the present approach and that in Ref. [34] is noticeable.

The effects of transverse cracking and splitting on the

axial, transverse and shear moduli as well as Poisson’s

ratio of typical CFRP and GFRP laminates were examined

on the basis of the new ECM/2-D shear lag approach. The

axial modulus value has appeared to be almost not

affected by splitting. The effect of splitting on the Poisson’s

ratio is more pronounced than on the shear modulus, and itslightly increases with an increase in the transverse

cracking density. For [0/90]s laminates, transversally

cracked and split under biaxial loading, the total reduction

in the transverse modulus is slightly larger than in the axial

one.

Acknowledgements

Financial support of this research by the Royal Society,

UK, Engineering and Physical Sciences Research Council


Fig. 9. Elastic moduli reduction ratios as functions of transverse crack density C 2 2s21 (cm1) for transversally cracked GFRP laminates with (solid lines)

and without (hatched lines) splitting: (a) [0/90]s; (b) [0/903]s. Splitting density 10 cm1.



(EPSRC/GR/L51348) and Ministry of Defence is gratefully

acknowledged. The authors would like to thank Professor

Glyn Davies, Imperial College of Science, Technology and

Medicine, London, UK, Professor Paul Smith, University of

Surrey, Guilford, UK, and Dr Neil McCartney, National

Physical Laboratory, Teddington, UK, for helpful discus-

sions. Also our special thanks to Dr McCartney and Dr

Catherine Henaff-Gardin, ENSMA, Futuroscope, France,

for providing numerical results of their models for compar-

ison purposes.

Appendix A

A.1. Derivation of shear lag parameters

Assuming that uk 3 x1 u

k 3 x2 0 k 1 2

the out-of-plane constitutive equations are

k

j3 u

k j

x3

1

Gk

j3

k

j3 k 1 2 A1


Fig. 10. Elastic moduli reduction ratios as functions of damage parameter Dmc2 for transversally cracked [0/90]s laminates with (solid lines, Dmc

1 Dmc2 ) and

without (hatched lines, Dmc1 0) splitting: (a) CFRP; (b) GFRP.



where Gk

j3 are the out-of-plane shear moduli of the k th

layers. As was already noted, these elastic constants are

not undergoing reduction due to matrix cracking. For the

inner layer, substitution of Eq. (10) into Eq. (A1) and

repeated integration with respect to x3 across the layer

thickness yields

u2

j u2

j x3h2

j h2

3 G2

j3

A2

For the outer layer, substitution of Eq. (10) into Eq. (A1) and

integrating with respect to x3 across the thickness of the

shear layer (Fig. 4) leads to

ul

j ul

j x3h2

j

hs G

2 j3

h2 hs x3 h2

x23 h

22 2 h2 x3 h2 hs

A3

Integrating Eq. (A3) across the thickness of the shear layeragain yields

ul

j ul

j x3h2

j hs

3Gl

j3

j 1 2 A4

where ul

j are the displacements averaged across the thick-

ness of the shear layer. In the part of the outer layer h2

hs x3 h2 h7 free from the out-of-plane shear, Fig. 4,

the displacements ul

j are constant across the thickness and

can be found from Eq. (A3) by putting x3 h2 hs as

u

l

j ul

j x3h2

j hs

2Gl

j3A5

The displacements, averaged across the whole thickness of

the outer (constraining) layer are then

ul

j 1

hl

hs ul

j h1 hs ul

j A6

Finally, the continuity of displacements at the interface due

to the perfect bonding between the layers in the considered

region should be taken into account, i.e.

ul

j x3h2 u

l j x3h2

A7

Combining Eqs. (A2), (A4)–(A7) together yields Eq. (11)for the interface shear stresses

j with the shear lag

parameter K j given by Eq. (12).

A.2. Laminate constants

L11

K 1

h1

S 111 b1

S 112 S

211 b1S

212

L12

K 2

h1

S 166 S

266

111

K 1

h1

1 S 211 b1S

212

1

22 K 1

h1

1 S 212 b1S

222

1

12

K 2

h11 S

2

66

L21

K 2

h1

S 122 a1S

112 S

222 a1

S 212

L22

K 1

h1

S 166 S

266

2

11 K 2

h1

1 S 112 a1S

111

2

22 K 2

h1

1 S 122 a1S

112

112

K 1

h1

1 S 166

a1 S

112 S

212

S 111 S

211

b1 S

112 S

212

S 122 S

222

h1

h2

K 1, K 2 are the shear lag parameters, Eq. (12).

A.3. Laminate constants

k 1 hk

L

k 1

k 1 2 k

2 hk

L

k 2

k 1 2

11 Q

111 S

211 b1S

212 Q

112 S

212 b1S

222

21

1

Q

222 S

122 a1S

112 Q

212 S

112 a1S

111

12 Q1

66 S 266 2

2 1

Q

266 S

166

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Stiffness degradation in cross-ply laminates damaged by transverse

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