Analysis of composite laminates with intra- and interlaminar damage

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Progress in Aerospace Sciences 41 (2005) 152–173

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Analysis of composite laminates with intra- andinterlaminar damage

Maria Kashtalyana, Costas Soutisb,�

aSchool of Engineering and Physical Sciences, University of Aberdeen, Fraser Noble Building, Aberdeen AB24 3UE, UKbAerospace Engineering, The University of Sheffield, Faculty of Engineering, Sir Frederick Mappin Building, Mappin Street,

Sheffield S1 3JD, UK

Abstract

Failure process of composite laminate under quasi-static or fatigue loading involves sequential accumulation of intra-

and interlaminar damage. Matrix cracking parallel to the fibres in the off-axis plies is the first intralaminar damage

mode observed. These cracks are either arrested at the interface or cause interlaminar damage (delamination) due to

high interlaminar stresses at the ply interface. This paper summarises recent theoretical modelling developed by the

authors on stiffness property degradation and mechanical behaviour of general symmetric laminates with off-axis ply

cracks and crack-induced delaminations. Closed-form analytical expressions are derived for Mode I, Mode II and the

total strain energy release rates associated with these damage modes. Dependence of strain energy release rates on crack

density, delamination area and ply orientation angle in balanced and unbalanced symmetric laminates is examined and

discussed. Also, stiffness degradation due to various types of damage is predicted and analysed.

r 2005 Elsevier Ltd. All rights reserved.

Contents

1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

1.1. Intralaminar damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

1.2. Interlaminar damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

2. Stress analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

3. Stiffness degradation due to intra- and interlaminar fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

3.1. Stiffness degradation due to off-axis ply cracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

3.2. Stiffness degradation due to crack-induced delamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

4. Onset and growth of intra- and interlaminar damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

4.1. Predicting onset and growth of off-axis ply cracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

5. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

e front matter r 2005 Elsevier Ltd. All rights reserved.

erosci.2005.03.004

ing author. Tel.: +4411 42227706; fax: +44 11 42227729.

ess: [email protected] (C. Soutis).

www.elsevier.com/locate/paerosci

ARTICLE IN PRESSM. Kashtalyan, C. Soutis / Progress in Aerospace Sciences 41 (2005) 152–173 153

1. Background

Failure of glass- and carbon-fibre reinforced plastic

(GFRP and CFRP) laminates subjected to static or

cyclic tensile loading acting in the plane of reinforce-

ment, and also under thermal fatigue, is a complex

process. It involves sequential accumulation of various

types of intra- and interlaminar damage, which gradu-

ally lead to the loss of the laminate’s stiffness and load-

carrying capability. The main damage mechanisms,

exhibited in composite layered plates, are matrix

cracking, delamination, fibre debonding, and fibre

breakage.

Damage mechanisms in composite laminates can be

studied theoretically following two approaches. Using

the continuum damage mechanics approach, various

types of damage are accounted for via the damage

tensor [1–5]. A composite is then described as a

continuum with mechanical properties depending on

the damage tensor. Using the damage micromechanics

approach, stress analysis of the damaged composite is

carried out in the explicit presence of damage. Various

types of damages are analysed directly with the aim to

predict their onset and growth, and also their effect on

the properties of the laminate. While for homogeneous

isotropic materials it is often possible to obtain exact

solutions within the linear elasticity theory, stress

analysis of damaged composite laminates is approximate

in the majority of cases. If interaction between various

types of damage is especially complex, stress field can

only be determined by numerical methods such as the

finite-element method [6,7].

1.1. Intralaminar damage

The first type of damage observed during the initial

stages of the failure process is the interlaminar damage

in the form of matrix cracks running parallel to the

fibres in off-axis plies of the laminate. Matrix cracking

initiates long before the laminate loses its load-carrying

capacity. It gradually reduces the stiffness and strength

of the laminate [8] and changes its coefficients of thermal

expansion [9], moisture absorption [10] and the natural

frequency [11]. Cracked matrix may cause leaks in

laminated composite pressure vessels. Matrix cracking

triggers development of other more harmful damage

mechanisms. Stress concentration near the crack tip at

the ply interface may cause either delamination [12] or

matrix cracking, or sometimes both, in the adjacent ply

[13–15]. Delaminations may result in fibre breakage in

the primary load-bearing plies [14] and lead to the loss of

the load-carrying capacity of the whole laminate.

Studies of matrix cracking have been focusing

predominantly on transverse cracks, i.e. matrix cracks

in the 90�-plies of a laminate. The laws of transverse

cracking in GFRP and CFRP laminates are in many

respects similar. As a rule, cracks in the matrix occur at

equal distances from each other [13] and immediately

propagate from edge to edge, cleaving the entire

thickness of the damaged ply. Under quasi-static

loading, the strain corresponding to the cracking onset

decreases with an increased ply thickness [15]. Under

cyclic loading, the cycle number corresponding to the

beginning of cracking increases with an increased

loading amplitude [16]. The degree of transverse

cracking is characterized by crack density, i.e., the

number of cracks per unit length. After cracking has

begun, the crack density abruptly increases with the

applied load. The rate of crack density increase

gradually decreases and the matrix cracking comes to

saturation state, which is sometimes called a character-

istic damaged state. The features of transverse cracking

of the matrix in ½90n=0m�s cross-ply CFRP laminates

were studied by Highsmith and Reifsnider [8] and Smith

et al. [17]. It was established that transverse cracking in

the outer 90�-plies begins at lower strains than in the

inner plies with a double thickness. However, the

saturation crack density is lower in this case. Moreover,

cracks are staggered rather than aligned in the outer

90�-plies.

The overwhelming majority of studies investigating

behaviour and properties of composite laminates with

matrix cracks assume that cracks are equally spaced and

therefore the analysis can be restricted to a representa-

tive segment of the laminate, containing one crack. The

analysis of a cross-ply laminate element with a

transverse crack is usually reduced to a plane problem

in the plane xOz (Fig. 1).

Shear-lag-based models remain the most commonly

used one for calculating the reduced stiffness properties

of laminates with transverse cracks. To eliminate the

dependence on the z-coordinate, it is assumed that the

stress sz, averaged across the layer thickness, is equal to

zero, and variation in the transverse displacement with

the longitudinal coordinate may be neglected [18]. The

dependence of in-plane displacements on the transverse

coordinate is assumed either linear [19,20] or quadratic

[7,21–23]. The latter is equivalent to the assumption that

the shear stresses depend linearly on the transverse

coordinate [18]. Some shear-lag models assume that the

shear stresses due to transverse cracking act only within

a thin resin-rich layer adjacent to a 90�-ply [8,24,25].

Zhang et al. [26] assumed that shear stresses in ½0m=90n�slaminates vary linearly throughout the thickness of the

90�-ply and one mth the thickness of the 0�-ply and

equal zero in the other parts. For cross-ply laminates

with a thick outer 0�-ply, Berthelot [27] assumed that the

dependence of the longitudinal displacements on the

transverse coordinate is quadratic in the 90�-ply and is

linear in the 0�-ply, but with the coefficient of

proportionality being an exponential function of the

density of transverse cracks.

ARTICLE IN PRESS

x

z

x

z

y

90°

0°

0°

transverse crack

Fig. 1. Cross-ply composite laminate with transverse matrix cracks.

M. Kashtalyan, C. Soutis / Progress in Aerospace Sciences 41 (2005) 152–173154

Hashin [28] was the first to solve a plane stress

problem for a cross-ply laminate with transverse cracks

based on the variational principles. He assumed that the

stresses sx in each ply depend only on x and do not

depend on z. The stresses determined based on this

hypothesis satisfy the equilibrium equations, the bound-

ary conditions, and the continuity conditions at inter-

faces, and the unknown constants can be determined

from the principle of minimum additional energy. The

tensile stresses sx obtained by the variational and shear-

lag methods are qualitatively close. The fundamental

difference is observed for the shear stress t at the

interface between plies. Within the framework of the

shear-lag methods, it turns out to be non-zero for x ¼ s,

which contradicts the assumption that the crack faces

are free from load. Moreover, the variational methods,

in contrast to the shear-lag methods, allow us to

determine the transverse stress sz.

Hashin’s variational approach was further developed

by Nairn [29], Varna and Berglund [30,31], and

Berglund and Varna [32]. The model of Varna and

Berglund [30] is a further development of Hashin’s

model. The authors assume that in a 90�-ply the stress sx

depends on the transverse coordinate z, the shear stress

sxz is a linear function of z, and the stress sz is a

quadratic function of z. In a 0�-ply, there may be an

inhomogeneous stress distribution described by an

exponential function with an unknown shape parameter.

The problem is reduced to a differential equation of the

fourth order with constant coefficients derived in

minimizing the additional energy. This equation and

its solution contain an unknown shape parameter, which

is calculated in the subsequent minimization of the

additional energy. The refined model developed by the

same authors [32] admits the inhomogeneity of the stress

distribution in both plies.

McCartney [33,34] assumed that the stresses sx in

each ply depend only on x and do not depend on z.

However, according to his approach, the problem is

reduced to a system of recurrent relations and an

ordinary differential equation of the fourth order. Also

plies are divided into subplies of smaller thickness, since

the elastic relations in the transverse direction are

satisfied in the averaged sense [35]. McCartney showed

that stresses and displacements determined by his

method are in agreement with those determined by the

variational method based on Reissner’s variational

theorem. Schoeppner and Pagano [36] directly used

Reissner’s variational principle. These approaches are

compared by McCartney et al. [37]. The variational

principles were used by Nairn [29] and Nairn and Hu

[12] to study transverse cracking in outer 90�-plies. Both

codirectional and non-codirectional cracks were

considered. Viscoelastic analyses of transverse cracking

in cross-ply composite laminates were made in [38,39].

In evaluating the effect of transverse cracks on the

stiffness of cross-ply laminates, many authors consid-

ered only the longitudinal elastic modulus [8,15,40–43].

Hashin [28] obtained the exact lower bound for the

longitudinal elastic modulus. Nairn and Hu [12,44] used

the variational principles to show that with the same

density of transverse cracks the stiffness of ½90n=0m�slaminates decreases more than that of ½0m=90n�slaminates whose 90�-plies are inner and adjacent. Both

co-directional and non-codirectional cracks were con-

sidered.

For cross-ply laminates with both transverse and

longitudinal cracks, a representative segment could be

defined by intersecting pair of transverse and long-

itudinal cracks [45–48]. Kashtalyan and Soutis [49–53]

suggested to analyse cross-ply laminates with damage in

both plies using the Equivalent Constraint Model.

Instead of the damaged laminate, two ECM laminates

are considered and analysed simultaneously. In the first

laminate, 0� layers contain damage explicitly, while 90�

plies are replaced with equivalent homogeneous ones

with reduced stiffness properties. These reduced stiffness

properties are assumed to be known from the analysis of

the second laminate, in which 90� layer contains damage

explicitly, while the damages 0� plies are replaced with

equivalent homogeneous ones with reduced properties,

assumed to be known from the analysis of the first


laminate. Thus, problems for both laminates are inter-

related.

Angle-ply laminates exhibit much more complex

morphologies of intralaminar damage than cross-ply

laminates. Comprehensive observations of sequential

accumulation of matrix cracks in off-axis plies have been

reported for quasi-isotropic ½0=45=� 45=90�s and ortho-tropic carbon/epoxy and glass/epoxy laminates [54–57].

It was found that longitudinal strain for matrix cracking

initiation decreases with increasing ply orientation angle

[56]. Also, ply stresses normal to the fibres at crack

formation were found to become progressively smaller

as the ply orientation angle increased [56].

Stress fields in the cracked off-axis plies of angle-ply

laminates were examined by means of finite element

method [58] and analytically [59]. Application of the

Equivalent Constraint Model to quasi-isotropic lami-

nates with matrix cracking in all but 0� layers was

presented by Zhang and Herrmann [60].

Soutis and Kashtalyan [61,62] predicted analytically

strain energy release rate associated with matrix crack-

ing in the y-layer of unbalanced symmetric ½0=y�scomposite laminate using a 2-D shear lag stress analysis

and the Equivalent Constraint Model (ECM). Compar-

ison of theoretical predictions with experimental data

for glass/epoxy laminates, obtained by Crocker et al.

[56], showed that a quadratic mixed mode fracture

criterion can successfully predict the cracking onset

strain for ply orientation angles 75�pyp90�.

1.2. Interlaminar damage

Matrix cracks, developing in the off-axis plies of the

laminate, are either arrested at the interface or cause

interlaminar damage leading to delamination due to

high interlaminar stresses at the ply interface. Fig. 2

summarises schematically types of intra- and interlami-

nar damage observed in cross-ply and angle-ply

composite laminates: transverse and longitudinal matrix

cracks in cross-ply laminates (Fig. 2a) transverse and

longitudinal crack tip delaminations in cross-ply lami-

nates (Fig. 2b) off-axis ply cracks in balanced symmetric

laminates (Figs. 2c–d); uniform (Fig. 2c) and partial

(Fig. 2d) local delaminations induced by angle ply

matrix cracks; crack induced edge delaminations

(Fig. 2e).

Studies of delaminations induced by matrix cracking

have been focusing predominantly on delaminations

caused by transverse cracks, i.e. matrix cracks in the 90�-

plies of a laminate. Crossman and Wang [63] made

comprehensive observations of transverse cracking and

delamination in balanced symmetric ½�25=90n�s; n ¼

0:5; 1; 2; 3; 4; 6; 8 graphite/epoxy laminates. A significant

reduction in the delamination onset strain was noted for

the laminates with nX4. A transition from edge

delamination to local delaminations growing from the

tip of a matrix crack in the 90�-ply occurred between

n ¼ 3 and 4. The onset and growth of edge delamination

in ½ð�30Þ2=90=90�s graphite/epoxy laminates under statictension and tension-tension fatigue loading was studied

by O’Brien [64]. Stiffness loss was monitored simulta-

neously with delamination growth and found to

decrease linearly with delamination size.

Armanios et al. [65] applied a shear deformation

theory and sublaminate approach to analyse local

delaminations originating from transverse cracks in

CFRP ½�25=90n�s laminates. Predictions of their model,

which also takes into account hygrothermal effects, are

in reasonable agreement with delamination onset strain

data by Crossmann and Wang [63].

Nairn and Hu [12,44] used two-dimensional varia-

tional approach to analyse crack tip delaminations in

½ðSÞ=90n�s laminates, where ðSÞ denotes a balanced

sublaminate, e.g. ð�ymÞ. They predicted that matrix

cracking should reach some critical density before

delamination initiates. The critical crack density for

delamination initiation is determined by material

properties, laminate structure as well as fracture

toughnesses for matrix cracking and delamination and

is nearly independent of the properties of the supporting

sublaminate ðSÞ.

Zhang et al. [22,66] used a 2-D improved shear lag

analysis to predict the strain energy release rate for edge

and local delaminations in balanced symmetric

½�ym=90n�s laminates. For edge delamination, they were

able to capture a zigzag delamination pattern, i.e. edge

delamination switching from one ðy=90Þ interface to

another through a matrix crack, and improve O’Brien’s

formula for strain energy release rate for edge delamina-

tion [64] incorporating the effect of matrix cracking. For

local delaminations, they obtained the strain energy

release rate as a function of crack density and

delamination area. Their predictions for delamination

onset strain agree well with experimental data of

Crossman and Wang [63] and capture the transition

from edge to local delamination quite accurately.

Initiation and growth of local delaminations from the

tips of transverse cracks in cross-ply ½0=90n�s n ¼ 2; 4; 6carbon/epoxy laminates under static tension was exam-

ined by Takeda and Ogihara [67]. Delamination was

noted to grow more rapidly and extensively in the

laminates with thicker 90� plies. Ogihara and Takeda

[68] used a modified shear lag method featuring

interlaminar shear layer to predict strain energy release

rate for transverse crack tip delaminations in cross-ply

½0=90n�s laminates and to model interaction between

transverse cracking and delamination. However, the

effect of cracking/delamination interaction was found to

be negligible in prediction of delamination growth.

Henaff-Gardin et al. [47,48] and Kobayashi et al. [43]

observed damage development in carbon/epoxy cross-

ply laminates under thermal cycling. The first damage

ARTICLE IN PRESS

transverse cracks

delaminationsdelaminations

transverse crack

longitudinal crack

off-axis crackdelamination

off-axis crack delamination

edge delaminations

ply cracks

(a) (b)

(c) (d)

(e)

Fig. 2. Types of intra- and interlaminar damage observed in cross-ply and angle-ply composite laminates: (a) transverse and

longitudinal matrix cracks in cross-ply laminates; (b) transverse and longitudinal crack tip delaminations in cross-ply laminates; (c) off-

axis ply cracks in balanced symmetric laminates and uniform local delaminations; (d) off-axis ply cracks in balanced symmetric

laminates and partial local delaminations; (e) matrix-crack induced edge delamination.


mode observed consisted of matrix cracks in 0� and 90�

plies. The first damage mode observed consisted of

matrix cracks in 0� and 90� plies. Most of the matrix

cracks spanned the entire width or length of the

specimen. Then delaminations initiated between 0� and

90� plies along the pre-existing cracks in 0� ply.

More recently, Selvarathinam and Weitsman [69,70]

observed and modelled, by means of finite elements

and shear lag methods, delaminations induced by

matrix cracking in cross-ply laminates under environ-

mental fatigue. By comparing strain energy release

rates associated with matrix cracking and delamina-

tion, they were able to explain the extensive delamina-

tions and reduced crack densities that arise under

immersed fatigue conditions, as compared with fatigue

in air.


Zhang et al. [71] studied delaminations induced by

transverse cracking at the ðf=90Þ interfaces in

½. . . =ji=fm=90n�s laminates loaded in tension using a

sublaminate-wise first-order shear deformation theory.

In particular, they were interested in the constraining

effect of the immediate neighbouring plies and remote

plies on stiffness reduction and strain energy release rate

for delaminations. It was found that the strain energy

release rate for local delamination and stiffness reduc-

tion of the constrained transverse plies largely depends

on a local lay-up configuration of a damaged laminate.

Kashtalyan and Soutis [49–51] examined the effect of

crack tip delaminations on stiffness reduction for cross-

ply ½0m=90n�s laminates with local delaminations along

transverse as well as longitudinal cracks. It was

established that reduction in the laminate shear modulus

and Poisson’s ratio is much more significant than in the

axial modulus. For balanced symmetric ½�ym=90n�s, the

effect of constraining ply orientation angle y on

reduction of the laminate in-plane stiffness properties

was also examined.

Delaminations induced by angle ply matrix cracks in

carbon/epoxy ½02=y2=�y2�s; y ¼ 20�; 25�; 30� laminates

subjected to tension fatigue loading were observed by

O’Brien and co-workers [55,72–74]. Matrix cracks

formed near the stress free edge and delaminations,

bounded by the free edge and the crack, developed in the

y=ð�yÞ interface. They were termed partial local

delaminations.

Using a quasi-3D finite element (FE) analysis,

Salpekar and O’Brien [73] found that the strain energy

release rate for uniform local delamination calculated

from O’Brien [72] expression matched the value

obtained by FE analysis in the laminate interior.

O’Brien and Hooper [55] and O’Brien [74] observed

matrix crack induced delaminations in symmetric angle-

ply ½02=y2=�y2�s carbon/epoxy laminates under quasi-

static and fatigue tensile loading ðy ¼ 15�; 20�; 25�; 30�Þ.Delaminations occurred in the ðy=� yÞ interface,

bounded by the cracks in the ð�yÞ-ply and the stress

free edge. The laminated plate theory and a quasi-3D

finite element analysis were used to examine stresses in

the ð�yÞ-ply. For the considered range of ply orienta-

tions, stresses normal to the fibres were found to be

compressive and shear stresses along the fibres to be

high in the laminate interior, while near the free edge

high tensile stresses normal to the fibres were present.

Two closed form expressions for strain energy release

rate were derived on the basis of simple load shearing

rules: one for a local delamination growing from an

angle ply matrix crack with a uniform delamination

front across the laminate width, and one for apartial

local delamination growing from an angle ply matrix

crack and bounded by the free edge.

Salpekar and O’Brien [75] used a 3-D FE analysis to

study matrix crack induced delaminations in ð0=y=� yÞs

graphite/epoxy laminates (y ¼ 15�; 45�) loaded in ten-

sion. For ð0=45=� 45Þs laminate, the strain energy

release rate for local delamination growing uniformly

in the ð45=� 45Þ interface from the matrix crack in the

(�45�)-ply was found to be higher near the laminate

edge than in the interior of the laminate.

Later, Salpekar et al. [76] computed strain energy

release rates associated with local delamination originat-

ing from matrix cracks and bounded by the free edge in

ð0=y=� yÞs and ðy=� y=0Þs graphite/epoxy laminates

using a 3-D FE method. The total strain energy release

rate was calculated using three different techniques:

the virtual crack closure technique, the equivalent

domain integral technique, and a global energy balance

technique.

Kashtalyan and Soutis [62] theoretically modelled

local delaminations growing uniformly from the tips of

matrix cracks in an angle-ply laminate loaded in tension.

They obtained closed-form expression for strain energy

release rates, associated with these delaminations, as

linear functions of the first partial derivatives of the

effective elastic properties of the damaged layer with

respect to delamination area. Strain energy release rate

dependent of delamination area and crack density, thus

taking into account the cumulative effect of damage.

The total strain energy release rate depends linearly on

crack density both in balanced ½02=y2=�y2�s and

unbalanced ½02=y2�s laminates. The dependence on

delamination area is linear in balanced and non-linear

in unbalanced laminates. Comparison with results by

Salpekar and O’Brien [73] showed that O’Brien’s closed-

form expression [72] for uniform local delamination

significantly overestimates the value of strain energy

release rate. For the same ply orientation angle, crack

density and delamination area, delamination-induced

changes in stiffness properties are much more significant

in unbalanced laminates than in balanced laminates.

In all above studies of crack induced delaminations it

was assumed that delamination surfaces, like matrix

crack surfaces, are stress-free. Besides that, delamina-

tions were assumed to behave in a self-similar manner,

i.e. boundary conditions prescribed at the delaminated

surfaces were assumed to be the same for small and large

delaminations. More recently, Ashkantala and Talreja

[77] and Berthelot and Le Corre [78,79] examined

transverse crack tips delaminations in cross-ply lami-

nates with shear friction between the delaminated plies.

While Berthelot and Le Corre [78] assumed the

magnitude of the interlaminar shear stress at the

delaminated interface to be constant, i.e. independent

of delamination length, Ashkantala and Talreja [77]

considered both linear and cubic polynomial shear stress

distribution at the delamination interface. Selvarathi-

nam and Weitsman [69,70] observed and modelled, be

means of finite elements and shear lag methods,

delaminations induced by matrix cracking in cross-ply

ARTICLE IN PRESSM. Kashtalyan, C. Soutis / Progress in Aerospace Sciences 41 (2005) 152–173158

laminates under environmental fatigue, with delamina-

tion surfaced loaded with hydrostatic pressure.

2. Stress analysis

In Fig. 3 a schematic of a symmetric ½ðSÞ=f�s laminateis shown, consisting of the outer sublaminate ðSÞ and the

inner f-layer damaged by matrix cracks and local

delaminations growing from their tips at the ðSÞ=finterface. The outer sublaminate ðSÞ, or layer 1, may

consist either of a single layer or a group of layers and

can also be damaged (in this case it needs to be replaced

in the analysis with an equivalent homogeneous layer

with reduced stiffness properties). The laminate is

referred to the global Cartesian coordinate system xyz

and local coordinate system x1x2x3, with the axis x1

directed along the fibres in the damaged f-layer, or layer2. The laminate is subjected to in-plane biaxial tension

sx and sy. Since the laminate is symmetric, no coupling

exists between in-plane loading and out-of-plane defor-

mation. Matrix cracks are assumed to be spaced

uniformly, with crack spacing 2s and span the whole

width of the laminate. Local delaminations are assumed

to be strip shaped, with strip width 2‘, Fig. 3.Due to the periodicity of damage, the stress analysis

may be carried out over a representative segment

containing one matrix crack and two crack tip

delaminations. Due to symmetry, it can be further

x

y

x2

2s

x1

off-axis crack

2l

delamination

2h2(�)

(S) h1

�y

�y

�x�x

�

Fig. 3. Front and edge view of a ½ðSÞ=f�s laminate subjected to

biaxial tensile loading and damaged by matrix cracks and

crack-induced delaminations. Local ðx1x2x3Þ and global ðxyzÞ

co-ordinate systems for the damaged f-layer (front view in the

negative x3 z direction).

restricted to one quarter of the representative segment,

Fig. 4, referred to the local coordinate system x1x2x3.

Let f ~sð1Þg and f~�ð1Þg denote the in-plane microstressesand microstrains in the layer 1, and f ~sð2Þg and f~�ð2Þgdenote the in-plane microstresses and microstrains in the

layer 2 (i.e. stresses and strains averaged across the

respective layer thickness). Since it is assumed that there

is no frictional contact between the layers in the locally

delaminated portion of the representative segment

ð0ojx2jo‘; jx3joh2Þ,the in-plane microstresses in the

delaminated portion are ~sð2Þ22 ¼ ~sð2Þ12 ¼ 0, i.e. this region is

stress-free. Assumption of stress-free crack tip delami-

nation surfaces, and the resulting implication that the

portion of the damaged ply bounded by matrix crack

and delamination surfaces is stress-free, has been widely

used in the studies of delaminations. Besides that,

delaminations are assumed to behave in a self-similar

manner, i.e. the boundary conditions prescribed at the

delaminated surfaces were assumed to be the same for

small and large delaminations.

In the perfectly bonded region ð‘ojx2josÞ of the

representative segment, they are determined from the

equilibrium equations

d

dx2~sð2Þj2 �

tj

h2¼ 0 j ¼ 1; 2, (1)

where tj are the interface shear stresses and h2 is the

thickness of the f-layer.By averaging the out-of-plane constitutive equations

for both layers across the layer thickness, the interface

shear stresses tj can be expressed in terms of the in-plane

displacements and shear lag parameters Kij as

tj ¼ Kj1ð ~uð1Þ1 � ~uð2Þ1 Þ þ Kj2ð ~u

ð1Þ2 � ~uð2Þ2 Þ. (2)

The shear lag parameters K11;K22;K12 K21 are

determined assuming that the out-of-plane shear stresses

~sðkÞj3 vary linearly with x3 (Fig. 5), see Appendix A.

Substitution of Eqs. (2) into Eqs. (1) and subsequent

x3

x2

loff-axis plycrack

s - l

delamination

�, or layer 2

(S), or layer 1

Fig. 4. A quarter of the representative segment containing a

matrix crack and delamination.

ARTICLE IN PRESS

(1)13 ,

x3

h1

h2

x2

layer 1

layer 2

� (1)23�

(2)13 ,� (2)

23�

�1,�2

Fig. 5. Variation of out-of-plane shear stresses.

M. Kashtalyan, C. Soutis / Progress in Aerospace Sciences 41 (2005) 152–173 159

differentiation yields

d2

dx2~sð2Þj2 þ Kj1ð~g

ð1Þ12 � ~gð2Þ12 Þ þ Kj2ð~�

ð1Þ22 � ~�ð2Þ22 Þ ¼ 0,

j ¼ 1; 2. ð3Þ

The strain differences ð~�ð1Þ22 � ~�ð2Þ22 Þ and ð~gð1Þ12 � ~gð2Þ12 Þ, in-

volved in Eqs. (3), can be expressed in terms of stresses

~sð2Þ12 ; ~sð2Þ22 using the constitutive equations for both layers,

the laminate equilibrium equations are given below

wf ~sð1Þg þ f ~sð2Þg ¼ ð1þ wÞ½T �fsg, (4a)

½T � ¼

cos2f sin2f 2 sinf cosf

sin2f cos2f �2 sinf cosf

� sinf cosf sinf cosf cos2f� sin2f

264

375,(4b)

fsg ¼ fsx; sy; 0gT; w ¼ h1=h2 (4c)

and the assumption of the generalised plane strain

condition

~�ð1Þ11 ¼ ~�ð2Þ11 . (5)

In the local coordinate system x1x2x3, the layer 2 is

orthotropic,

~�ð2Þ11

~�ð2Þ22

~gð2Þ12

8>><>>:

9>>=>>; ¼

Sð2Þ

11 Sð2Þ

12 0

Sð2Þ

12 Sð2Þ

22 0

0 0 Sð2Þ

66

26664

37775

~sð2Þ11

~sð2Þ22

~sð2Þ12

8>><>>:

9>>=>>;, (6a)

while the layer 1 is anisotropic

~�ð1Þ11

~�ð1Þ22

~gð1Þ12

8>><>>:

9>>=>>; ¼

Sð1Þ

11 Sð1Þ

12 Sð1Þ

16

Sð1Þ

12 Sð1Þ

22 Sð1Þ

26

Sð1Þ

16 Sð1Þ

26 Sð1Þ

66

26664

37775

~sð1Þ11

~sð1Þ22

~sð1Þ12

8>><>>:

9>>=>>;, (6b)

where ½SðkÞ� is the compliance matrix for the kth layer.

Finally, Eqs. (3) can be reduced to a system of two

coupled second order ordinary differential equations

(see Appendix B)

d2 ~sð2Þ12

dx22

� N11 ~sð2Þ12 � N12 ~s

ð2Þ22 � P11sx � P12sy ¼ 0, (7a)

d2 ~sð2Þ22

dx22

� N21 ~sð2Þ12 � N22 ~s

ð2Þ22 � P21sx � P22sy ¼ 0. (7b)

Here Nij and Pij are laminate constants depending on

the layer compliances SðkÞ

ij , layer thickness ratio w, shearlag parameters K11;K22;K12 and angle f (Appendix B).

Eqs. (7a) and (7b) can be uncoupled at the expense of

increasing the order of differentiation, resulting in a

fourth order non-homogeneous ordinary differential

equation

d4 ~sð2Þ22

dx42

� ðN11 þ N22Þd2 ~sð2Þ22

dx22

� ðN21N12 � N11N22Þ ~sð2Þ22 þ ½N11ðP21 þ aP22Þ

� N21ðP11 þ aP12Þ�sx ¼ 0. ð8Þ

Here a ¼ sy=sx is the biaxiality ratio. The boundary

conditions for Eq. (8) are prescribed at the stress-free

boundary between locally delaminated and perfectly

bonded portions of the representative segment

~sð2Þ22 jx2¼�‘ ¼ 0 ~sð2Þ12 jx2¼�‘ ¼ 0. (9)

Finally, the in-plane microstresses can be expressed in

the following form:

~sð2Þ11 ¼ a22 ~sð2Þ22 þ a12 ~s

ð2Þ12 þ bxsx þ bysy, (10a)

~sð2Þj2 ¼ Ajcosh l1ðx2 � sÞ

cosh l1ðs � ‘Þþ Bj

cosh l2ðx2 � sÞ

cosh l2ðs � ‘Þþ Cj

� �sx

j ¼ 1; 2, ð10bÞ

where coefficients a22; a12;bx and by are given in

Appendix B, lj are the roots of the characteristic

equation and Aj ;Bj and Cj are constants depending on

Nij and Pij , see Appendix C.

In cross-ply and balanced laminates the outer

sublaminate ðSÞ is orthotropic, with compliances Sð1Þ

16 ¼

Sð1Þ

26 ¼ 0 and stiffnesses Qð1Þ

45 ¼ 0. In this case shear lag

coefficients K12 K21 ¼ 0 vanish, and equilibrium

equations are reduced to two uncoupled seconded order

differential equations. Details of this case are given

elsewhere [49,51,80].

ARTICLE IN PRESS

0

50

100

150

200

15 30 45 60 75 90

Ply orientation angle, degrees

Nor

mal

str

ess,

MP

a

uniaxial

0.5

1

2

Yt=57MPa

-25

0

25

50

75

100

15 30 45 60 75 90

Ply orientation angle, degrees

She

ar s

tres

s, M

Pa

uniaxial

0.5

1

2

S=71 MPa

(a)

(b)

Fig. 6. Ply stresses in the y-ply of the AS4/3506-1 graphite/

epoxy ½02=y2�s laminate as function of ply orientation angle yunder uniaxial ðsy=sx ¼ 0Þ and biaxial ðsy=sx ¼ 0:5; 1; 2Þ tensileloading: (a) stresses normal to the fibres; (b) shear stress. The

applied stress is sx ¼ 100MPa.


Fig. 6 shows the in-plane stresses in the y-ply of the

½02=y2�s laminate as function of ply orientation angle yunder uniaxial ðsy=sx ¼ 0Þ and biaxial ðsy=sx ¼

0:5; 1; 2Þ tensile loading as calculated from the classical

laminated plate theory. The applied stress is

sx ¼ 100MPa, the material system is AS4/3506-1

graphite/epoxy and itsengineering properties are as

follows [55]: axial modulus E11 ¼ 135GPa, transverse

modulus E22 ¼ 11 GPa, in-plane shear modulus G12 ¼

5:8 GPa, major Poisson’s ratio n12 ¼ 0:301. Nominal plythickness is t ¼ 0:124mm.The in-plane stresses were transformed into the local

coordinate system, Fig. 3, in order to determine stresses

normal to the fibres, ~s22, and shear stresses along the

fibres, ~s12, since these stresses contribute directly to the

formation of matrix cracks. Under uniaxial tension,

stresses normal to the fibres in the y-ply of the ½02=y2�slaminate are compressive for y smaller than 25� and

tensile for y greater than 25�, Fig. 6a. Under biaxial

loading, stresses in the ðyÞ ply normal to the fibres are

tensile. For all considered biaxiality ratios a ¼ sy=sx,

the highest normal stresses are observed for y ¼ 15�.

They decrease significantly as y increases and at y ¼ 60�

reach almost a constant value that is almost independent

of the biaxiality ratio. For 15�pyp60�, the magnitude

of normal stresses increases with increasing biaxiality

ratio, sometimes exceeding the value of the transverse

tensile strength of the material. Under uniaxial tension,

relatively high shear stresses are present along the fibres,

Fig. 6b, which may contribute to the formation of

matrix cracking in the y-ply of the ½02=y2�s laminate.

3. Stiffness degradation due to intra- and interlaminar

fracture

The reduced stiffness properties of the mth ply can be

determined by applying the laminate plate theory to the

ECM m laminate after replacing the explicitly damaged

mth ply with an equivalent homogeneous one. In the

local coordinate system x1x2x3, the constitutive equa-

tions of the ‘equivalent’ homogeneous layer are

fsð2Þg ¼ ½Qð2Þ�f�ð2Þg (11)

In the local coordinates, the modified in-plane stiffness

matrix ½Qð2Þ� of the homogeneous layer equivalent to the

damaged one is related to the in-plane stiffness matrix

½Qð2Þ� of the undamaged layer as

½Qð2Þ� ¼ ½Qð2Þ�

�

ðQð2Þ

12 Þ2=Q

ð2Þ

22L22 Qð2Þ

12L22 0

Qð2Þ

12L22 Qð2Þ

22L22 0

0 0 Qð2Þ

66L66

266664

377775. ð12Þ

Here LðmÞ22 ;L

ðmÞ66 are the in-situ damage effective functions

(IDEFs) [26,66]. They can be expressed in terms of

lamina macrostresses and macrostrains as

L22 ¼ 1�sð2Þ22

Qð2Þ

12 �ð2Þ11 þ Q

ð2Þ

22 �ð2Þ22

,

L66 ¼ 1�sð2Þ12

Qð2Þ

66 gð2Þ12

. ð13Þ

The lamina macrostresses fsð2Þg and macrostrains f�ð2Þgare obtained by averaging, respectively, microstresses

f ~sð2Þg, Eqs. (10), and microstrains f~�ð2Þg, Eqs. (6a), acrossthe length of the representative segment. The lamina

ARTICLE IN PRESS

0.5

0.6

0.7

0.8

0.9

1.0

0 10 20 30Crack density (cracks/cm)

Nor

mal

ised

stif

fnes

s pr

oper

ty

Axial modulus

Transverse modulus

Shear modulus

Poisson's ratio

Fig. 7. Normalised stiffness properties of ½0=90�s (filled

symbols) and ½0=89�s (open symbols) glass/epoxy laminates as

a function of crack density Cmc in the inner ply (cracks/cm).


macrostresses sð2Þij are

sð2Þ11 ¼ a22sð2Þ22 þ a12s

ð2Þ12 þ bxsx þ bysy, (14a)

sð2Þj2 ¼ Aj2Dmc

l�1ð1� DldÞtanh

l�1ð1� DldÞ

Dmc

�

þ Bj2Dmc

l�2ð1� DldÞtanh

l�2ð1� DldÞ

Dmc

þ Cj2ð1� DldÞ�sx; j ¼ 1; 2, ð14bÞ

where Dmc ¼ h2=s denotes relative crack density and

Dld ¼ ‘=s denotes relative delamination area. The

macrostrains in the individual homogeneous layers and

the laminate are assumed to be equal

�ð1Þ11 ¼ �ð2Þ11 ¼ �11; �ð1Þ22 ¼ �ð2Þ22 ¼ �22; gð1Þ12 ¼ gð2Þ12 ¼ g12.

ð15Þ

Using the constitutive equations for layer 1, Eq. (6b),

and equations of the global equilibrium of the laminate,

Eq. (4), the lamina macrostrains in the layer 2 are

f�ð2Þg ¼ ½Sð1Þ�w�1ðð1þ wÞ½T �fsg � fsð2ÞgÞ, (16)

where the transformation matrix ½T � is given by Eq. (4b).

Thus, the lamina macrostresses, Eq. (14), and macro-

strains, Eq. (16), are determined as explicit functions of

the damage parameters Dmc;Dld.

Finally, the modified stiffness matrix ½Q�2 of the

‘equivalent’ homogeneous layer in the global coordi-

nates xyz can be obtained from the modified stiffness

matrix ½Qð2Þ� in the local coordinates, Eq. (12) as

½Q�2 ¼ ½T ��1½Qð2Þ�½T ��T (17)

where the transformation matrix ½T � is given by Eq. (4b).

The extension stiffness matrix ½A� of the ‘equivalent’

laminate in the global coordinates xyz can then be

determined as

½A� ¼X

k

½Q�khk; k ¼ 1; 2, (18)

where ½Q�1 is the in-plane stiffness matrix of layer 1, or

the outer sublaminate, in the global coordinates.

3.1. Stiffness degradation due to off-axis ply cracking

In this subsection, predictions of stiffness properties

as function of damage are presented and discussed for

balanced ½�y�s glass/epoxy laminates and unbalanced

½02=y2�s graphite/epoxy laminates. The results are pre-

sented in terms of the laminate stiffness properties, that

is the axial modulus Ex, transverse modulus Ey, shear

modulus Gxy, major Poisson’s ratio nxy, as well as

shear–extension coupling coefficients Zxy;x and Zxy;y. The

shear–extension coupling coefficients Zxy;j ¼ gxy=�j char-

acterise shearing in the xy plane caused by normal stress

in the jth direction (j ¼ x; y).

To validate the developed approach, a limiting case of

a cross-ply [0/90] laminate was considered (Kashtalyan

and Soutis, [52]). Cross-ply laminates cannot be

analysed using the developed approach for angle-ply

½y1=y2�s laminates directly, because when y1 ¼ 0 and

y2 ¼ 90, the system of differential equations, Eq. (9),

becomes uncoupled, and the solution, given by Eq. (12),

is no more valid. However, it works for any y2 close

enough to 90�. Fig. 7 shows normalised (i.e. referred to

their value in the undamaged state) stiffness properties

versus crack density in the inner ply for a ½0=89�sglass–epoxy laminate, obtained using the current meth-

od. They appear to be in a good agreement with results

for a cross-ply ½0=90�s laminate obtained using the

ECM/2-D shear lag model [49–53]. The material proper-

ties of a unidirectional E-glass 1200tex fibre reinforced

MY750/HY917/DY063 epoxy composite are as follows:

EA ¼ 45:6GPa, ET ¼ 16:2GPa, GA ¼ 5:83GPa, nA ¼

0:278, GT ¼ 5:79GPa, single ply thickness t ¼ 0:25mm.Fig. 8 shows the normalised stiffness properties of two

angle-ply ½y1=y2�s glass/epoxy laminates as a function of

the crack density in the inner ð�yÞ ply. Crack density

C2 ¼ ð2sð2ÞÞ�1 in the inner ð�yÞ layer varies from 0 (no

matrix cracking) to 30 cracks/cm, which corresponds to

variation in the damage parameter Dmc2 from 0 to 1.5.

In the ½30=�30�s laminate, Fig. 8a, the reduction of

the axial modulus Ex is bigger than that of the

transverse modulus Ey, while for the ½55=�55�s laminatethe opposite is true, Fig. 8b. In both laminates, the

reduction of the shear modulus Gxy is smaller than that

of Ex and Ey, in contrast to the cross-ply ½0=90�slaminate, where the reduction in the shear modulus and

the Poisson’s ratio is bigger than that of the axial

modulus, Fig. 7. It is also worth noting that in angle-ply

laminates matrix cracking may actually increase the

ARTICLE IN PRESS

0.00

0.03

0.06

0.09

0.12

0.15

0 10 20 30

Axial

Transverse

0.00

0.02

0.04

0.06

0.08

0.10 Axial

Transverse

Crack density (cracks/cm)

0 10 20 30


She

ar e

xten

sion

cou

plin

g co

effic

ient

sS

hear

ext

ensi

on c

oupl

ing

coef

ficie

nts

(a)

(b)

Fig. 9. Shear extension coupling coefficients of angle-ply

½y=�y�s glass/epoxy laminates as a function of crack density

Cmc in the inner ð�yÞ layer (cracks/cm) for: (a) ½30=�30�slaminate; (b) ½55=�55�s laminate.

0.5

0.7

0.9

1.1

1.3

1.5

0 10 20 30

Axial modulus

Transversemodulus

0.6

0.7

0.8

0.9

1.0

1.1Axial modulusTransverse modulusShear modulusPoisson's ratio



Nor

mal

ised

stif

fnes

s pr

oper

tyN

orm

alis

ed s

tiffn

ess

prop

erty

(a)

(b)

Fig. 8. Normalised stiffness properties of angle-ply ½y=�y�sglass/epoxy laminates as a function of crack density Cmc in the

inner ð�yÞ layer (cracks/cm): (a) ½30=�30�s laminate; (b)

½55=�55�s laminate.


Poisson’s ratio—a phenomenon, not observed in cross-

ply laminates. In the ½30=�30�s laminate the increase inthe Poisson’s ratio occurs for all crack densities, while in

the ½55=�55�s laminate, it is observed only at higher

crack densities, Fig. 8b.

In the undamaged state, angle-ply ½�y�s laminates arebalanced and orthotropic and exhibit no coupling

between extension and shear. When matrix cracking

occurs, the laminate becomes unbalanced, resulting in

coupling between extension and shear, reflected by non-

zero shear–extension coupling coefficients Zxy;x and Zxy;y.

Dependence of shear–extension coupling coefficients on

the crack density is shown in Fig. 9 for ½30=�30�s and½55=�55�s glass/epoxy laminates. When 0�oyo45�, the

Zxy;x coefficient increases more rapidly with the crack

density than the Zxy;y, while for 45�oyo90� the opposite

is true.

Fig. 10 shows the normalised stiffness properties with

for ½02=552�s and ½02=752�s AS4/3506-1 laminates as a

function of crack density Cmc ¼ ð2s2Þ�1. It may be seen

that in both angle-ply laminates the most significantly

reduced properties are transverse and shear moduli. In a

½02=552�s laminate (Fig. 10a), a slight increase of the

Poisson’s ratio with the crack density is observed.

Ought to their unbalanced configuration, ½02=y2�slaminates exhibit shear extension coupling characterised

by axial Zxy;x ¼ gxy=�x and transverse Zxy;y ¼ gxy=�y

shear–extension coefficients. Fig. 11 shows variation of

the shear–extension coupling coefficients with the crack

density Cmc in a ½02=752�s laminate. While axial

shear–extension coupling coefficient is almost unaffected

by matrix cracking, the transverse one is increased by

theabsolute value.

3.2. Stiffness degradation due to crack-induced

delamination

Fig. 12 shows normalised stiffness properties of T800H/

3631 carbon/epoxy ½0=90n�s; n ¼ 2; 4; 6 cross-ply lami-

nates containing transverse cracks and delaminations.

ARTICLE IN PRESS

0.8

0.9

1

1.1

0 2 4Crack density (cracks/cm)

Nor

mal

ised

stif

fnes

s pr

oper

ty

Axial modulus

Transverse modulus

Shear modulus

Poisson's ratio

Axial modulus

Transverse modulus

Shear modulus

Poisson's ratio

0.8

0.9

1

Nor

mal

ised

stif

fnes

s pr

oper

ty

1 3 5

0 2 4Crack density (cracks/cm)

1 3 5

(a)

(b)

Fig. 10. Normalised stiffness properties of angle-ply ½02=y2�sgraphite/epoxy laminates as a function of crack density Cmc in

the inner y layer (cracks/cm): (a) ½02=552�s laminate; (b)

½02=752�s laminate.

-1.8

-1.2

-0.6

0

0.6

0 3Crack density (crack/cm)

She

ar e

xten

sion

cou

plin

g co

effic

ient

s

Axial

Transverse

5421

Fig. 11. Shear extension coupling coefficients of angle-ply

½02=752�s graphite/epoxy laminate as a function of crack densityCmc in the inner layer (cracks/cm).


The axial modulus Ex, shear modulus Gxy and Poisson’s

ratio nxy, normalised by their values in the undamaged

state, are plotted against the transverse crack density

Cmc ¼ ð2s2Þ�1. The relative delamination area is

Dld ¼ 10%, which corresponds to ‘=s ¼ 0:1. For the

axial modulus, predictions are compared to experimen-

tal data obtained by Takeda and Ogihara [67] and

appear to be in acceptable agreement. However,

predictions show that reduction in shear modulus and

Poisson’s ratio due to crack tip delamination is more

significant (no available experimental data).

Fig. 13 shows the variation of the normalised stiffness

properties of the ½02=302=�302�s AS4/3506-1 graphite/

epoxy laminate with the relative delamination area

Dld ¼ ‘=s. Matrix crack density in the inner ð�30�Þ-ply

is assumed equal to C ¼ 2 cracks=cm. Values at Dld ¼ 0

indicate stiffness properties of the laminate at this crack

density without delaminations. It can be seen that local

delaminations further decrease the laminate moduli and,

for the considered lay-up, increase the Poisson’s ratio

(Fig. 13a). In balanced ½02=y2=�y2�s laminates uniformlocal delaminations result in an increase in the absolute

value of the axial shear–extension coupling coefficient

for yo45� and of the transverse shear–extension

coupling coefficient fory445�. However, all shear–ex-

tension coupling coefficients are significantly smaller

than those for unbalanced laminates (see later Fig. 13b).

Fig. 14 shows the normalised stiffness properties in

the ½02=302�s AS4/3506-1 graphite/epoxy laminate. Axialmodulus Ex, transverse modulus Ey, shear modulus Gxy

and major Poisson’s ratio nxy normalised by their value

for the undamaged laminates are plotted as function of

the relative delamination area Dld in Fig. 14a. The axial/

transverse shear–extension coupling coefficients that

characterise shearing in the xy plane caused by, res-

pectively, axial/transverse stress are plotted in Fig. 14b.

Matrix crack density in the 30�-ply is assumed equal to

C ¼ 2cracks=cm, values at Dld ¼ 0 indicate residual

stiffness properties of the laminates at this crack density

without delaminations. It can be seen that reduction of

ARTICLE IN PRESS

0.4

0.6

0.8

1

0 5 10 15


Nor

mal

ised

stif

fnes

s pr

oper

ty

Axial modulus (experiment)

Axial modulus (prediction)

Shear modulus

Poisson's ratio



Shear modulus

Poisson's ratio



Shear modulus

Poisson's ratio

0.2

0.4

0.6

0.8

1


Nor

mal

ised

stif

fnes

s pr

oper

ty

0.2

0.4

0.6

0.8

1


Nor

mal

ised

stif

ness

pro

pert

y

(a)

(b)

(c)

Fig. 12. Normalised stiffness properties of T800H/3631 cross-

ply laminates as a function of crack density Cmc: (a) ½0=902�s;(b) ½0=904�s; (c) ½0=906�s. Transverse delamination area

Dld ¼ 10%.

0.7

0.9

1.1

1.3

0 30 60 90

Relative delamination area (%)

0 30 60 90


Nor

mal

ised

stif

fnes

s pr

oper

ty

Axial modulusTransverse modulusShear modulusPoisson's ratio

-0.01

0.01

0.03

0.05

0.07

She

ar e

xten

sion

cou

plin

g co

effic

ient

s

Axial

Transverse

(a)

(b)

Fig. 13. Stiffness properties of AS4/3506-1 ½02=302=�302�slaminate as a function of relative delamination area Dld: (a)

normalised moduli and Poisson’s ratio; (b) shear-extension

coupling coefficients. Matrix crack density Cmc ¼ 2 cracks/cm.


the laminate moduli and, for the considered lay-up,

increase the Poisson’s ratio due to local delaminations

are more significant in the unbalanced ½02=302�s laminatethan in the balanced ½02=302=�302�s laminate with the

same orientation of the damaged ply. Matrix cracking

and crack tip delaminations are expected to amplify the

shear–extension coupling exhibited in the undamaged

unbalanced ½02=y2�s laminates. As in balanced

½02=y2=�y2�s laminates, crack tip uniform local delami-

nations in unbalanced laminates result in an increase in

the absolute value of the axial shear–extension coupling

coefficient for yo45� and of the transverse shear–exten-

sion coupling coefficient for y445�.

4. Onset and growth of intra- and interlaminar damage

The concept of the ‘equivalent’ laminate can be used

to calculate strain energy release rates for intra- and

interlaminar damage modes.

The total strain energy release rate G associated with a

particular damage mechanism is equal to the first partial

derivative of the total strain energy U stored in the

damaged laminate with respect to the total damage area

for this damage mode provided the applied strains f�gare fixed and the areas covered by other damage modes

remain unchanged

G ¼ �qU

qA

��f�g

. (19)

The strain energy release rates Gmc and Gld associated,

respectively, with matrix cracks and local delaminations

ARTICLE IN PRESS

0.5

0.7

0.9

1.1

1.3

1.5

0 30 60 90

0 30 60 90


Nor

mal

ised

stif

fnes

s pr

oper

ty

Axial modulusTransverse modulusShear modulusPoisson's ratio

-1.2

-1

-0.8

-0.6

-0.4

-0.2


She

ar e

xten

sion

cou

plin

g co

effic

ient

s

Axial

Transverse

(a)

(b)

Fig. 14. Stiffness properties of AS4/3506-1 ½02=302�s laminate

as a function of relative delamination area Dld: (a) normalised

moduli and Poisson’s ratio; (b) shear-extension coupling

coefficients. Matrix crack density Cmc ¼ 2 cracks/cm.


growing from the tips of matrix cracks can be effectively

calculated using the ‘equivalent’ laminate, in which the

damaged ply is replaced with an equivalent constraint

layer with degraded stiffness properties.

In the global coordinates, the total strain energy

stored in the laminate element with a finite gauge length

L and width w is

U ¼wL

2

Xk

ðzk � zk�1Þðf�g þ f�thermalk g þ f�hygrok gÞT

�½Q�kðf�g þ f�thermalk g þ f�hygrok gÞ, ð20Þ

where f�thermalk g and f�hygrok g are, respectively, residual

thermal and residual hygroscopic strains in the laminate

due to the temperature and moisture difference between

the stress-free and actual state, and ½Q�k is the in-plane

reduced stiffness matrix of layer k in the global

coordinates.

Since the area of a single crack is equal to

amc ¼ 2h2w=j sinfj, the total area covered by all cracks

is

Amc ¼ amcCmcL ¼ LwDmc=j sinfj. (21)

Likewise, since the area of a single crack tip delamina-

tion is equal to ald ¼ 2‘w=j sinfj, Fig. 3, the total

delamination area is equal to

Ald ¼ 2aldCL ¼ 2LwDld=j sinfj. (22)

If hygrothermal effects are neglected [80,81], the strain

energy release rates for matrix cracking and crack–tip

delaminations, calculated from Eqs. (19)–(22), are:

Gmcð�;DmcÞ ¼ �h2f�gT q½Q�2

qDmc f�gj sinfj, (23a)

Gldð�;Dmc;DldÞ ¼ �h2

2f�gT

q½Q�2

qDldf�gj sinfj. (23b)

Under uniaxial strain �xx, Eqs. (23) simplify to

Gmcð�xx;DmcÞ ¼ �h2�

2xx

qQxx;2

qDmc j sinfj, (24a)

Gldð�xx;Dmc;DldÞ ¼ �

h2

2�2xx

qQxx;2

qDldj sinfj. (24b)

Calculation of the in-plane axial stiffness Qxx;2 using Eq.

(16) and the transformation formulae given by Eq. (18),

yields the strain energy release rates in terms of the in

situ damage effective functions (IDEFs) L22;L66 and

stiffness properties of the undamaged material Qð2Þ

ij as

follows:

�
for off-axis ply cracking:
Gmcð�xx;DmcÞ

¼ h2 �2xx

Qð2Þ2

12

Qð2Þ

22

cos4fþ 2Qð2Þ

12 sin2fcos2f

"

þQð2Þ

22 sin4f

!qL22

qDmc

þ4Qð2Þ

66 sin2f cos2f

qL66

qDmc

#j sinfj. ð25aÞ

�
for crack-induced delamination:
Gldð�xx;Dmc;DldÞ

¼h2

2�2xx

Qð2Þ2

12

Qð2Þ

22

cos4fþ 2Qð2Þ

12 sin2f cos2f

"

þQð2Þ

22 sin4f

!qL22

qDld

þ4Qð2Þ

66 sin2f cos2f

qL66

qDld

#j sinfj. ð25bÞ

The first partial derivatives of IDEFs that appear in

Eqs. (25) are explicit functions of the damage

parameters Dmc and Dld and can be calculated

analytically.


tips of matrix cracks, O’Brien [72] suggested a simple

closed-form expression for the strain–energy release

1

2

3

4

0 3

Crack density (cracks/cm)N

orm

alis

ed s

trai

n en

ergy

rel

ease

rat

e (M

J/m

2 )

45

60

75

90

0

4

8

12

Nor

mal

ised

str

ain

ener

gy r

elea

se r

ate

(MJ/

m2 )

45

60

75

90

21 54

0 3


21 54

(a)

(b)

Fig. 15. Normalised strain energy release rate Gmc=�2xx for

matrix cracking in a ½0=y�s laminate as a function of crack

density Cmc in the y layer: (a) carbon/epoxy laminate; (b) glass/epoxy laminate.

For local delaminations growing uniformly from the

rate, based on simple load shearing rules and the

classical laminated plate theory. It gives the strain

energy release rate that depends only on the laminate

lay-up and thickness, the location of the cracked ply and

subsequent delaminations, the applied load and the

laminate width, and is independent of delamination size

and matrix crack density. In the nomenclature of this

paper it is given by

Gld

�2xx

¼NE

2

xh

2m

1

ðN � nÞEld

�1

NEx

� �, (26)

where h is the laminate thickness, N is the number of

plies, n is the number of cracked plies, Ex and Eld are,

respectively, the laminate axial modulus and the

modulus of the locally delaminated sublaminate as

calculated from the laminated plate theory. Parameter

m has a value of 2 if the cracked ply is in the interior of

the laminate, corresponding to local delamination on

either side of matrix crack and that of m ¼ 1 if the

cracked ply is a surface ply.

Later, O’Brien [74] showed that this simple closed-

form expression is valid for the total strain energy

release rate associated with uniform local delamination

growing from an angle ply matrix crack. It is worth

noticing that the strain energy release rate given by Eq.

(26) is independent from the delamination size. Also, the

effect of matrix cracking is not taken into account when

calculating the laminate modulus Ex. Nairn and Hu [44]

established that O’Brien’s expression for strain energy

release rate applies only to delaminations induced by

isolated matrix cracks, i.e. when crack density is very

small and the influence of neighbouring cracks is

negligible. For crack densities, at which delaminations

are observed to initiate, strain energy release rate

depends both on delamination area and crack density.

It is worth noting that the expressions derived in this

chapter, Eqs. (25), give strain energy release rates that

depend crack density and, for delaminations, also on

delamination area.

Fig. 15 shows normalised strain energy release rate

Gmc=�2xx for off-axis ply cracking, calculated using Eq.

(23b), as a function of crack density Cmc for, respec-

tively, carbon/epoxy and glass/epoxy ½0=y�s laminates.

Results are presented for four ply orientation angles:

45�, 60�, 75� and 90�. It may be seen that in carbon/

epoxy angle-ply laminates Gmc=�2xx decreases faster with

the crack density than in a cross-ply ½0=90�s laminate.

For smaller crack densities, thevalues of Gmc=�2xx in a

½0=75�s carbon/epoxy laminate are higher than in a

cross-ply laminate (Fig. 15a). It is also worth noticing

that the glass/epoxy angle-ply laminates exhibit signifi-

cantly higher levels of normalised strain energy release

rates for matrix cracking than carbon/epoxy ones

(Fig. 15b).

Fig. 16 shows the normalised strain energy release rate

Gld=�2xx, calculated from Eq. (23b), as a function of the

ARTICLE IN PRESS

0.3

0.32

0.34

0.36

0.38

0 0.8 1.6 2.4 3.2 4Normalised delamination width

Nor

mal

ised

str

ain

ener

gy r

elea

se r

ate

(MJ/

m2 ) s=40t

s=20t

Fig. 16. Normalised strain energy release rate Gld=�2xx for

uniform local delamination in a cracked ½02=252=�252�s AS4/3506-1 laminate as a function of normalised delamination

length ‘=t. Matrix crack spacing s ¼ 20t and 40t.

0

0.5

1

1.5

2

2.5

0 20 40 60 80 100


Nor

mal

ised

str

ain

ener

gy r

elea

se r

ate

(M

J/m

2 )

45

60

75

90

1

1.5

2

2.5

0 3Crack density (cracks/cm)

Nor

mal

ised

str

ain

ener

gy r

elea

se r

ate

(MJ/

m2 )

45

60

75

90

5421

(a)

(b)

Fig. 17. Normalised strain energy release rate Gld=�2xx for

uniform local delamination in a cracked AS4/3506-1 ½02=y2�slaminate: (a) as a function of relative delamination area Dld

(crack density 1 crack/cm); (b) as a function of crack density

Cmc in the y-ply (relative delamination area Dld ¼ 0, i.e. onset

of delamination).


delamination length normalised by the single ply

thickness ‘=t. The laminate lay-up is ½02=252=�252�s,and crack half-spacings are s ¼ 40t and 20t. This is

equivalent to the crack densities of approximately C ¼ 1

and 2 cm�1, respectively. It can be seen that the present

approach gives the strain energy release rate for uniform

local delamination that depends both on crack density

and delamination length. The result of Eq. (26) for the

same lay-up is found equal to 12:7MJ=m2 provided

shear–extension coupling and bending–extension cou-

pling are taken into account [74].Still, it is much higher

than our predictions, since the model of Eq. (26) is for a

single isolated matrix crack and associated local

delamination and does not account for the cumulative

effect of multiple cracking and local delaminations as

illustrated in Fig. 2.

Fig. 17 shows the normalised strain energy release rate

Gld=�2xx associated with uniform local delaminations

induced by off-axis ply cracking in a graphite/epoxy

½02=y2�s laminate. Dependence on the relative delamina-

tion area Did is shown in Fig. 17a, and on the crack

density in Fig. 17b. Results are presented for four

different ply orientations angles: 45�, 60�, 75� and 90�. It

may be seen that strain energy release rate non-linearly

depends on delamination area and almost linearly on the

crack density. While in a cross-ply ½02=902� laminate thevalue of Gld=�2xxjDld¼0 at the delamination onset is almost

independent of crack density, in ½02=y2�s it strongly

depends on it. Also, in ½02=602�s and ½02=752� laminates itis significantly higher than in a cross-ply ½02=902�laminate, suggesting lower delamination onset strains.

ARTICLE IN PRESS

3

)

as-cut edges

polished edges

notched edges


4.1. Predicting onset and growth of off-axis ply cracking

Even under the uniaxial loading, damage development

in the off-axis plies of general symmetric laminates

always occurs under mixed mode conditions due to

shear–extension coupling. It is therefore important in

the calculation of the total strain energy release rate to

be able to separate Modes I and II contributions.

For a ½ðSÞ=f�s laminate with damaged f-layermodelled by an ‘equivalent’ laminate, the total strain

energy release rate for off-axis ply cracks and crack-

induced local delaminations is equal to the first partial

derivative of the portion of the total strain energy stored

in the ‘equivalent’ homogeneous layer with respect to

damage area

Gmc ¼ �qU ð2Þ

qAmc

��f�g

, (27a)

Gld ¼ �qU ð2Þ

qAld

��f�g;C

. (27b)

In the local coordinates (Fig. 3), this portion of the total

strain energy can be separated into extensional and

shear parts

U ð2Þ ¼ Uð2ÞI þ U

ð2ÞII ¼ Lwh2ðs

ð2Þ11 �

ð2Þ11 þ sð2Þ22 �

ð2Þ22 Þ

þ Lwh2sð2Þ12 g

ð2Þ12 . ð28Þ

Under uniaxial strain �xx, strains and stresses in the

‘equivalent’ homogeneous layer are

f�ð2Þg ¼ fcos2f; sin2f; 2 cosf sinfgT�xx,

fsð2Þg ¼ ½Qð2Þ�fcos2f; sin2f; 2 cosf sinfgT �xx, ð29Þ

where the modified stiffness matrix ½Qð2Þ� of the

‘equivalent’ homogeneous layer in the local coordinates

is given by Eq. (12). Substitution of Eqs. (22), (28) and

(29) into Eq. (27) gives Modes I and II contributions

into the total strain energy release rate as follows:

2

ain

(%

prediction
�
0

1

45 60 75 90Ply orientation angle (degrees)

Cra

ckin

g on

set s

tr

Fig. 18. Off-axis ply cracking onset strain as a function of ply

orientation angle y in glass/epoxy ½0=y�s laminates.

for off-axis ply cracking:

GmcI ¼ �

qUð2ÞI

qAmc ¼ �2xxf mc1 ðDmcÞ, (30a)

f mc1 ðDmcÞ ¼ h2Q

ð2Þ2

12

Qð2Þ

22

cos4fþ 2Qð2Þ

12 sin2fcos2f

þQð2Þ

22 sin4f

!qLð2Þ

22

qDmc j sinfj, ð30bÞ

GmcII ¼ �

qUð2ÞII

qAmc ¼ �2xxf mc2 ðDmcÞ, (31a)

f mc2 ðDmcÞ ¼ 4h2Qð2Þ

66

qLð2Þ66

qDmc cos2fjsin3fj, (31b)

�
for crack-induced delaminations:
GldI ¼ �

qUð2ÞI

qAld¼ �2xxf 1ðD

ldÞ, (32a)

f 1ðDldÞ ¼

h2

2

Qð2Þ2

12

Qð2Þ

22

cos4fþ 2Qð2Þ

12 sin2f cos2f

þQð2Þ

22 sin4f

!qLð2Þ

22

qDldj sinfj, ð32bÞ

GldII ¼ �

qUð2ÞII

qAld¼ �2xxf 2ðD

ldÞ, (33a)

f 2ðDldÞ ¼ 2h2Q

ð2Þ

66

qLð2Þ66

qDldcos2fjsin3fj. (33b)

These expressions can be used with appropriate

fracture criteria to estimate the onset of local

delamination in an already cracked laminate. The

resulting total strain energy release rate Gld ¼ GldI þ

GldII coincides with Eq. (25).

To predict the development of the off-axis ply cracks

in the y -layer of a ½0=y�s laminate, it is suggested to use amixed mode fracture criterion

GI

GIC

� �M

þGII

GIIC

� �N

¼ 1, (34)

where GIC and GIIC are respectively Modes I and II

interlaminar fracture toughnesses. The exponents M


and N depend on the material system. Following

Rikards et al. [82], for a glass/epoxy system they can

be taken as M ¼ 1;N ¼ 2.

Fig. 18 shows predicted and experimentally observed

cracking onset strains for ½0=y�s glass/epoxy laminates.

Crocker et al. (1997) measured cracking onset strains in

specimens with as-cut, polished and notched edges. At

that, an independence of strain at onset of crack

propagation in notched samples on the notch depth

was observed. To predict cracking onset strains, GmcI and

GmcII values are calculated from Eqs. (30) and (31), and

cracking onset strain �xx is found as a root of the

following equation:

�4xx

f mc2 ðDmcÞ

GIIC

� �2

þ �2xx

f mc1 ðDmcÞ

GIC

� �¼ 1

when Dmc ! 0. ð35Þ

Since the exact GIC and GIIC critical values for the

considered glass/epoxy system are not known, predic-

tions are made using typical for glass/epoxy systems

values of GIC ¼ 200 J=m2 and GIIC ¼ 1500 J=m2. Com-

parison with limited experimental data shows that the

mixed mode fracture criterion, Eq. (9), can successfully

predict the initiation of matrix cracking for ply

orientation angles 75�pyp90�. For 45�pyp75�, mea-

sured strains are much higher than predictions. Also,

they increase steeply as y decreases. Further work is

required to develop an appropriate fracture or failure

criterion that captures initiation and development of

matrix cracks in off-axis plies of composite laminates

reinforced by glass or carbon fibres, especially for

yo75�.

Further work is required to validate theoretical

predictions. For the lay-ups, damage modes and loading

conditions examined in this study the experimental data

are currently not available.

5. Concluding remarks

The fracture process of composite laminates subjected

to static or fatigue tensile loading involves a sequential

accumulation of intra- and interlaminar damage, in the

form of transverse cracking, splitting and delamination,

prior to catastrophic failure. Matrix cracking parallel to

the fibres in the off-axis plies is the first damage mode

observed. It triggers development of other harmful resin-

dominated modes such as delaminations. Since a

damaged lamina within the laminate retains certain

amount of its load-carrying capacity, it is important to

predict accurately the stiffness properties of the laminate

as a function of damage as well as progression of

damage with the strain state. Incorporation of analytical

models of stiffness degradation and damage progression

into a finite element code will constitute the most

effective tool for progressive failure modelling of

composite plates with more complex configurations,

e.g. holes, notches and other stress concentrators.

In multidirectional laminates subjected to in-plane

tensile or thermal loading matrix cracks parallel to the

fibres develop in several off-axis plies. Comprehensive

observations of sequential accumulation of matrix

cracks in quasi-isotropic and balanced carbon/epoxy

and glass/epoxy laminates under quasi-static and fatigue

tensile loading have been extensively reported in the

literature. Concurrent matrix cracking in the adjacent

off-axis plies is an extremely complex problem to model

and has been analysed in the literature mostly using

finite elements method. A theoretical model that

describes damage development under complex loading

conditions does not yet exist.

Here, analytical modelling of off-axis ply cracking

and crack tip delaminations in balanced and unbalanced

angle-ply composite laminates subjected to in-plane

tensile loading is presented and discussed. A 2-D

shear-lag analysis is used to determine ply stresses in a

representative segment and the equivalent laminate

concept is applied to derive expressions for Modes I,

and II and the total strain energy release rate associated

with uniform local delaminations. These expressions can

be used with appropriate fracture criteria to estimate the

onset and growth of damage in off-axis plies.

To calculate strain energy release rates for off-axis ply

cracking and uniform local delaminations growing along

matrix cracks in balanced and unbalanced angle-ply

laminates, the damaged layer of the laminate is replaced

with an equivalent homogeneous one with effective

elastic properties. Closed form expressions for strain

energy release rate associated with matrix cracking and

crack induced uniform local delaminations have been

derived, representing them as linear functions of the first

partial derivatives of the effective elastic properties of

the damaged layer with respect to appropriate damage

parameters. Dependence of strain energy release rates

and the laminate stiffness properties on delamination

area, crack density and ply orientation angle has been

examined. It appears that matrix cracking and delami-

nation area influence the strain energy release rate value

significantly.

Comparison with results obtained by O’Brien [72,74]

shows that O’Brien’s closed-form expression for uniform

local delamination significantly overestimates the value

of the total strain energy release rate leading to lower

theoretical strains for the initiation of local delamination

and therefore over-conservative designs. Also, it gives

the total strain energy release rate as independent of

delamination area and does not take into account the

cumulative effect of damage.

It is found, in particular, that the reduction due to

matrix cracking of the laminate axial and transverse

moduli is more significant in angle-ply than in cross-ply


laminates, while for the shear modulus, the opposite is

true. Matrix cracking in angle-ply CFRP and GFRP

laminates can result in an increase in the Poisson’s

ratio—a phenomenon, not observed in cross-ply lami-

nates. Also, matrix cracking in angle-ply laminates

introduces coupling between extension and shear.

In near future work, the analytical predictions will be

compared to numerical (finite element) and experimental

data, which for the lay-ups, damage modes and loading

conditions examined in this study are currently not

available.

Acknowledgements

Financial support of this research by the Engineering

and Physical Sciences Research Council (EPSRC/GR/

L51348 and EPSRC/GR/A31001/02) and the British

Ministry of Defence is gratefully acknowledged.

Appendix A

Variation of the out-of-plane shear stresses has the

form

sð2Þj3 ¼tj

h2x3; 0pjx3jph2; j ¼ 1; 2, (A.1a)

sð1Þj3 ¼tj

h1ðh � x3Þ; h2pjx3jph. (A.1b)

Constitutive equations for the out-of-plane shear

stresses

sðkÞ13

sðkÞ23

8<:

9=; �

QðkÞ55 Q

ðkÞ45

QðkÞ45 Q

ðkÞ44

24

35 qqx3

uðkÞ1

uðkÞ2

8<:

9=;; i ¼ 1; 2.

(A.2)

After substituting Eq. (A.2) into Eqs. (A.1), multiplying

them by x3 and by h � x3, respectively, and integrating

with respect to x3 we get

h1

3

t1t2

( )¼

Qð1Þ

55 Qð1Þ

45

Qð1Þ

45 Qð1Þ

44

24

35 ~uð1Þ1

~uð1Þ2

8<:

9=;�

V1

V2

( )0@

1A,

(A.3a)

h2

3

t1t2

( )¼

Qð2Þ

55 0

0 Qð2Þ

44

24

35 V1

V2

( )�

~uð2Þ1

~uð2Þ2

8<:

9=;

0@

1A.

(A.3b)

Here fVg ¼ fuð1Þgjx3¼h2 ¼ fuð2Þgjx3¼h2 are the in-plane

displacements at the interface. After rearranging

Eqs. (A.3) become

~uð1Þ1

~uð1Þ2

8<:

9=;�

~uð2Þ1

~uð2Þ2

8<:

9=; ¼

h1

3

Qð1Þ

55 Qð1Þ

45

Qð1Þ

45 Qð1Þ

44

264

375�10

B@

þh2

3

Qð2Þ

55 0

0 Qð2Þ

44

264

375�11CA t1

t2

( ).

ðA:4Þ

Inversion of Eq. (A.4) leads to

t1t2

( )¼

K11 K12

K21 K22

" #~uð1Þ1

~uð1Þ2

8<:

9=;�

~uð2Þ1

~uð2Þ2

8<:

9=;

0@

1A, (A.5)

with

½K � ¼h1

3

Qð1Þ

55 Qð1Þ

45

Qð1Þ

45 Qð1Þ

44

24

35�1

þh2

3

Qð2Þ

55 0

0 Qð2Þ

44

24

35�1

0B@

1CA

�1

.

(A.6)

Appendix B

On referring to the constitutive equations, Eqs. (6),

the generalised plane strain condition, Eq. (5), becomes

Sð1Þ

11 ~sð1Þ11 þ S

ð1Þ

12 ~sð1Þ22 þ S

ð1Þ

16 ~sð1Þ12 ¼ S

ð2Þ

11 ~sð2Þ11 þ S

ð2Þ

12 ~sð2Þ22 . (B.1)

Using the laminate equilibrium equations, Eqs. (4),

stresses in the constraining layer (layer 1) can be

excluded, so that the microstress component ~sð2Þ11 is

given by

~sð2Þ11 ¼ a22 ~sð2Þ22 þ a12 ~s

ð2Þ12 þ bxsx þ bysy, (B.2)

a22 ¼ �Sð1Þ

12 þ wSð2Þ

12

Sð1Þ

11 þ wSð2Þ

11

; a12 ¼ �Sð1Þ

16

Sð1Þ

11 þ wSð2Þ

11

,

bx ¼ð1þ wÞðS

ð1Þ

11 cos2fþ S

ð1Þ

12 sin2f� S

ð1Þ

16 sinf cosfÞ

Sð1Þ

11 þ wSð2Þ

11

,

by ¼ð1þ wÞðS

ð1Þ

11 sin2fþ S

ð1Þ

12 cos2fþ S

ð1Þ

16 sinf cosfÞ

Sð1Þ

11 þ wSð2Þ

11

.

Strain differences are expressed in terms of stresses as

~gð1Þ12 � ~gð2Þ12

~�ð1Þ22 � ~�ð2Þ22

8<:

9=; ¼ �

1

w

L11 L12

L21 L22

" #~sð2Þ12

~sð2Þ22

8<:

9=;

þ1

w

M11 M12

M21 M22

" #sx

sy

( ). ðB:3Þ


Here

L11 ¼ Sð1Þ

66 þ a12Sð1Þ

16 þ wSð2Þ

66 ,

L12 ¼ Sð1Þ

26 þ a22Sð1Þ

16 ,

L21 ¼ Sð1Þ

26 þ a12Sð1Þ

12 þ wa12Sð2Þ

12 ,

L22 ¼ Sð1Þ

22 þ a22Sð1Þ

12 þ wðSð2Þ

22 þ a22Sð2Þ

12 Þ, ðB:4aÞ

M11 ¼ ð1þ wÞbðSð1Þ

16 þ a12Sð2Þ

11 Þcos2f

þ ðSð1Þ

26 þ a12Sð1Þ

12 Þsin2f� ðS

ð1Þ

66 þ a12Sð1Þ

16 Þ

� sinf cosfc,


12 þ a22Sð1Þ

11 Þcos2f

þ ðSð1Þ

22 þ a22Sð1Þ

12 Þsin2f� ðS

ð1Þ

26 þ a22Sð1Þ

16 Þ

� sinf cosfc,


16 þ a12Sð2Þ

11 Þsin2f

þ ðSð1Þ

26 þ a12Sð1Þ

12 Þcos2fþ ðS

ð1Þ

66 þ a12Sð1Þ

16 Þ

� sinf cosfc,


12 þ a22Sð1Þ

11 Þsin2f

þ ðSð1Þ

22 þ a22Sð1Þ

12 Þcos2fþ ðS

ð1Þ

26 þ a22Sð1Þ

16 Þ

� sinf cosfc. ðB:4bÞ

Substitution into the equilibrium equations, Eqs. (3),

yields the following coupled 2nd order differential

equations

d2

dx2

~sð2Þ12

~sð2Þ22

8<:

9=;�

1

h1

K11 K12

K21 K22

" #L11 L12

L21 L22

" #~sð2Þ12

~sð2Þ22

8<:

9=;

0@

þM11 M12

M21 M22

" #sx

sy

( )1A ¼ 0, ðB:5Þ

or

d2

dx2

~sð2Þ12

~sð2Þ22

8<:

9=;�

N11 N12

N21 N22

" #~sð2Þ12

~sð2Þ22

8<:

9=;

þP11 P12

P21 P22

" #sx

sy

( )¼ 0, ðB:6Þ

where ½N� ¼ h�11 ½K �½L� and ½P� ¼ h�11 ½K �½M�, with ma-

trices ½K �; ½L� and ½M� defined by Eqs. (A.6), (B.4a) and

(B.4b), respectively.

Appendix C

A1 ¼l21 � N22

N21A2; B1 ¼

l22 � N22

N21B2,

C1 ¼ �C2N22 þ P21 þ aP22

N21, ðC:1Þ

A2 ¼ �ðP21 þ aP22ÞðN21N12 � N11N22Þ þ Rl22

ðl22 � l21ÞðN21N12 � N11N22Þ, (C.2)

B2 ¼ðP21 þ aP22ÞðN21N12 � N11N22Þ þ Rl21

ðl22 � l21ÞðN21N12 � N11N22Þ, (C.3)

C2 ¼R

N21N12 � N11N22,

R ¼ N11ðP21 þ aP22Þ � N21ðP11 þ aP12Þ. ðC:4Þ

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Analysis of composite laminates with intra- and interlaminar damage

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