Damage analysis of laminated composites using a new coupled …webpages.iust.ac.ir/bijan_mohammadi/Bijan_Mohammadi... · 2015-10-07 · doi: 10.1111/j.1460-2695.2010.01456.x Damage

doi: 10.1111/j.1460-2695.2010.01456.x

Damage analysis of laminated composites using a new coupledmicro-meso approach

A. FARROKHABADI 1, H . HOSSEINI -TOUDESHKY 1 and B. MOHAMMADI 2

1Aerospace Engineering Department, Amirkabir University of Technology, Tehran, Iran, 2School of Mechanical Engineering, Iran University of Scienceand Technology, Narmak, Tehran, Iran

Received in final form 18 February 2010

A B S T R A C T In this study, the simplicity and strong physical meaning of micromechanics approachand capability of mesomechanics approach for damage analysis of structures with complexloadings are employed to develop a new micro-meso approach. For this purpose, a newmicromechanics model is developed to predict the matrix cracking initiation and evolutionin laminated composites. These damage initiation and evolution are replaced with thedamage criteria and flow rule in the continuum damage approach, respectively. Theresults of this procedure are used in the FEM damage analyses of laminated compositesto predict constitutive response of layered composites. It is shown that, the predictedstress distribution and strain energy in a lamina unit cell are in good agreement with thefinite element results. Furthermore, it is shown that the predicted stress–strain behavioursare in good agreement with the available experimental results for various laminates withdifferent lay-ups.

Keywords damage analysis; FEM; mesomechanics; micromechanics; transversecracking.

I N T R O D U C T I O N

The fracture process of composite laminates under mono-tonic tension loading is in the form of matrix ply cracking,edge delamination and local delamination prior to catas-trophic failure.1 The first failure mode observed in thelaminated composite materials is matrix cracking in thetransverse plies. The immediate effect of matrix crack-ing is to cause degradation in the elastic properties ofthe laminate including changes in all effective elasticitymodulus, Poisson’s ratios and thermal expansion coeffi-cients. Transverse cracks in composite laminates appearunder tensile loading and grow until reaching a satu-ration state.2 In laminates under general loading con-dition, after matrix cracking, by increasing the appliedloading, two distinct types of delamination called edgedelamination and local delamination (or transverse crack-tip delamination) can be initiated and propagated. Edgedelamination initiates at load-free edges of the compositeplate due to interlaminar stresses whereas local delamina-tion originates from the matrix crack tip because of large

Correspondence: H. Hosseini-Toudeshky. E-mail: [email protected]

inter-laminar stress concentration at the crack tip.3 Withinitiation and growth of delaminations, the strength andstiffness of composite laminates are significantly im-paired.4 The matrix-dominated failure modes can bedetrimental to the strength of the laminate because theycan cause fibre breakage in the primary load-bearing plies(0 ◦ plies).1

In the last decades, a great deal of attention has beenfocused on the investigating of matrix crack initiation andmultiplication and its influence on the thermo-mechanicalbehaviour of laminates. The developed degradation mod-els of composite materials exist on different scales, fromthe fibre scale to the structure scale, including what isknown as the ‘meso’ scale of the elementary layer.

Two well-known approaches for composite damageanalysis are ‘continuum damage mechanics’ and ‘microme-chanics of damage’.5 In micromechanics methods the stressdistribution of a composite in the presence of certaintypes of damage is analysed, which must be described onthe basis of experimental observations, e.g. matrix crack-ing of 90◦ in cross-ply laminates, or matrix cracking to-gether with delamination. Among the micromechanicsapproaches, the most applicable approaches are shear lag

420 c© 2010 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 33, 420–435

Fatigue & Fracture of Engineering Materials & Structures

DAMAGE ANALYSIS OF LAMINATED COMPOSITES 421

approach,6–8 variational principles9−12 and stress transfermechanics.13

Reifsnider6 introduced a shear stress transfer layer ofunknown thickness and stiffness between plies to evaluatestiffness reduction due to cracks, and assumed that thislayer carries only shear stress while the plies carry onlytensile stress. This method is called ‘shear lag’. In shearlag analysis the normal stress in external load directionis constant over the ply thickness and shear stresses de-velop only within a boundary layer of unknown thicknessbetween plies. The modified shear-lag model was also de-veloped which assumes existence of a thin interlaminaradhesive layer between the neighbouring layers, capableto transfer not only the interlaminar shear stress but alsointerlaminar normal stress.7 This approach was furtherdeveloped by Kashtalyan and Soutis.8

Variational approach was established by Hashin.9 Hesolved a plane stress problem for a cross-ply laminate withtransverse cracks based on the variational principles. Theobtained stresses based on this hypothesis satisfy the equi-librium equations, the boundary conditions and the con-tinuity conditions at interfaces. The unknown constantscan be determined from the principle of minimum addi-tional energy. Hashin’s variational approach was furtherdeveloped by Nairn, Varna and Berglund and Berglundand Varna.10,11 Nairn (1989) considered thermal stressesand applied Hashin’s variational approach to the prob-lem of transverse microcracks and delamination near thetransverse crack tip.

A generalized plane strain model of stress transfer formultiaxially loaded cross-ply laminates has been devel-oped by McCartney to predict the dependence of thethermoelastic constants on the matrix crack density, in-cluding the through-thickness properties.13 He assumedthat the axial stress in each ply depends only on the ax-ial direction and do not depends on through the thicknessdirection. However, according to this approach, the prob-lem is reduced to a system of recurrent relations and anordinary differential equation of the fourth order. Alsoplies are divided into the sub-plies of smaller thickness,because the elastic relations in the transverse direction aresatisfied in the averaged sense.

As a general sense, the advantages of the available mi-cromechanical based approaches are their simplicity andstrong physical meaning, but some of them are mainlylimited to the analysis of cross-ply laminates under uni-axial tensile loading condition. Among the mentionedmicromechanics approaches, the shear lag is a simplemethod, which reduced to a single differential equationand yield to reasonable predictions of stiffness reduction.However, a major disadvantage of this approach is thatthe parameters of the shear transfer layer are unknownand must be determined by fitting to the experimentalresults. The variational approaches are mainly limited to

the analysis of cross-ply laminates. The McCartney stresstransfer approach is also another useful method whichis applied for multiaxially loaded composite laminates.13

The major disadvantage of this approach is that it doesn’tconsider delamination induced by micro cracks.

In continuum damage mechanics approach all damagestates are generalized into a damage parameter tensorwhich explains the state of damage containing matrixcracking, fibre breakage, interfacial debonding, or pliesdelamination. The composite is then treated as a contin-uum field under the states of stress and strain and the goalis to find constitutive relations between the stress, strainand damage.5

Ladeveze et al.14–16 developed an approach using con-tinuum damage mechanics in the mesoscale. The mainconcept of this damage mesomodel is the use of an in-termediary scale related to the scale of the laminates.On this intermediary scale, which is called ‘meso’ scale,the material is described by means of two basic meso-constituents of the single layer and interface. Thus, thelaminated structure is described as a stacking sequence ofhomogeneous layers throughout the thickness and inter-laminar interfaces. For both of the two basic constituents,i.e. the ply and the interface, material models are intro-duced using the internal variables framework for speci-fying the material’s state. Continuum damage mechanicsare used to describe the degradation of the meso con-stituents. Meso damage indicators are linked to the stiff-ness variation of the meso constituents. Aside from fibrebreakage, the damage mechanisms of matrix micro crack-ing, fibre/matrix debonding and delamination are con-sidered. An important point is that the state of damageis assumed to be uniform within the thickness of eachmeso-constituent but can vary from one layer to the nextin the laminate. The ‘interface’ is a surface entity whichdepends on the relative orientations of the adjacent plies;it transfers displacements and normal stresses from oneply to adjacent one. The advantage of this procedure isthat damage mechanisms which can be very complex onthe structure’s scale can be quite simple on the scale of thebasic constituents. Mesomechanics models are also capa-ble of predicting different damage mechanisms and theirevolution until final failure. In addition, these models areapplicable to industrial structures subjected to complexloading and provide a general formulation and can betransposed more easily into the commercial software. Butthis approach requires additional empirical constants andneeds specific tests for each kind of lay-ups which are nec-essary to obtain the non-associated flow rule parametersin damaged condition.

In this paper, a new micro-meso approach benefitedfrom both strong physical meaning of micromechanics ap-proach and capability of mesomechanics approach for theanalysis of structures with complex loadings is developed.

c© 2010 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 33, 420–435

422 A. FARROKHABADI et al.

More details of this method are presented in the next sec-tion. This method is used to obtain the stress distributionand strain energy in two lamina unit cell with differentmaterial properties. The results of which are comparedwith the finite element results. Furthermore, the effectsof damage initiation and growth on the stress–strain be-haviours of various laminates with different lay-ups arepredicted and the obtained results are compared with theavailable results in the literature.

G E N E R I C P R O C E D U R E S T A T E M E N TI N P R E S E N T S T U D Y

In this study, the simplicity and strong physical meaningof micromechanics approach and capability of mesome-chanics approach for the analysis of structures with com-plex loadings are employed to develop a new micro-mesoapproach. In order to study the nonlinear behaviour oflaminates due to damage growth, the appropriate evo-lution laws which describe the evolution of the damagestate is necessary. In mesomechanics approach, the rela-tion between the conjugate forces of damage and damageparameters are obtained by performing the specific testsfor each lay-up configuration. By reviewing of differentmicro-mechanic approaches, it is obvious that microme-chanics approaches have the capability to obtain the re-lation between the strain energy release rate and damageparameters in an analytical way. In order to consider thedamage growth in micromechanics approaches, the en-ergy methods are applied and the strain energy release ratedue to the damage growth is compared with the micro-cracking fracture toughness. This advantage is used forpredicting the evolution of the damage state in mesome-chanics approach in this study. Using this new approach,the major disadvantage of meso modelling which is therequirement of specific tests for various lay-ups to ob-tain the non-associated flow rule parameters in damagedcondition can be improved. By the way, standard testsand known parameters such as finite fracture toughnessof lamina are applied to obtain the evolution of damagein laminates.

To study the damage growth in laminated compositeswith different lay-ups, the damage growth in each laminahas to be considered separately and independent of otherlayers using appropriate evolution laws. For this purpose,a single lamina under a general stress field on all sides isconsidered which is prone to damage growth. Followingthat, using the micromechanics approach, governing re-lations such as stress field, displacements field and etc.,are extracted for any specific crack density. In contrast tothe previous micromechanical approaches, in this new ap-proach, the stiffness degradation is obtained for a ‘singlelamina under multi-axial loading conditions’. The pre-viously developed micromechanical models couldn’t be

directly used for this purpose. By introducing of the newmicromechanics approach, the stiffness degradations canbe obtained for any loading and crack density. For thispurpose, a generalized plane strain model is developed fororthotropic composite lamina including transverse crackwhich satisfies the stress field continuity.

The stress boundary conditions of each unit cell are ob-tained from the finite element analysis of laminates ineach step of loading. Having these boundary conditionsand using the above mentioned procedure the stress andstrain fields of the cracked unit cell are obtained and thestrain energy can be calculated. By comparing the calcu-lated strain energy with micro-cracking fracture tough-ness, the evolution of crack density in each unit cell canbe recognized. Crack density evolution in each unit cellcauses degradation in thermo-elastic constants of lamina.The obtained stiffness reductions are used for evolutionof damage parameters of each lamina. Following this pro-cedure, the damage growth can be predicted accordingto a new evolution law based on the micromechanics ap-proach for uniformly cracked composite laminates withany lay-up configuration under any loading combination.Contrary to the previous continuum damage mechanicsapproach, in the purposed approach the evolution law isnot obtained from the non-standard and lay-up dependenttest results and it only needs the micro-cracking fracturetoughness which is well known in standard material char-acterization.

G E O M E T R Y A N D B A S I C F I E L D E Q U A T I O N S

In this study the tri-axial deformation of orthotropic com-posite lamina is considered. Due to the symmetry aboutthe mid-plane of the lamina, half of the lamina is onlyconsidered as shown in Fig. 1. A global coordinates ischosen having the origin at the centre of the lamina asshown in this figure. The x, y and z axis of the coordinatedefine the axial or longitudinal, in-plane transverse andthrough-thickness directions respectively.

The mid-plane of the lamina is specified by z = 0 and theexternal surfaces by z = ±h where 2h is the thickness of thelamina. It is assumed that the matrix cracks are uniformlydistributed along the transverse direction surfaces with 2Ldistance from each other (on the planes of y = ±L). Therepresentative volume element of the lamina is specifiedby |y| < L, |x| < W and |z| < h. The displacement field ofthe lamina is assumed to be in the generalized plane-straincondition such that:

u = f (y, z) + xεcx v = v(y, z) w = w(y, z), (1)

where εcx is a uniform transverse strain for the cracked

lamina that its value can be determined using the effectivetransverse stress which is applied to the lamina and usuallyit is a specified value.



Fig. 1 Typical uniform transverse crack in an orthotropic lamina.

Fig. 2 Typical orthotropic lamina comprise a thick damagedsub-lamina containing transverse crack and thin undamagedsub-lamina.

S T R E S S A N D D I S P L A C E M E N T F I E L D SF O R A C R A C K E D L A M I N A

The basic assumption of the analysis for cracked laminais that after crack formation the lamina is divided by twosub-lamina with different thicknesses as shown in Fig. 2.The thick sub-lamina contains transverse crack and thethin sub-lamina is considered to be undamaged. After

crack formation it assumed that at the interface betweenthe two sub-lamina, two shear stress components are ap-peared in y and x directions (φ′(y) and ψ ′(y)). These shearstress components are considered as a function of in-planetransverse direction y. The shear stress components oflamina caused by transverse crack (σ yz and σ xz) along thethickness is also considered as the following piecewise lin-ear forms for each sub-lamina. These perturbation shearstresses are function of in-plane transverse and throughthe thickness directions and independent of axial direc-tion;

σ 1yz = φ′(y)

zh1

0 ≤ z ≤ z1

σ 2yz = −φ′(y)

z − z2

h2z1 ≤ z ≤ z2

(2)

σ 1xz = ψ ′(y)

zh1

0 ≤ z ≤ z1

σ 2xz = −ψ ′(y)

z − z2

h2z1 ≤ z ≤ z2,

(3)

where the superscripts and subscripts of 1 and 2 denote thethick and thin sub-lamina respectively. φ(y) and ψ(y), arefunctions of y which have to be determined, and they areidentically zero for the case of undamaged lamina. Thederivatives of these functions related to y are denotedby primes. Also 2h1 and h2 are thicknesses of thick andthin sub-lamina respectively. z1 and z2 are located at theinterface of two sub-lamina and top surface of laminarespectively as shown in Fig. 2. The form of the Eqs (2)and (3) ensures that the stress components σ yz and σ xz arecontinuous across the interfaces. From Eqs (2) and (3) itis clear that φ′(y) and ψ ′(y) define the distributions of σ yz

and σ xz on the interface of the two sub-lamina.Substituting of the stresses from (2) and (3) in to the

equilibrium equations, σ yy, σ xy and σ zz are obtained foreach sub-lamina for 0 ≤ z ≤ h:

σ 1yy = σ 0

yy − φ(y)/h1 0 ≤ z ≤ z1

σ 2yy = σ 0

yy + φ(y)/h2 z1 ≤ z ≤ z2(4)

σ 1xy = σ 0

xy − ψ(y)/h1 0 ≤ z ≤ z1

σ 2xy = σ 0

xy + ψ(y)/h2 z1 ≤ z ≤ z2(5)

σ 1zz = z − z1

2h1[−(z − z1 + 2h1)φ′′(y)] + h1

2φ′′(y) + σ 0

zz

0 ≤ z ≤ z1

σ 2zz = z − z2

2h2[(z − z2)φ′′(y)] + σ 0

zz z1 ≤ z ≤ z2,

(6)

where σ 0yy , σ 0

xy and σ 0zz are the uniform axial, shear and

transverse through the thickness stresses for undamaged



lamina subjected to the same applied effective stressesconditions.

Using the elastic constitutive law and the strain-displacement relations along with the equations of (2) to(6), the displacement field can be obtained.

u1 = z − z1

h1(z + h1)

(1

2Gxzψ ′(y)

)+ U1(y) + xεc

x

0 ≤ z ≤ z1

u2 = − (z − z2)2

h2

(1

2Gxzψ ′(y)

)+ U2(y) + xεc

x

z1 ≤ z ≤ z2,

(7)

where ui (y, zi ) = Ui (y) + εcx x is the axial displacement on

the interface and Ui (y) is appeared from the integrationwith respect to z.

U1(y) = U0(y) + h1

2Gxzψ ′(y) 0 ≤ z ≤ z1

U2(y) = U1(y) + h2

2Gxzψ ′(y) z1 ≤ z ≤ z2.

(8)

Similarly, for transverse displacements the following re-lations are derived.

v1 = z − z1

h1(z + h1)

(1

2Gyzφ′(y)

)+ (z − z1)3

24h1

×(

1Ez

− ν2xz

Ex

)(z + 3h1)φ′′′(y) − 1

2(z − z1)2

×[(

1Ez

− ν2xz

Ex

)h1

2φ′′′(y)

−(

νyz

Ey− νxzνxy

Ex

)φ′(x)

h1

]− (z − z1)W′

1(y) + V1(y) 0 ≤ z ≤ z1

v2 = − (z − z2)2

h2[a11φ

′(y)] + (z − z1)3

24h2

(1Ez

− ν2xz

Ex

)

× (z − z2)φ′′′(y) − 12

(z − z2)2

×[(

νyz

Ey− νxzνxy

Ex

)φ′(y)

h2

]− (z − z2)W′

2(y) + V2(y) z1 ≤ z ≤ z2,

(9)

where vi (y, zi ) = Vi (y) is the transverse displacement atthe interface z = zi , which is obtained by integration withrespect to z. Vi(y) components are defined by the following

equations:

V1(y) = h1[a11φ′(y)] + h3

1

8

(1Ez

− ν2xz

Ex

)φ′′′(y)

+ h21

2

[(1Ez

− ν2xz

Ex

)h2

2φ′′′(y)

− 1h1

(νyz

Ey− νxzνxy

Ex

)φ′(y)

]

− h1W′1(y) + V0(y) 0 ≤ z ≤ z1

V2(y) = h2[a11φ′(y)] + h3

2

24g11φ

′′′(y)

+ h2

2

[(νyz

Ey− νxzνxy

Ex

)φ′(y)

]

− h2W′2(y) + V1(y) z1 ≤ z ≤ z2.

(10)

Similarly, for through the thickness displacements, thefollowing relations are derived.

w1 = − (z − z1)2

6h1

(1Ez

− ν2xz

Ex

)(z + 2h1)φ′′(y) + (z − z1)

×[(

1Ez

− ν2xz

Ex

)h1

2φ′′(y) −

(νyz

Ey− νxzνxy

Ex

)

×φ(y)h1

+ εz

]+ W1(y) 0 ≤ z ≤ z1

w2 = (z − z2)2

6h2

(1Ez

− ν2xz

Ex

) [(z − z2)φ′′(y)

] − (z − z2)

×[(

νyz

Ey− νxzνxy

Ex

)φ(y)h2

+ εz

]+ W2(y)

z1 ≤ z ≤ z2,

(11)

where wi (y, zi ) = Wi (y) is through the thickness displace-ment on the interface at z = zi , which is obtained by inte-gration with respect to z. The Wi(y) components are alsodefined as the following equations:

W1(y) = 2h21 + 3h1h2

6

(1Ez

− ν2xz

Ex

)φ′′(y)

−(

νyz

Ey− νxzνxy

Ex

)φ(y) + h1εz 0 ≤ z ≤ z1

W2(y) = 2h21 + 3h1h2 + h2

2

6

(1Ez

− ν2xz

Ex

)

× φ′′(y) + (h1 + h2)εz z1 ≤ z ≤ z2,

(12)



where εz is the through-thickness strain in undamagedlamina. Having the above stress and displacement equa-tions for the damaged lamina, and combining with theconstitutive equations satisfies the equilibrium equationsexcept the following stress–strain relations for any func-tions of φ(y) and ψ(y) in each sub-lamina:

εyy = ∂v

∂y=

(νyz

Ey− νxzνxy

Ex

)σzz

+(

1Ey

− ν2xy

Ex

)σyy − νxyε

cx (13)

εyx = ∂v

∂x+ ∂u

∂y= 1

Gxyσxy . (14)

The values of stresses and displacements are functionof through the thickness direction and by imposing intorelations (13) and (14), these relations couldn’t be solved.Thus the average values of stresses and displacementsalong of each sub-lamina thickness are employed and thefollowing equations are obtained:

εyy = ∂v

∂y=

(νyz

Ey− νxzνxy

Ex

)σzz

+(

1Ey

− ν2xy

Ex

)σyy − νxyε

cx (15)

εyx = ∂v

∂x+ ∂ u

∂y= 1

Gxyσxy , (16)

where σzz, σyy and σxy are the average values of stressesfor each sub-lamina. Using the average values, the stressesand displacements in Eqs (2) to (6) are obtained asfollows:

σ 1yy = σ 1

yy = −φ(y)h1

+ σ 0yy

σ 2yy = σ 2

yy = φ(y)h2

+ σ 0yy

σ 1yx = σ 1

yx = −ψ(y)h1

+ σ 0yx

σ 2yx = σ 2

yx = ψ(y)h2

+ σ 0yx

σ 1zz =

(h1

3+ h2

2

)φ′′(y) + σ 0

zz

σ 2zz = h2

6φ′′(y) + σ 0

zz

u1 = −2h1

31

2Gxzψ ′(y) + εc

x x + U1(y)

u2 = h2

31

2Gxzψ ′(y) + εc

x x + U1(y)

v1 = −2h1

31

2Gyzφ′(y) − h3

1

30

(1Ez

− ν2xz

Ex

)φ′′′(y)

− h21

6

[(1Ez

− ν2xz

Ex

)h1

2φ′′′(y)

−(

νyz

Ey− νxzνxy

Ex

)φ′(y)

h1

]

+ h1

2W′

1(y) + V1(y)

v2 = −h2

31

2Gyzφ′(y) − h3

2

120

(1Ez

− ν2xz

Ex

)φ′′′(y)

− h22

6

[(νyz

Ey− νxzνxy

Ex

)φ′(y)

h2

]

+ h2

2W′

2(y) + V2(y). (17)

By substituting the average stresses and displacementsfrom above relations into (15) and (16) and using the func-tions ofUi (y), Vi (y) and Wi (y)from (8), (10) and (12) foreach sub-lamina and subtracting from each other, the fol-lowing homogeneous differential equations are obtained:

Fφ′′′′(y) + Gφ′′(y) + Hφ(y) = 0

Sψ ′′(y) + Kψ(y) = 0.(18)

The coefficients of F, G, H, S and K are material depen-dent constants. These differential equations can be solvedby imposing appropriate boundary conditions using stan-dard techniques. The boundary conditions on crackedsurfaces of lamina at y = ±L are stress free conditions asfollow:

σyy (±L, z) = 0 σyx(±L, z) = 0 σyz(±L, z) = 0. (19)

Now, the strains in the thin sub-lamina must be defined.This sub-lamina is undamaged and its strains are equal tothe lamina strains after crack formation. The undamagedthin sub-lamina surface on y = ±L are subjected to theaveraged boundary displacements as follow:

v2(±L) = ±εcy L w2(±L, z) = ±γ c

xy L + εcx x. (20)

By substituting v2 and w2 in relations (20), εcy and γ c

xycan be defined.

C A L C U L A T I O N O F T H E R M O - E L A S T I CC O N S T A N T S

Estimation of the elastic constants is based on deriving thestress and displacement fields from the solution of differ-ential equations, the parameters of σy , σz, τxy and εc

x arealready specified as initial conditions. The correspondingeffective through-thickness strain εc

z, the transverse strainεc

y and the in-plane shear strain γ cxy after the crack forma-

tion are calculated by imposing the appropriate boundary



conditions of sub-lamina. Thus the stress–strain relationsfor cracked lamina can be written as follows:

εcx = σx

Ex(ρ)− νyx(ρ)

Ey (ρ)σy − νzx(ρ)

Ez(ρ)σz + αx(ρ)�T

εcy = −νxy (ρ)

Ex(ρ)σx + σy

Ey (ρ)− νzy (ρ)

Ez(ρ)σz + αy (ρ)�T

εcz = −νxz(ρ)

Ex(ρ)σx − νyz(ρ)

Ey (ρ)σy + σz

Ez(ρ)+ αz(ρ)�T

γ cxy = τxy

Gxy (ρ). (21)

In these equations the constants Ex(ρ), Ey (ρ), . . . , αz(ρ)and Gxy (ρ) are thermo-elastic constants of cracked lam-ina that are unknown. For this purpose, the stress–strainrelations and ply crack closure for constrained uni-axialloading in axial, transverse and through the thickness di-rections are used and three independent constants areobtained which will be used for deriving the damagedproperties of lamina. In continue, in this part, ply crackclosure conditions for the whole lamina (thin and thicksub-lamina) in different directions is derived indepen-dently.

At first, ply crack closure for the constrained uni-axialloading is considered such that σx = σz = τxy = 0. It fol-lows from stress–strain relations (21), by setting εx = εc l

x ,εy = εc l

y , εz = εc lz and σy = σ c l

y such that

εc lx = −υyx(ρ)

Ey (w)σ c l

y + αx(ρ)�T

εc ly = σ c l

y

Ey (ρ)+ αy (ρ)�T

εc lz = −υyz(ρ)

Ey (ρ)σ c l

y + αz(ρ)�T. (22)

Similarly, for undamaged lamina it follows that:

εc lx = −υyx

Eyσ c l

y + αx�T

εc ly = σ c l

y

Ey+ αy�T

εc lz = −υyz

Eyσ c l

y + αz�T. (23)

Considering the equivalent strain components for un-damaged and damaged lamina in Eqs (26) and (27) and bycombining of strain equations in the similar directions,the following relations are obtained.

σ c ly = − αz(ρ) − αz

υyzEy

− υyz(ρ)Ey (ρ)

�T = −αy (ρ) − αy1

Ey (ρ) − 1Ey

�T

= − αx(ρ) − αxυyxEy

− υyx (ρ)Ey (ρ)

�T. (24)

By comparing the relations of (24), the following rela-tionships are obtained.

αz(ρ) − αzυyzEy

− υyz(ρ)Ey (ρ)

= αy (ρ) − αy1

Ey (ρ) − 1Ey

= αx(ρ) − αxυyxEy


= k1. (25)

This equation indicates that the k1 parameter is a laminaconstant that is dependent of the damage parameter ρ andis such that σ 0

y = −k1�T.Similarly, by considering ply crack closure for con-

strained uni-axial loading for in-plane direction of x, suchthat σy = σz = τxy = 0 and following the above approachit can be shown that, by setting the εx = εc l

x , εy = εc ly ,

εz = εc lz and σx = σ c l

x

σ c lx = − αz(ρ) − αz

υxzEx

− υxz(ρ)Ex (ρ)

�T = − αy (ρ) − αyυyxEy


�T

= −αx(ρ) − αx1

Ex (ρ) − 1Ex

�T.

(26)

It then follows from (26) that:

αz(ρ) − αzυxzEx

− υxz(ρ)Ex (ρ)

= αy (ρ) − αyυyxEy


= αx(ρ) − αx1

Ex (ρ) − 1Ex

= k2. (27)

The k2 constant is thus another lamina constant depen-dent of the damage parameter ρ such that σ 0

x = −k2�T.The ratio of k1 over k2 is also defined as:

υxzEx

− υxz(ρ)Ex (ρ)

υyzEy

− υyz(ρ)Ey (ρ)

=υyxEy


1Ey (ρ) − 1

Ey

=1

Ex (ρ) − 1Ex

υyxEy


= k, (28)

where k is a lamina constant that is dependent of thedamage parameter ρ and independent of and the thermalexpansion coefficients.

Finally, considering the crack closure conditions for thelamina subjected to the uni-axial tension loading in theout-of-plane direction z, in which σx = σy = τxy = 0, andsetting εx = εc l

x , εy = εc ly , εz = εc l

z and σz = σ c lz , it can be

shown that:

σ c lz = −αz(ρ) − αz

1Ez(ρ) − 1

Ez

�T = −αy (ρ) − αyυyzEy

− υyz(ρ)Ey (ρ)

�T

= −αx(ρ) − αxυxzEx

− υxz(ρ)Ex (ρ)

�T. (29)

It then follows from the above relations that:

αz(ρ) − αz1

Ez(ρ) − 1Ez

= αy (ρ) − αyυyzEy

− υyz(ρ)Ey (ρ)

= αx(ρ) − αxυxzEx

− υxz(ρ)Ex (ρ)

= k3. (30)

The k3 constant is thus another lamina constant depen-dent of the damage parameter ρ such that σ 0

z = −k3�T.



The ratio of k1 over k3 is also defined as:

1Ez(ρ) − 1

Ez

υyzEy

− υyz(ρ)Ey (ρ)

=υyzEy

− υyz(ρ)Ey (ρ)

1Ey (ρ) − 1

Ey

=υxzEx

− υxz(ρ)Ex (ρ)

υyxEy


= k′, (31)

where the constant k′ is independent of the thermalexpansion coefficients and dependent of the damageparameterρ. The relations (28) and (31) are not of coursecompletely independent. By substituting the k and k′ fromthe (28) and (31) into (21), the elastic properties parame-ters of damaged lamina couldn’t be obtained (three equa-tions and two unknown). For this purpose, a macroscopicdamage parameter D(ρ) is introduced:

D(ρ) = Ey

Ey (ρ)− 1. (32)

The parameter of D(ρ) is borrowed from the field ofcontinuum damage mechanics and is equal to zero valuefor undamaged lamina.

By substituting of D(ρ) from (32) in to (28) and (31), thefollowing set of independent relations are obtained.

νyx

Ey− νyx(ρ)

Ey (ρ)= k

D(ρ)Ey

1Ex

− 1Ex(ρ)

= k2 D(ρ)Ey

νyz

Ey− νyz(ρ)

Ey (ρ)= k′ D(ρ)

Ey

1Ez

− 1Ez(ρ)

= k′2 D(ρ)Ey

νxz

Ex− νxz(ρ)

Ex(ρ)= kk′ D(ρ)

Ey. (33)

Using these relations and Eq. (32), the effective thermo-elastic constants of a damaged lamina (Ex(ρ), Ey(ρ), Ez(ρ),υyx(ρ), υxz(ρ), υyz(ρ)) can be calculated in terms of thedamage dependent parameter of D(ρ) and the corre-sponding values for kand k′.

As far as, the shear strain of the damaged sub-lamina isalready obtained and the shear stress of the damaged sub-lamina isn’t changed after the crack formation, the shearmodulus of the damaged lamina could be also defined asfollow:

Gxy (ρ) = γxy

γ cxy

Gxy . (34)

The following scalar damage parameter can be definedanalogous to Eq. (32):

D′(ρ) = Gxy

Gxy (ρ)− 1. (35)

The equations of (32)–(35) show how the effective elasticconstants of the damaged lamina may be calculated interms of k, k′, D(ρ) and D′(ρ) in order to characterizethe multi-axial behaviour of the homogenized damagedlamina.

In continue, it is explained that how damage formationin a composite lamina can be applied to the structuralfeatures at the macroscopic scale. One method of achiev-ing this objective is to employ the continuum damagemechanics method. One continuum damage mechanicsapproach, applied to structures through the use of FEA,is based on mesomechanics method.14 In the present ap-proach, the energy density (complementary energy den-sity corresponding to the Gibbs energy density) is writtenas the following form:

E(D, D′) ≡ 12

⎡⎢⎢⎢⎢⎣

σ 0xx

σ 0yy

σ 0zz

σ 0xy

⎤⎥⎥⎥⎥⎦

t

×

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1Ex(ρ)

−vxy (ρ)Ex(ρ)

−vxz(ρ)Ex(ρ)

0

−vxy (ρ)Ex(ρ)

1Ey (ρ)

−vyz(ρ)Ey (ρ)

0

−vxz(ρ)Ex(ρ)

−vyz(ρ)Ey (ρ)

1Ez(ρ)

0

0 0 01

Gxy (ρ)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎣

σ 0xx

σ 0yy

σ 0zz

σ 0xy

⎤⎥⎥⎥⎥⎦ ,

(36)

where E(D, D′) is strain energy of the damaged laminaand D and D′ are functions of the damage parameter ρ.Substituting the elastic constants of damaged lamina fromequations (32)–(35) in the above relation leads to:

E(D, D′) ≡ 12

⎡⎢⎢⎢⎢⎣

σ 0xx

σ 0yy

σ 0zz

σ 0xy

⎤⎥⎥⎥⎥⎦

t

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1Ex

− k2 D(ρ)Ey

kD(ρ)Ey

− νyx

Eykk′ D(ρ)

Ey− νzx

Ez0

kD(ρ)Ey

− νyx

Ey

D(ρ) + 1Ey

k′ D(ρ)Ey

− νyz

Ey0

kk′ D(ρ)Ey

− νzx

Ez

−vyz(ρ)Ey (ρ)

1Ez

− k′2 D(ρ)Ey

0

0 0 0D′(ρ) + 1

Gxy

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎣

σ 0xx

σ 0yy

σ 0zz

σ 0xy

⎤⎥⎥⎥⎥⎦ . (37)



The stress–strain relations and the ‘thermodynamicforces Y are defined by

ε = ∂ E(D, D′)∂σ

Y = ∂ E(D, D′)∂D

. (38)

The stress–strain relations for the damaged lamina are inthe form of ε = Sσ . The obtained compliance matrix fordamaged lamina shows that the behaviour of the damagedlamina can be described by stress–strain relations that arein the same form as those for an equivalent undamagedlamina. The thermo-elastic constants are only affected bythe damage, resulting that in any localized region of astructure the damaged lamina properties can be replacedby its homogeneous equivalent lamina.

P R E D I C T I N G O F M A T R I X C R A C K F O R M A T I O N

Early works on micro-crack formation in composite lami-nates assumed that the micro-cracks form when the stressin the 90◦ plies reaches the transverse strength of the plymaterial.5 Similarly, the first-ply failure models assumethe ply transverse strength or some multi-axial stress statecriterion determines micro-crack formation. In more re-cent works, ‘energy methods’ or ‘fracture mechanics ap-proach’ is used to predict the crack formation.5,17–19 Mostof the energy methods use a discrete model or a finitefracture mechanics model in which the micro-crack oc-curs when the total energy released by the formation ofthat micro-crack reaches to the critical energy release ratefor micro-cracking, Gmc , or the micro-cracking fracturetoughness. For this purpose, the stress analysis is requiredto calculate the total energy released by the formation ofthe complete micro-crack, Gm. Once this energy releaserate is known, the obtained value of Gm is equated bythe micro-cracking fracture toughness, and the resultingequation is solved at each applied load step.

According to the above explanations and considering theinitiation of new cracked planes per unit length in a laminaand using the energy balance, the fracture criterion forcrack formation can be defined as follows:20

1A

(∫σε(ρ) dV −

∫σεdV

)− Gmc 〉0. (39)

In this relation, A is the crack surface area, which is de-fined by lamina thickness multiply by the unit cell width.The expressions

∫σε(ρ) dV and

∫σεdV define the strain

energy at cracked lamina with ρ crack density and strainenergy of undamaged lamina, respectively. It is worth tomention that in this study, the effects of transverse cracksas an initial damage mode on the laminates response areonly considered in the analyses.

N O N L I N E A R F I N I T E E L E M E N T C O D E

In this part, a developed nonlinear finite element code21,22

based on the continuum damage approach and layer-wisetheory is used. It has been used as a source code and thenecessary modifications are applied as explained in theflowchart of the program in Fig. 3. In this code, 8 nodesfull layer-wise elements are used. In these elements, it isassumed that the displacements, material properties andelement geometry can be approximated by a sum of conve-niently separable interpolation functions. The transversestrains are assumed as a piecewise continuous distributionthrough the laminate thickness.

Using this element and defining the material properties,stacking sequence and boundary conditions, the stressesand strains are obtained by linear analysis. For modellingof the nonlinear response due to damage initiation andgrowth, a precise damage flow rule needs to be consid-ered. For this purpose, the developed micromechanicsapproach is used and the constitute relations are extendedusing the proposed generalized plane strain model. Thisprocedure is employed in the extended subroutine for de-veloping a proposed flow rule of damage and numericalintegration of constitute relations. The stress boundaryconditions on each unit cell are obtained from the finiteelement analysis of laminates at each loading step. In thefinite element code, each element is considered as a unitcell, which the initial unit cell dimension is equal to the el-ement size. The dimensions of each unit cell are changedby crack density evolution. By increasing the crack den-sity and using the obtained stress–strain fields of each unitcell, the strain energy release rate of recent unit cells arecalculated to find the possibility of new crack formation.The energy-based criterion is used to predict the ma-trix crack formation. By comparing the calculated strainenergy with micro-cracking fracture toughness the evo-lution of crack density in each unit cell can be recognized.If the calculated strain energy release rate is greater thanthe micro-cracking fracture toughness, the assumed crackdensity is valid. Otherwise, the crack density must be in-creased and solution procedure is repeated. Crack densityevolution in each unit cell causes degradation in thermo-elastic constants of lamina. By performing this process, thecrack density can be determined and the elastic propertiesof the unit cell can be updated. This stiffness reductionis used to predict the matrix crack formation in mesome-chanics based continuum damage mechanics method bydefining the damage parameters for each element of anylamina through the thickness of laminate.

This process is repeated at any specified load step foreach element. Following that, the residual forces are cal-culated for the entire laminate to satisfy the convergencecriteria. Then, the damage growth can be predicted ac-cording to a new evolution law based on the microme-chanics approach for uniformly cracked laminates with



Fig 3 Flow chart structure of progressive damage analysis in developed nonlinear finite element code.

any lay-up configuration under any loading combination.Fig. 3 depicts the procedure of the explained progressivedamage analysis in this code.

R E S U L T S A N D D I S C U S S I O N S

In this section, the developed micromechanical model isused to analyse the stress distribution of cracked singlelayer unit cells and to compare the results with the ob-tained results from commercial finite element code fortypical lamina with different initial conditions. Whenthe initial elastic stresses and crack density of the lam-ina are known, the perturbation stresses can be obtainedby this study. The obtained stiffness degradations includ-ing transverse stiffness, shear stiffness and Poisson’s ratioversus matrix cracking are also compared with the avail-

able results from literature. Furthermore, the strain en-ergy of the cracked lamina is compared with the obtainedresults from commercial finite element code. Finally, thestress–strain behaviour and the damage initiation are com-pared with the available experimental results for variouslaminates.

Unit cell verification

The geometry of the first considered lamina is similar tothat shown in Fig. 2 and its material is Carbon/Epoxywith the properties listed in Table 1. The lamina thick-ness is 0.127 mm. The initial stresses and strains are: σ yy =160 MPa, σ yx = 100a, σ zz = 85 MPa, εc

x = 0 and crackdensity = 1.66 (1/mm). The obtained stress distributionfor the cracked lamina is presented in Fig. 4 where y = 0 is

Table 1 Composite material properties23−27

T300/976 Ex = 139.178 GPa Ey = 9.71 GPa Gxy = 5.512 GPa νxy = 0.29AS4/3501-6 Ex = 139.178 GPa Ey = 9.85 GPa Gxy = 5.24 GPa νxy = 0.30Carbon/Epoxy Ex = 144 GPa Ey = 9.58 GPa Gxy = 4.785 GPa νxy = 0.31Graphite/Epoxy Ex = 45.6 GPa Ey = 16.2 GPa Gxy = 5.83 GPa νxy = 0.278



0

40

80

120

160

0.30.250.20.150.10.050

y(mm)

σy(

MP

a)

FEM: z=2h1/7

Present Study: z=2h1/7

ANSYS: z=4h1/7


(a)

0

20

40

60

80

100

120

0.30.250.20.150.10.050

y(mm)

σyx

(MP

a)

FEM: z=2h1/7


FEM: z=2h1/7


(b)

-250

-190

-130

-70

-10

50

110

0.30.250.20.150.10.050

y(mm)

σz(

MP

a)

FEM: z=5h1/7


(c)

-30

10

50

90

130

0.30.250.20.150.10.050

y(mm)

σyz

(MP

a)

FEM: z=5h1/7


(d)

Fig. 4 Stress distributions in Carbon/Epoxylamina by this study and ANSYS FE code,(a) σ yy, (b) σ yx, (c) σ zz, (d) σ yz.



Table 2 Strain energy compare between this study andcommercial finite element code

σ yy = 160,σ yy = 160, σ yx = 100,

Loading (MPa) σ yy = 160 σ yx = 100 σ zz = 85

FEM strain energy(×10−3 N m)

1.33 1.90 1.99

This study strainenergy (×10−3 N m)

1.27 2.02 1.89

Difference percentage 4% 5.9% 5%

at the middle of the surface between the two neighbour-ing matrix cracks with the distance of 0.6 mm. This figureshows that the σ yy, σ yz and σ yx stresses have zero values atcracked surfaces (stress free condition) and they have theuniform value of equal to the applied stress at the areas,which are far from the cracked surface affects. The ob-tained results from the commercial FEM code have alsobeen shown in the same figure. The differences betweenthe stresses obtained from the analytical solution of thiswork with those obtained from FEM near the stress freesurfaces are due to the considered stress free surfaces asboundary conditions and assuming of the uniform stressesalong the lamina thickness which are both different fromthe conditions in FEM analysis. The peak values of σ zz

and σ yz in this figure, which have been predicted by thecurrent investigation are reasonable and they are due tothe existence of transverse crack and may cause delamina-tion which will be considered in the future works.

Furthermore, the strain energy of the unit cell con-taining the transverse crack density is calculated andcompared with the results of commercial FEM code inTable 2. This table shows that the differences betweenthe results of this study and FEM is negligible and are lessthan 6% for various loadings.

For the second case, the lamina’s material isGraphite/Epoxy and its thickness is 0.26 mm. But the ini-tial stresses and strains are: σ yy = 110 MPa, σ yx = 70 MPa,σ zz = 50 MPa, εc

x = 0 and crack density = 1 (1/mm). Theobtained stress distribution for the cracked lamina is givenin Fig. 5. The results explanation and differences betweenthe analytical and FEM results are almost the same as theabove presented discussions for the first case.

Elastic properties reduction

As discussed in Section ‘Calculation of thermo-elasticconstants’, using the relations of (32)–(34), the effectivethermo-elastic constants of a damaged lamina can be eval-uated as a function of crack density. Then, having thereduced material properties of damaged lamina, the stiff-ness reduction of any laminates can be obtained. The

resulting stiffness reduction for a Graphite/Epoxy cross-ply laminate with 0.26 mm ply thickness is shown inFig. 6 and compare with the obtained results by Kash-talyan and Soutis.23 In this figure, the initial moduli oflaminate are defined by superscript of 0. This figure showsthat the predicted stiffness reductions are in good agree-ments with the results obtained by Kashtalyan et al. forany crack density.

Stress–strain response of laminates

As discussed before, the calculated unit cell strain energycan be used to predict the composite laminate propertieshaving a uniform distribution of matrix cracks in 90◦ ori-entation. For this purpose, the developed nonlinear finiteelement code based on the continuum damage approach isused for progressive damage analysis of the laminates. Topredict the matrix crack formation, the explained energy-based generalized framework is used. For some laminatesbefore the final failure occurs, fibres are broken. In thisstudy the fibre breakage and its effect on the stiffnessreduction is not considered by any failure criterion. Theresulting stress–strain behaviours for a cross-ply laminatesis shown in Fig. 7. This figure shows the predicted and ex-perimental results for [0/90]s Graphite-Epoxy laminateswith 0.26 mm ply thickness and fracture energy of Gmc =165 J/m2.24 The predicted stress–strain behaviour is in agood agreement with the experimental results up to theapplied stress of about 84% of the final failure load ofthe laminate for both initiation and propagation of thetransverse cracks. The crack initiation point in this fig-ure may be interpreted as the position of changing theslope of the curve. Experimental failure load and failurestrain of this laminate under uniaxial loading are 610 MPaand 2.7%, respectively. It is worth to mention that thesepredictions were made using basic information about thefracture toughness of the matrix material which assumedto be equal to the measured mode-I fracture toughness.

Figure 8 shows the stress–strain behaviour for an angleply Graphite/Epoxy laminate of [45/−45]s under biaxialloading. In this case the ratio of axial stress over the trans-verse stress is also equal to 1.0. This figure shows thatthe predicted stress–strain behaviour is in a good agree-ment with the experimental results26 for both initiationand propagation of the transverse cracks up to the appliedstress of about 80% of the final failure load of the lami-nate. Experimental failure stress and failure strain of thislaminate under biaxial loading are 430 MPa and 2.5%,respectively.

Figure 9 shows the stress–strain behaviour for a quasiisotropic AS4/3501–6 laminate of [90/−45/45/0]s underbiaxial loading. The properties of plies material is listedin Table 1. The lamina thickness is 0.13 mm and fractureenergy is Gmc = 175 J/m2.27 This figure shows that the



0

20

40

60

80

100

120

140

0.50.40.30.20.10

y (mm)

σ y (

MP

a)

FEM: z=2h1/7


FEM: z=4h1/7


(a)

0

10

20

30

40

50

60

70

0.50.40.30.20.10

y(mm)

σ yx (

Mpa

)

FEM: z=2h1/7


FEM: z=4h1/7


(b)

-220

-170

-120

-70

-20

30

80

0.50.40.30.20.10

y(mm)

σ zz(

MP

a)

FEM: z=5h1/7


(c)

-20

0

20

40

60

80

100

120

0.50.40.30.20.10

y(mm)

σ yz(

MP

a)

FEM: z=5h1/7


(d)

Fig. 5 Stress distributions inGraphite/Epoxy lamina by this study andANSYS FE code, (a) σ yy, (b) σ yx, (c) σ zz, (d)σ yz.



0

0.2

0.4

0.6

0.8

1

43.532.521.510.50

Crack Density(1/mm)

Ela

stic

Red

ucti

on P

rope

rty

Rat

io

Kashtalyan & Soutis [23]



Present Study

Present Study

Present Study0xyxy

0xyxy

0xx

0xyxy

0xyxy

0xx

/

G/G

E/E

/

G/G

E/E

υυ

υυ

Fig. 6 Elastic reduction properties ratio versus transverse crackdensity for [0/90]s Graphite/Epoxy laminate.

0

100

200

300

400

500

600

700

0.0240.020.0160.0120.0080.0040axial strain

axia

l str

ess

(MP

a)

[0/90]s:Experiment [24]

[0/90]s:present study

θ+

Fig. 7 Axial stress versus axial strain behaviour predicted by thisstudy for Graphite/Epoxy cross-ply laminates.

0

50

100

150

200

250

300

350

400

450

0.0240.020.0160.0120.0080.0040transverse strain

tran

sver

se s

tres

s(M

Pa)

[45/-45]s: Experiment [26]- Sx/Sy=1/1

Present Study

θ+

Fig. 8 Axial stress versus axial strain behaviour for biaxial loadingpredicted by this study for [45/−45]s Graphite/Epoxy laminate.

predicted stress–strain behaviour is in an excellent agree-ment with the experimental results26 for both initiationand propagation of the transverse cracks up to the appliedstress of about 95% of the final failure load of the lam-inate. Experimental final failure stress and strain of thislaminate under biaxial loading are 711 MPa and 1.45%,respectively.

Figure 10 shows the stress–strain behaviour for[−35/35]s Graphite-Epoxy laminates under the uniax-ial tension loading. Again in this case, the predictedstress–strain behaviour is in a good agreement with theexperimental results26 for both initiation and propaga-tion of the transverse cracks up to the applied stress ofabout 78.5% of the final failure load of the laminate. Ex-perimental final failure stress and strain of this laminate

0

100

200

300

400

500

600

700

0.0150.01250.010.00750.0050.00250

axial strain

axia

l str

ess

(MP

a)

[90/-45/45/0]s:Experiment[26]

Present Study

θ+

Fig. 9 Axial stress versus axial strainbehaviour for biaxial loading predicted bythis study for [90/−45/+45/0]s AS4/3501-6laminate.



0

100

200

300

400

500

0.060.050.040.030.020.010

axial strain

axia

l str

ess

(MP

a)

[-35/35]s:Experiment [26]

Present study

θ+

Fig. 10 Axial stress versus axial strain behaviour for axial loadingpredicted by this study for [−35/35]s Graphite/Epoxy laminate.

0

20

40

60

80

100

0.0090.00750.0060.00450.0030.00150

axial strain

axia

l str

ess(

MP

a)

[60/60/90/-60/-60]s:Experiment [27]

[60/60/90/-60/-60]s: Present Study

θ+

Fig. 11 Axial stress versus axial strain behaviour for biaxial loadingpredicted by this study for [602/90/−602]s T300/976 laminate.

under the uniaxial transverse tension are 470 MPa and6.3%, respectively.

Finally the stress–strain behaviour for [602/90/−602]s

laminates under uniaxial loading is also obtained and de-picted in Fig. 11. The plies material is T300/976 with theproperties listed in Table 1. The lamina thickness is 0.13mm and fracture energy is Gmc = 157 J/m2.27 This figurealso shows that the predicted stress–strain behaviour is ina good agreement with the experimental results27 for bothinitiation and propagation of the transverse cracks up tothe applied stress of about 70% of the final failure loadof the laminate. Experimental final failure applied stressand strain of this laminate under the uniaxial tension are72 MPa and 1% respectively.

C O N C L U S I O N

In this study, a new micromechanic approach was de-veloped to study the progressive damage of compositelaminates. It was employed to predict the matrix crackformation in mesomechanics based continuum damagemechanics method. For this purpose a generalized planestrain unit cell was formulated for orthotropic compositelamina including transverse cracks. Using this model, thestress distribution and strain energy of lamina contain-ing crack density was obtained and applied in a developednonlinear finite element code to predict the occurrenceof transverse cracks in the laminated composites. It wasshown that, the predicted stress–strain behaviours, andthe damage initiation are in good agreement with theexperimental results for transverse cracks in various lam-inates. The major advantage of the proposed approach isconsidering the saturation due to transverse cracks.

R E F E R E N C E S

1 Zhang, J., Soutis, C. and Fan, J. (1994) Strain energy releaserate associated with local delamination in cracked compositelaminates. Composites 25, 851–862.

2 McCartney, L. N. (1992) Theory of stress transfer in a 0-90-0cross ply laminate containing a parallel array of transversecracks. J. Mech. Phys. Solids. 40, 27–68.

3 Zhang, J., Soutis, C. and Fan, J. (1994) Effects of matrixcracking and hygrothermal stresses on the strain energyrelease rate for edge delamination in composite laminates.Composites 25, 27–35.

4 Rebiere, J. L. and Gamby, D. (2004) A criterion for modelinginitiation and propagation of matrix cracking anddelamination in cross-ply laminates. Comp. Sci. Tech. 64,2239–2250.

5 Nairn, J. A. (2000) Matrix microcracking in composites. In:Comprehensive Composite Materials, Vol. 2 (Edited by A. Kellyand C. Zweben), Elsevier Science, Amsterdam, TheNetherlands. pp. 403–432.

6 Kashtalyan, M. and Soutis, C. (2002) Analysis of localdelaminations in composite laminates with angle-ply matrixcracks. Int. J. Solids Struct. 39, 1515–1537.

7 Zhang, C. and Zhu, T. (1996) On inter-relationships ofelastic moduli and strains in cross-ply laminated composites.Comp. Sci. Tech. 56, 135–146.

8 Kashtalyan, M. and Soutis, C. (2005) Analysis of compositelaminates with intra- and interlaminar damage. Prog. Aerosp.Sci. 41, 152–173.

9 Hashin, Z. (1986) Analysis of stiffness reduction of crackedcross-ply laminates. Eng. Fracture Mech. 25, 771–778.

10 Nairn, J. A. (1989) The strain energy release rate ofcomposite micro-cracking: a variational approach. J. Comp.Mater. 23, 1106–1129.

11 Nairn, J. A. and Hu, S. (1992) The formation and effect ofouter-ply microcracks in cross-ply laminates: a variationalapproach. Eng. Fracture Mech. 41, 203–221.

12 Zhang, H. and Minnetyan, L. (2006) Variational analysis oftransverse cracking and local delamination in [θm/90n]slaminates. Int. J. Solids Struct. 43, 7061–7081.



13 McCartney, L. N. (2000) Model to predict effects of triaxialloading on ply cracking in general symmetric laminates.Comp. Sci. Tech. 60, 2255–2279.

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Damage analysis of laminated composites using a new coupled …webpages.iust.ac.ir/bijan_mohammadi/Bijan_Mohammadi... · 2015-10-07 · doi: 10.1111/j.1460-2695.2010.01456.x Damage

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