Page 1

Hindawi Publishing Corporation

International Journal of Aerospace Engineering

Volume 2009, Article ID 486063, 22 pages

doi:10.1155/2009/486063

Research Article

AThree-DimensionalPly FailureModelforComposite Structures

Maur´ ıcioV.Donadon,S´ ergio FrascinoM. de Almeida,

MarianoA.Arbelo,andAlfredo R.de Faria

Department of Mechanical Engineering, Aeronautical Institue of Technology (ITA), CTA-ITA-IEM, Prac ¸a Mal. Eduardo Gomes 50,

12228-900 S˜ ao Jos´ e dos Campos-SP, Brazil

Correspondence should be addressed to Maur´ ıcio V. Donadon, donadon@ita.br

Received 21 October 2008; Revised 28 January 2009; Accepted 23 June 2009

Recommended by Ever Barbero

A fully 3D failure model to predict damage in composite structures subjected to multiaxial loading is presented in this paper. The

formulation incorporates shear nonlinearities effects, irreversible strains, damage and strain rate effects by using a viscoplastic

damageable constitutive law. The proposed formulation enables the prediction of failure initiation and failure propagation by

combining stress-based, damage mechanics and fracture mechanics approaches within an unified energy based context. An

objectivity algorithm has been embedded into the formulation to avoid problems associated with strain localization and mesh

dependence. The proposed model has been implemented into ABAQUS/Explicit FE code within brick elements as a userdefined

material model. Numerical predictions for standard uniaxial tests at element and coupon levels are presented and discussed.

Copyright © 2009 Maur´ ıcio V. Donadon et al. This is an open access article distributed under the Creative Commons Attribution

License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

cited.

1.Introduction

Damage in composite structures is a very complex phe-

nomenon which can occur through a different number of

failure mechanisms such as fibre breakage, fibre buckling,

matrix cracking, fibre-matrix debonding and delamination

either combined or individually. The increasing computa-

tional resources have allowed reliable prediction of such

phenomenon with a certain degree of accuracy by using the

finite element method. However, there is still a lot of work

to be done in this field to better understand the physics

of failure in order to improve numerical failure models for

composite materials. The damage modelling in composites

can be broadly divided into four different approaches:

(i) failure criteria approach,

(ii) fracture mechanics approach,

(iii) plasticity approach,

(iv) damage mechanics approach.

Failure criteria approaches were initially developed for

unidirectional materials and restricted to the static regime.

They are divided into two categories: interactive and non-

interactive criteria [1]. Noninteractive criteria assume that

the failure modes are decoupled and specific expressions

are used to identify each failure mechanism. In the stress

based criteria, for example, each and every one of the stresses

in the principal material coordinates must be less than

the respective strengths; otherwise, fracture is said to have

occurred. In a similar fashion, the maximum strain failure

criteriastatesthatthefailureoccursifoneofthestrainsinthe

principal material coordinates exceed its respective failure

strain.

On the other hand, interactive criteria assume an

interaction between two or more failure mechanisms and

they describe the failure surface in the stress or strain

space. Usually stress or strain polynomial expressions are

used to describe the boundaries for the failure surface or

envelope. Any point inside the envelope shows no failure

in the material. Several interactive failure criteria can be

found in the open literature, such as the Tsai and Wu [2]

criterion, Tsai-Hill criterion, Hoffman criteria among others

[1]. Nevertheless, the disadvantage in using such polynomial

criteria is that they do not say anything about the damage

mechanisms themselves, therefore modified versions have

been used to distinguish between failure modes. Hashin [3]

proposed a three-dimensional failure criteria for unidirec-

tional composites. In his model four distinct failure modes

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2 International Journal of Aerospace Engineering

associated with fibre failure in tension or compression and

matrix cracking in tension or compression are modelled

separately. Engblom and Havelka [4] proposed the use of a

combination of the Hashin [3] and Lee [5] failure criteria.

They used the Hashin criterion to detect in-plane failure and

the Lee criteria to predict delaminations. The degradation

was performed by reducing the stresses associated with

each failure mechanism to zero in the composite material

constitutive law.

Shivakumar et al. [6] used the Tsai-Wu failure criterion

and maximum stress criterion to model low-velocity impact

damage in composites. In this approach the Tsai-Wu crite-

rion was used in order to determine whether the damage

has occurred and the maximum stress criteria was used to

identify the failure mode. Good agreement was obtained

between computed and experimental damage areas.

F.-K. Chang and K.-Y. Chang [7, 8] proposed a pro-

gressive in-plane damage model for predicting the residual

strength of notched laminated composites. Three in-plane

failure modes are considered: matrix cracking, fibre-matrix

shearing and fibre breakage. Their model is currently avail-

ableinthematerialmodellibraryoftheLS-DYNA3Dexplicit

finite element code. In their model the shear stress-shear

strain relation is assumed to be nonlinear and an expression

proposed by Hahn and Tsain [9, 10] is used to represent

composite behaviour in shear. For fibre breakage and/or

fibre-matrix shearing the degree of property degradation

within the damaged area depends on the size of the damage

predicted by the fibre failure criterion. F.-K. Chang and

K.-Y. Chang [7] proposed a property reduction model for

fibre failure based on the micromechanics approach for fibre

bundle failure. It is postulated that for fibre failure both Ey

and νyxare reduced to zero, but Exand the shear modulus

Gxydegenerate according to the Weibull distribution.

Choi et al. [11] investigated the low velocity impacts

on composite plates using a line-nose impactor. They used

the Chang-Chang failure criterion to detect the initiation of

matrixcrackinganddelamination.Goodagreementbetween

experimental and numerical results was achieved.

Davies and Zhang [12] studied the low-velocity impact

damage in carbon/epoxy composite plates impacted by

an hemispherical impactor. The plates tested were quasi-

isotropic having a symmetrical lay-up. Different impact

energy levels, thickness, dimensions and boundary condi-

tions were considered. The finite element modelling was

carried out using plate elements.

The Chang-Chang failure criteria were used for the in-

plane damage predictions and in general good agreement

was obtained between the computed force histories and

experiments.The Chang-Chang failure criteria have shown a

reasonable performance for in plane damage predictions in

composite structures and therefore, their applications have

been widely reported in the literature [13–18].

The prediction of the onset of delamination depends on

the interlaminar stress state and interlaminar strength of

the laminate. Kim and Soni [19] used the distribution of

interlaminar normal stress, and averaged that stress along

a ply thickness distance in all cases they studied. They

assumed that failure occured when the average of the normal

interlaminar stress value over the fixed distance reached the

interlaminar tensile strength.

Jenetal.[20]developedamodelbasedonboundarylayer

theory to predict initiation and propagation of delamination

in a composite laminate containing a central circular hole.

The Hashin-Rotem failure criterion was adopted in their

model to predict the loading and location at which the

initiation of delamination occurs.

Brewer and Lagace [21] proposed a quadratic stress

criterion for initiation of delamination using the approach

suggested by Kim and Soni [19]. According to their criterion

only out-of-plane stresses contribute to delamination and

they assume that the predicted stresses should be indepen-

dent of the sign of the interlaminar shear stress. A similar

criteria was also proposed by Liu et al. [22] to predict

delamination in composite laminates.

The disadvantage in using stress based criteria for

composite materials is that the scale effects relating to the

length of cracks subject to the same stress field cannot be

modelled correctly [13, 23]. In the failure criteria approach

either the position and size of the cracks are unknown. For

these reasons the fracture mechanics approach may be more

attractive. Fracture mechanics considers the strain energy at

the front of a crack of a known size and compares the energy

with critical quantities such as critical strain energy release

rate.

The fracture mechanics approach has been used to

predict compression after impact strength of composite

laminates [24–27]. In such models the damaged area is

replaced by an equivalent hole and the inelastic deformation

associated with fibre microbuckling that develops near the

hole edge is replaced with a equivalent crack loaded on

its faces by a bridging traction which is linearly reduced

with the crack closing displacement. The diameter of the

hole is obtained from X-radiographs and/or ultrasonic C-

scan images. The results showed good correlation between

analytical and experimental values.

Another potential application of the fracture mechanics

approach is its indirect use to predict progressive delam-

ination in composites [28–31]. In such models stress-

displacement constitutive laws describe the interfacial mate-

rialbehaviourandfracturemechanicsconceptsareused.The

area defined by the constitutive relationship is equal to the

fracture energy or energy release rate and once the stresses

have been reduced to zero the fracture energy has been

consumed and the crack propagates. Linear and quadratic

interaction relationships were assumed to describe the crack

propagation in mixed-mode delamination. Comparisons

were made with experimental and closed-form results and

good agreement was obtained.

However, the fracture mechanics approach cannot be

easily incorporated into a progressive failure methodology

because its application requires an initial flaw. A possible

solution is to use a hybrid approach by using a stress or

strain-based criterion for the failure initiation and a fracture

mechanics approach for the failure propagation.

The plasticity approach is suitable for composites

that exhibit ductile behaviour such as boron/aluminium,

graphite/PEEK andother thermoplasticcomposites.

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International Journal of Aerospace Engineering3

Vaziri et al. [32] proposed an orthotropic plane stress

material model that combines the classical flow theory

of plasticity with a failure criterion. In their work the

material constitutive law is assumed to be elastic-plastic

and it has two stages. The first stage is the post-yield and

pre-failure where an orthotropic plasticity model is used

to model the nonlinear material behaviour. The second

stage is the postfailure where brittle or ductile failure modes

start to occur. Favourable agreement is obtained between

experiment and the model results.

The damage mechanics approach has been investigated

by many researchers in recent years and its application to

damage modelling in composites has shown to be efficient.

The method was originally developed by Kachanov [33] and

Rabotnov [34] and it has the potential to predict different

composite failure modes such as matrix cracking, fibre

fracture and delamination.

Ladeveze and Dantec [35] proposed an in-plane model

basedondamagemechanicstopredictmatrixmicrocracking

and fibre/matrix debonding in unidirectional composites.

Two internal damage variables were used to degrade the

ply material properties, one of them associated with the

transverse modulus and another with the in-plane shear

modulus.Alinearelastic-damagebehaviourwasassumedfor

tensile and compressive stresses and a plasticity model was

developed to account for the inelastic strains in shear. Dam-

age evolution laws associated with each failure mechanism

were introduced which relate the damage variables to strain

energy release rates in the ply. Tension, compression and

cyclic shear tests were performed to determine the constants

required in the damage-development laws. Comparisons

with experiments were performed by the authors and a good

correlation between numerical and experimental results was

obtained.

Johnson [36] applied the model suggested by Ladeveze

and Dantec for the prediction of the in-plane damage

response of fibre reinforced composite structures during

crash and impact events. The damage model was imple-

mented for shell elements into the PAM-CRASH explicit

finiteelementcode.Anexperimentalprogrammewascarried

out in order to validate the model. Low velocity impact

tests were performed using a drop test rig. The plates were

simply supported on a square steel frame and they were

impacted at their centre using a hemispherical impactor

with 50mm diameter. The mass of the impactor was 21kg.

The plates were fabricated from 16plies of carbon/epoxy

with a quasi-isotropic layup. Different energy levels were

considered to give a range of different failure modes

from rebound to full penetration. Based on force history

comparisons between experiments and numerical results the

model overpredicted the peak load. According to Johnson,

the discrepancy between measured and predicted peak load

isexplainedbytheneglecteddelaminationinthemodel.This

fact was demonstrated by the author in his subsequent work

[37] where the delamination was included in the damage

modelling using contact interface conditions.

Williams and Vaziri [38] implemented an in-plane

damage model based on continuum damage mechanics

(CDM) into LS-DYNA3D for impact damage simulation in

composite laminates. The model was originally developed

by Matzenmiller et al. [39]. The model considers three

damage parameters: fibre failure damage parameter, matrix

failure damage parameter and an extra damage parameter

to account for the effect of damage in shear response.

Individual stress based criteria for each failure mechanism

are used for the damage initiation. The failure criterion

defines certain regions in stress (or strain) space where

the damage state does not change. The damage growth

law adopted by Matzenmiller is a function of the strain

and it assumes an exponential form. The stress/strain curve

predicted by this damage function is a Weibull distribution

that can be derived from statistical analysis of the probability

of the failure of a bundle of fibres with initial defects. Impact

simulations with different energy levels were performed and

the performance of the proposed model was checked against

the Chang-Chang failure criteria and experimental results.

Some limitations of the model were pointed out by the

authors.Firstly,theresponseispredictedbyasingleequation

and, as a result the loading and postfailure responses cannot

be separated, thus restricting the versatility of the model.

Also, the dependence of the damage growth on the loading

rate as well as mesh size.

In his recent work Williams et al. [40] proposed a plane-

stress continuum damage mechanics model for composite

materials. The model was implemented for shell elements

into LSDYNA3D explicit finite element code. Also, the

model is an extension of their previous work [41] and

it deals with some issues related to the limitation of the

model suggested by Matzenmiller et al. [39]. The approach

adopted by the authors uses the sub-structuring concept

where one integration point is used for each sub-laminate

and composite laminate theory is used to obtain its effective

material properties. The model assumes a bilinear damage

growth law and the damage process has two phases, one

associated with matrix/delamination damage and another

with fibre breakage. The driving force for damage growth is

assumed to be a strain potential function and the threshold

values for each damage phase are experimentally deter-

mined. High-velocity and low-velocity impact simulations

were performed in order to assess the performance of the

model. Also, the results were compared with the model

proposed by Matzenmiller et al. [39] and the existing Chang-

Chang failure criteria. Pinho et al. [42] proposed a three-

dimensional failure model to predict failure in composites.

Their model is an extension of the plane-stress failure model

proposed by Davila and Camanho [43] and it accounts for

shear nonlinearities effects. The model was implemented

into LS-DYNA3D finite element code and the authors

obtained good correlation between numerical predictions

and experimental results for static standard in-plane tests.

However, neither strain rate effects nor strain localization

problems associated with irregular mapped meshes were

addressed by the authors.

This paper presents a formulation for a three-

dimensional ply failure model for composite laminates.

The proposed model is an extension of the authors previous

work [44]. The new version of the model incorporates

strain rate effects in shear by means of a semiempirical

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4 International Journal of Aerospace Engineering

viscoplastic constitutive law and combines a quadratic stress

based criterion with the Mohr-Coulomb failure criterion

to predict interfibre-failure (IFF) without knowing a priori

the orientation of the fracture plane. The new version of

the model also uses the Second Piola-Kirchhoff stress tensor

which potentially avoids material deformation problems.

High-order damage evolution laws are proposed to avoid

both numerical instabilities and artificial stress waves

propagation effects commonly observed in the numerical

response of Finite Element Codes based on Explicit Time

Integration Schemes. These laws compared to the widely

used bilinear law ensure smoothness at damage initiation

and fully damaged stress onsets leading to a more stable

numerical response. Iterative energy based criteria have

been also added to the previous version [44] to better

predict damage under multiaxial stress states. The model

also incorporates a three dimensional objectivity algorithm

[45] which accounts for orthotropic and crack directionality

effects, making the model to be completely free of strain

localization and mesh dependence problems.

2.Formulation

This section presents the failure model formulation. The

formulation is based on the Continuum Damage Mechanics

(CDM) approach and enables the control of the energy

dissipation associated with each failure mode regardless of

mesh refinement and fracture plane orientation by using a

smeared cracking formulation. Internal thermodynamically

irreversible damage variables were defined in order to

quantify damage concentration associated with each possible

failure mode and predict the gradual stiffness reduction

during the fracture process. The material model has been

implemented into ABAQUS explicit finite element code

within brick elements as an user defined material model.

2.1. 3-D Orthotropic Stress-Strain Relationship. The ortho-

tropic material law that relates second Piola-Kirchhoff stress

S to the Green-St. Venant strain e in the local material

coordinate system is written as

S = TtCTe,

(1)

where e = [e11,e22,e33,e12,e23,e31] is the Green-St. Venant

strain vector and T is the transformation matrix given by

T =

⎡

⎢

⎢

⎢

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

k2

l2

l20 0 02kl

k20 0 0

−2kl

00 0 1 0 0

0 0 0 k −l

0 0 l

0

0

k

0

−kl kl 0 0 0 k2−l2

⎤

⎥

⎥

⎥

⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

, (2)

where k = cos(β) and l = sin(β). β is the inplane rotation

angle around the Z-direction which defines the orientation

of the material axes with respect to the reference coordinate

system. The compliance matrix C−1is defined in terms of the

material axes as

C−1=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎢

⎢

⎢

⎢

1

E11

−ν12

E11

−ν13

E11

0

−ν21

E22

1

E22

−ν23

E22

0

−ν31

E33

−ν32

E33

1

E33

0

000

000

0

1

00

G12

0

0

1

0

000

G23

0

0

1

0000

G31

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎥

⎥

⎥

⎥

, (3)

where the subscripts denote the material axes, that is,

νij= νx?

Eii= Ex?

ix?

j

i.

(4)

Since C is symmetric,

νij

Eii

=νji

Ejj.

(5)

The Green-St. Venant strain tensor e is given by

e =1

2

?FtF −I?, (6)

where F is the deformation gradient and I is the identity

matrix. The Cauchy stress tensor σij can be determined in

termsofthesecondPiola-KirchhoffstresstensorSasfollows:

σij= J−1FSFt,(7)

where J is defined as the Jacobian determinant which is

the determinant of the deformation gradient F. The present

formulationwillpredictrealisticmaterialbehaviourforfinite

displacements and rotation as long as the strains are small.

2.2. Degraded Stresses. The degraded or damaged stresses are

defined as stresses transmitted across the damaged part of

the cross-section in a representative volume element of the

material. Based on the isotropic damage theory as originally

proposed by Kachanov [33], an orthotropic relationship

between local stresses acting on the damaged configuration

can be written in terms of the local effective stresses in the

undamaged configuration at ply level as follows:

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International Journal of Aerospace Engineering5

⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

σd

1

σd

2

σd

3

τd

12

τd

23

τd

13

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

=

⎡

⎢

⎢

⎢

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

(1 −d1(?1))

0

00000

?

1 −dtm

?

?t

m

??

0000

001000

000

?

1 −dtm

?

?t

m

???1 −d12

0

?γ12

??

00

000

?

1 − dtm

?

?t

m

??

0

000001

⎤

⎥

⎥

⎥

⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

σ1

σdcm

2

σdcm

3

τdcm

12

τdcm

23

τdcm

13

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

,(8)

where d1, dtm, and d12 are internal variables introduced to

quantifythedamageconcentrationwithintheRepresentative

Volume Element (RVE) [46, 47]. A detailed description on

the physical meaning and definition of these variables will

be given in the following sections. The stress components

with the superscript dc

the action fracture plane. Details about the determination of

the local action plane and the local degradation procedure is

given in Section 2.4.

Theproposed orthotropic

degraded and intact stresses ensures that the material

stiffness matrix is positive defined during the degradation

process. Moreover, it is physically based. By inspecting

carefully (8) one can notice that transverse compression

damage dc

σ3, τ23 and τ13. From the physical point of view, this

relationship incorporates the the main features observed in

the experiments, such as follows:

mare the stresses degraded locally on

relationshipbetween

m degrades the through-the-thickness stresses

(i) It accounts for fibre damage effects due to tensile

and compression loadings by means of the internal

variable d1.

(ii) It provides the coupling between shear induced

damage and matrix cracking during the degradation

process using the internal damage variables d12, dtm

and dc

(iii) It accounts for damage coupling between the normal

stress σ2, out-of-plane shear stress τ23 (transverse

shear cracking, which leads to delaminations) and

inplane shear stress τ12 for matrix cracking predic-

tions.

m.

2.3. Fibre Failure. The failure index to detect fibre failure in

tension is given by

Ft

1(σ1) =σ1

Xt

≥ 1.

(9)

In order to detect the catastrophic failure in compression

related to the total instability of the fibres, the maximum

stress criteria is used to detect damage initiation:

Fc

1(σ1) =|σ1|

Xc

≥ 1,

(10)

where Xt and Xc are the longitudinal strengths in tension

and compression, respectively. When one of criteria given

aboveismet,damagecommencesandgrowsaccordingtothe

damage evolution law proposed by Donadon et al. [44]:

??+

where dt

irreversible damagedueto fibrebreakagein tension andfibre

kinking in compression, respectively:

d1(?1) = dt

1(?+

11

?+dc

1(?−

1

??−

1

?−dt

1

??+

1

?dc

1

??−

1

?, (11)

1) and dc

1) are the contributions of the

dt

1

??+

??−

1

?= 1 −?t

?= 1 −?c

1,0

?+

1

?

?

1 +κ2

1,t

??+

??−

1

??2κ1,t

??2κ1,c

??+

??−

1

?−3??

?−3??

,

dc

1

1

1,0

?−

1

1 +κ2

1,c

11

(12)

with

κ1,t

??+

??−

1

?=

?=

?+

?t

1− ?t

1,f− ?t

?−

?c

1,0

1,0,

κ1,c

1

1− ?c

1,f− ?c

1,0

1,0,

(13)

where ?t

compression, respectively. ?+

max(|?1(t)|,?c

strain time history in tension (σ1 > 0) and compression

(σ1 < 0), respectively. ?t

tension and compression which are written as a function

of the tensile fibre breakage and compression fibre kinking

fracture toughnesses, respectively, as follows,

1,0and ?c

1,0are the failure strains in tension and

1 = max(?1(t),?t

1,0) are the maximum achieved strains in the

1,0) and ?−

1 =

1,fand ?c

1,fare the final strains in

?t

1,f=2Gt

fibre

Xtl∗,

?c

1,f=2Gc

fibre

Xcl∗

(14)

where Gt

nesses associated with fibre breakage in tension and com-

pression, respectively and l∗is characteristic element related

to the size of the process zone. Experimental procedures

and data reduction schemes to characterise the intralaminar

fracture toughness for composites can be found in [48].

The proposed damage evolution law d1(?1) results in a

thermodynamically consistent degradation procedure, reg-

ularizing damage and combining stress, damage mechanics

and fracture mechanics based approaches within an unified

way similarly to the approach proposed by Miami et al. [49]

fibreand Gc

fibreare the intralaminar fracture tough-

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6 International Journal of Aerospace Engineering

2.4. Interfibre Failure (IFF). The interfibre failure modes

consist of transverse matrix cracking either in tension or

compression. Based on the Worldwide Failure Exercise

experimental results [50] Pinho et al. [42, 51] found that

the failure envelope defined between the transverse stress

σ2and in-plane shear stress τ12is accurately described by a

quadratic interaction criteria. Thus, a failure index based on

theinteractivequadraticfailurecriteriongivenin[42,51]has

been used to predict tensile transverse matrix cracking. For

tensilematrixcrackingafailureindexbasedonaninteractive

quadratic failure criterion written in terms of tensile and

shear stresses is proposed in the following form:

Ft

2(σ2,τ23,τ12) =

?σ2

Yt

?2

+

?τ23

S23

?2

+

?τ12

S12

?2

≥ 1.

(15)

Once the criterion above is met, the proposed expression

for damage growth due to tensile matrix cracking is given by

dt

m

?

?t

m

?

= 1 −?t

m,0

?t

m

?

1+κ2

m,t

?

?t

m

??

2κm,t

?

?t

m

?

−3

??

(16)

with

κm,t

?

?t

m

?

=

?t

?t

m− ?t

m,f− ?t

m,0

m,0,(17)

where ?t

mis the defined as resultant strain which is given by,

?

?t

m=

?2

2+γ2

s,m

(18)

with

γs,m=

?

γ2

12+γ2

23,

(19)

?t

stress, respectively, that is,

m,0and σt

m,0are the damage onset resultant strain and

?t

m,0= ?t

m

???Ft

2=1,

σt

m,0= σt

m

??Ft

2=1.

(20)

In order to account for damage irreversibility effects

m= max(?t

maximumachievedresultantstraininthestraintimehistory.

The derivation of the resultant failure strain ?t

with tensile/shear matrix cracking is based on a power law

criterion, which accounts for interactions between energies

per unit of volume of damaged material within the RVE

subjectedtotensileandshearloadings.Thepowerlawenergy

criterion is given in the following form,

?t

m(t),?t

m,0) must be used in (16), where ?t

mis the

m,fassociated

?gtm

gtmc

?λ

+

?gs

gsmc

m

?λ

= 1

(21)

with λ = 1 for UD laminates. The resultant stress in the

transverse direction (or matrix direction) due to combined

tensile and shear loadings is given by

?

σt

m=

σ2

2+τ2

12+τ2

23=

?

σ2

2+?τt

m

?2.

(22)

The tensile and resultant shear stress components can be

written in terms of the resultant stress as follows:

σt

2= σt

τt= σt

mcos(θ),

msin(θ),

(23)

where θ is the angle defined between the resultant stress

and tensile normal stress in the transverse direction, that is,

θ = acos[max(0,σt

resultant shear strain components are given as follows:

2)/σt

m]. In a similar way the tensile and

?t

2= ?t

s,m= ?t

mcos(θ),

γt

msin(θ).

(24)

For the damage evolution law given by (16) the specific

fractureenergiesassociatedwithtensileandshearstressesare

respectively given by

gt

m=

??f

?γf

2

0σt

2d?2=

σt

m,0?t

m,fcos2(θ)

2

,

gs

m=

s,m

0

τtdγ =

σt

m,0?t

m,fsin2(θ)

2

,

(25)

where the area under the stress-strain curves defined by the

proposed polynomial damage evolution laws is identical to

the one defined by the widely used bilinear softening law

given in [42, 44]. Substituting (25) into (21) we obtain the

following expression for the final strain due to the combined

tensile and shear stress state,

?t

m,f=

2

σt

m,0

⎡

⎣

?cos2(θ)

gtmc

?λ

+

?sin2(θ)

gsmc

?λ⎤

⎦

−1/λ

(26)

where gtmcand gs

Theseenergiesarerelatedtotheintralaminarfracturetough-

nesses. By using a smeared cracking formulation [52] and

assuming that for UD laminates the values of intralaminar

toughnesses associated with tensile matrix cracking and

shearmatrixcrackingarecomparablewithmodeIandmode

II interlaminar fracture toughnesses, a relationship between

specific critical fracture energies and intralaminar fracture

toughnesses can be written as follows:

mcare the critical specific fracture energies.

gt

mc=GIc

l∗,

gs

mc=GIIc

l∗,

(27)

where l∗is the characteristic length associated with the

length of the process zone for each particular failure

mode. A detailed description about the characteristic length

calculation will be presented in the following section.

The failure index to detect matrix cracking in compres-

sion failure is based on the criterion proposed by Puck and

Sch¨ urmann [53, 54]. Their criterion is based on the Mohr-

Coulomb theory and it enables the prediction of fracture

planes for any given stress state related to Interfibre-Failure

Page 7

International Journal of Aerospace Engineering7

(IFF) Modes. This criterion is currently the state of the art

to predict transverse compression response of composite

laminates. The failure index to detect matrix cracking in

compression based on the failure criterion proposed by Puck

and Sh¨ urmann [53, 54] can be written as follows,

?

SA

Fc

2(σnt,σnl) =

σnt

23+μntσnn

?2

+

?

σnl

S12+μnlσnn

?2

≥ 1,

(28)

here the subscripts n, l and t refer to the normal and tangen-

tial directions in respect to the fracture plane direction. S12

is the inplane shear strength and SA

strengthinthepotentialfractureplane(ActionPlane),which

is given by [55],

?1 −sin?φ?

23is the transverse shear

SA

23=Yc

2 cos?φ?

?

(29)

with

φ = 2θf−900,

(30)

whereYcisthetransversecompressionstrength.Thefracture

angle can be determined either experimentally or alterna-

tively using (30), where θf maximises the failure criterion.

Following the Mohr-Coulomb failure theory the friction

coefficients can be determined as a function of the material

friction angle as follows:

μnt= tanφ = tan

?

2θf−900?

.

(31)

In the absence of experimental values an orthotropic

relationship for the friction coefficients can be used [54],

μnt

SA

23

=μnl

S12.

(32)

The stress components acting on the potential fracture

plane are written in terms of angle θf which defines the

orientation of the fracture plane in respect to through-

the-thickness direction (direction-3 in the local material

coordinate system):

?

σnt

θf

= −σ2mn+σ3mn+τ23

?

σnl

θf

= τ12m+τ13n,

?

where m = cos(θf) and n = sin(θf). θfis the rotation around

the local fibre direction (direction-1 in the local material

coordinate system). It is clear that in order to apply (28)

θf must be known. Many authors have defined θf ≈ 530

based on experimental results for standard compression tests

in UD laminates [42, 44, 51]. This is true and has been

σnn

θf

?

= σ2m2+σ3

?1 −m2?+2τ23mn,

?2m2−1?,

?1 −m2?+σ3m2−2mnτ23,

?

θf

= −τ12n+τ13m,

??

σtt

θf

?

= σ2

?

σlt

?

(33)

also confirmed by Puck and Schurmann [53, 54]. However,

θf ≈ 530is only valid for uniaxial compression loading.

This implies that θf changes for different stress states and

alternatives are needed in order to handle such a problem.

By examining (28) it is possible to see that the criterion has

also the potential of predicting transverse intralaminar shear

cracking for values of θfdifferent from 530. The transverse

intralaminar shear cracking is a very important failure mode

because it leads to delamination between adjacent layers. In

ordertacklethisproblemwehaveusedaniterativeprocedure

to compute the fracture plane orientation for a given stress

state. The procedure consists of incrementally varying θf

within the interval [−900≤ θf ≤ 900] for a given stress

state defined at ply level and check if the failure index for

matrix cracking in compression being reached. Once the

failure index is reached the local shear components σnland

σnt acting on the candidate fracture plane are degraded to

zero according to the following damage evolution law:

dc

m

??c

m

?= 1 −?c

m,0

?c

m

?

1+κ2

m,c

??c

m

??2κm,c

??c

m

?−3??

(34)

with

κm,c

??c

m

?=

?c

?c

m− ?c

m,f− ?c

m,0

m,0,(35)

where ?c

which is defined as

mis the resultant shear strain on the action plane

?c

m=

?

?

?2

nl+ ?2

nt,

σc

m=

σ2

nl+σ2

nt

(36)

with

?nl

?

θf

?

= γ12m+γ13n,

?nt

?

θf

?

= −?2mn+ ?3mn+γ23

?2m2−1?,

(37)

where ?c

stress, respectively, that is,

m,0andσc

m,0arethedamageonsetresultantstrainand

?c

m,0= ?c

σc

m

??Fc

2=1,

m,0= σc

m

??Fc

2=1.

(38)

The resultant final strain for transverse compression

failure is defined in terms of mode II interlaminar fracture

toughness as follows:

?c

m,f=2Gc

matrix

m,0l∗

σc

=2GIIc

σc

m,0l∗.

(39)

In order to account for damage irreversibility effects

m= max(?c

maximumachievedresultantstrainontheactionplaneinthe

strain time history. After degrading the shear stresses acting

on the potential fracture angle, the stresses are rotated back

?c

m(t),?c

m,0) must be used in (34), where ?c

mis the

Page 8

8 International Journal of Aerospace Engineering

to the local material coordinate system using the following

transformation:

?1 −dc

σdcm

3

= σnn

τdcm

m

σdcm

2

= m2σnn−2mnσnt

?1 − m2?+2mnσnt

12= mσnl

13= mnσnn+?2m2−1?σnt

τdcm

m

?1 −dc

??c

?1 − dc

??c

??c

m

??+σtt

m

??−nσlt,

m

??+mσlt.

?1 −m2?,

??+σttm2,

??c

m

?1 −dc

m

τdcm

??c

m

??−nmσtt,

23= nσnl

?1 −dc

m

m

(40)

2.5. In-Plane Shear Failure. The observed behaviour of glass

and carbon fibres laminates generally shows marked rate

dependence in matrix-dominated shear failure modes and

for this reason a rate dependent constitutive model has been

used to model the in-plane shear behaviour. The constitutive

model formulation is based on previous work carried out

by Donadon et al. [44, 56] and it accounts for shear

nonlineatities, irreversible strains and damage within the

RVE. The stress-strain behaviour for in-plane shear failure

is defined as follows,

τ12= αG12γ12

(41)

with

G12= G0

12+c1(e−c2γ12−1),

(42)

where G0

constants obtained from static/quasistatic in-plane shear

tests. α is the strain-rate enhancement given by the following

law,

12is the initial shear modulus and c1, c2are material

α = 1+e(˙ γ12/c3)

(43)

wherec3isanothermaterialconstantobtainedfromdynamic

in-plane shear tests. By decomposing the total shear-strain

into inelastic γi

shear-strain can be written in terms of the elastic and total

strain components as follows:

12and γe

12elastic components, the inelastic

γi

12= γ12−γe

12= γ12−τ12

?γ12

G0

12

?

.

(44)

Thefailureindexforin-planeshearfailureisbasedonthe

maximum stress criterion and it is given by:

F12(τ12) =|τ12|

S12

≥ 1.

(45)

The proposed damage evolution law for in-plane shear

failure is given by

d12

?γ12

?= 1 −γ12,0− γi

12,0

γ12− γi

12,0

?1+κ2

12

?γ12

??2κ12

?γ12

?−3??

(46)

with

κ12

?γ12

?=γ12−γ12,0−2γi

12,0

γ12,0−γi

12,0− γ12,f, (47)

where γ12,0 and γi

strain at failure (τ12= S12), respectively, that is,

12,0are the total strain and total inelastic

γ12,0= γ12

??S12,

γi

12,0= γi

12

???S12,

(48)

γ12,f is written in terms of the intralaminar toughness in

shear:

γ12,f =2Gshear

S12l∗.

(49)

In the absence of experimental results for Gshear, is

reasonable to assume Gshear= GIIcfor UD plies, where GIIcis

the mode II interlaminar fracture toughness.

2.6.

described in the previous sections relates the specific

energy within a Representative Volume Element (RVE)

with the fracture energy of the material for each particular

failure mode. Since finite elements are volume based, mesh

dependency problems will arise as a result of the mesh

refinement. The correction of the postfailure softening slope

according to the finite element size, as reported by Bazant

[52] seems an attractive solution for the problem. However

the approach has some limitations. Firstly, the crack growth

direction must be parallel to one edge of the finite element,

which is not the case for multidirectional composite

laminates where layers can have arbitrary directions.

Secondly, it cannot handle nonstructured meshes required

in most of the complex finite element models with geometric

discontinuities. In order to overcome such limitations and to

ensuretheobjectivityofthemodelforgeneralizedsituations,

a methodology originally developed by Oliver [57] has been

used and extended to handle composite layers [58]. The

dependence of the characteristic length on the fracture

energy as well as its mathematical expression, were derived

based in the work proposed by Oliver [57]. The method

ensures a constant energy dissipation regardless of mesh

refinement, crack growth direction and element topology so

that, it is still applicable to nonstructured meshes.

Objectivity Algorithm. Thesmearedformulation

2.6.1. Crack Modelling in the Continuous Medium. Imagine a

singular line in a two dimensional domain as a continuous

material line, across which displacements are continuous but

displacement gradients are discontinuous. The condition for

apointbelonging toasingularlinewithunitnormal natthis

point is that the determinant of the acoustic tensor in the n

direction be zero, that is [57],

?

Nonpositivematerialsbifurcate,producingsingularlinesand

the equation above permits their direction to be determined

at each point. In the context of standard finite elements of

C0continuity, a singular line can be modelled only by the

sides of the elements, these being the only points in the mesh

wheredisplacementgradientdiscontinuitiescanbeobtained.

det

niCijklnl

?

= 0.

(50)

Page 9

International Journal of Aerospace Engineering9

However, a crack produces not only displacement gradient

discontinuities but also displacement discontinuities. This

latter kind of discontinuity cannot be modelled by a C0finite

element mesh for finite levels of discretization. However, a

displacement discontinuity can be modeled as the limit of

two parallel singular lines Γ−and Γ+which tend to coincide

with each other. The band delimited by these lines is known

as singular band, and h is its width.

By assuming an orthogonal curvilinear coordinate sys-

tem (x?, y?) in the interior of the band, where y?coordinates

lines are parallel to the singular lines Γ−and Γ+, and x?are

the straight coordinates lines. Let u+(y?) and u−(y?) be the

displacement vectors on Γ+and Γ−the relative displacement

vector can be written as,

ω?y??= u+?y??− u−?y??

as a vector representing the displacement “jump” between

the two singular lines and

(51)

δ?y??= lim

h?→0ω?y??= lim

h?→0

?u+?y??− u−?y???,

(52)

If δ/ =0, the singular band is modelling a discontinuous

displacement field as the limit of a continuous one. This

allows a crack to be idealised as a limit (with mesh

refinement) of a band of finite elements where, by means of

some numerical mechanisms, the condition δ/ =0 is satisfied.

2.6.2. Displacement and Traction Vectors in the Singular Band.

Consider a singular band in the solid, with a width h

according to Figure 1.

Along a coordinate line x?, the displacement vector u can

be expanded from its value in the Γ−line using Taylor’s series

as

u?x?, y??= u−?y??+

and consenquently, for a point in the Γ+line

?∂u

∂x?

?−

Δx?+O?h2?,

(53)

u+?y??= u−?y??+

From the (51), (53), (54) we can write

?∂u

∂x?

?−

h?y??+O?h2?.

(54)

u?x?, y??= u−?y??+

u?x?, y??= u−?y??+Δx?

∼= u−?y??+φ?x?, y??ω?y??,

where φ is a function to be determined, which approximates

Δx?/h when h ?→ 0.

From (51) and (56) it can be seen that

?∂u

∂x?

hω?y??+0?h2?

?−

Δx?+O?h2?, (55)

(56)

φ = 0 = ⇒ Γ−,

φ = 1 = ⇒ Γ+.

(57)

The equilibrium across the singular band will be enforced by

assuming the traction vector t acting on the plane defined by

the normal n:

ti= σijnj, (58)

which is constant in the x?direction that is

t?x?, y??= t+?y??= t−?y??.

2.6.3. Energy Dissipation Within the Band. For a generic

deformation process which takes place over a time τ(0 ≤ τ ≤

∞) the specific energy dissipation (energy per unit volume)

within a closed domain Ω∗(see Figure 1) is given by

?∞

Foruniaxialdeformationprocessgfwouldbe,foragiven

point, the area under the stress-strain curve at that point. By

takingthelinearizedgeometricequationsgfcanbeexpressed

as,

?∞

2

∂xi

?∞

(59)

gf =

0σij

?x?, y?,τ?d?ij

?x?, y?,τ?=

?∞

0σij˙ ?ijdτ.

(60)

gf =

0σij1

?∂˙ ui

∂xj+∂˙ uj

?

dτ

=

0σij∂˙ ui

∂xjdτ −

?∞

0σij1

2

?∂˙ ui

∂xj+∂˙ uj

∂xi

?

dτ.

(61)

The last integrand in (61) is zero, being the product of a

symmetric and antisymmetric tensor, so that

?∞

gf =

0σij∂˙ ui

∂xjdτ =

?∞

0

∂

∂xj

?

σij˙ ui

?

dτ

(62)

where the Cauchy’s equations for quasistatic processes and

negligible body forces (∂σij/∂xi = 0) have been considered.

The total dissipated energy in the domain Ω∗is

??∞

W∗=

?

Ω∗gfdΩ∗=

?

Ω∗

0

∂

∂xj

?

σij˙ ui

?

dτ

?

dΩ∗.

(63)

By applying the Gauss’s theorem to (63) and using (58)

we obtain,

?

Γ∗

Owing to the infitesimal width of the band, the curvilin-

earintegralin(64)canbeevaluatedonlyonthelinesΓ∗+and

Γ∗−(see Figure 1):

?

Γ∗+∪Γ∗−

and taking into account (56) and (59) we obtain,

?

Γ∗+∪Γ∗−

?

Γ∗+∪Γ∗−

W∗=

??∞

0σijnj˙ uidτ

?

dΓ∗=

?

Γ∗

??∞

0ti˙ uidτ

?

dΓ∗(64)

W∗=

??∞

0ti˙ uidτ

?

dy?,

(65)

W∗=

??∞

??∞

0t−

i

?y?,τ?˙ u−

0t−

i

?y?,τ?dτ

?

dy?

+

i

?y?,τ?φ?x?, y??˙ ω?y?,τ?dτ

?

dy?.

(66)

Page 10

10 International Journal of Aerospace Engineering

Γ−

y’

h

Γ+

n

x’

(a)

Γ−

Γ∗−

y’

B

Γ+

Ω∗

Γ∗+

x’

A

Ω

Γ∗

(b)

Figure 1: Analysis within the singular band.

The first integral vanishes because the contributions on

Γ∗+and Γ∗−cancel each other out. Thus, the dissipated

energy on Ω∗is

?

Γ∗+∪Γ∗−

W∗=

?

φ?x?, y???∞

0t−

i

?y?,τ?˙ ω?y?,τ?dτ

?

dy?.

(67)

Now, if the case where Ω∗= Ω is considered; that is,

the whole band between points A and B in Figure 1, the total

energy dissipated within the band between points A and B is

?

A

W =

ΩgfdΩ =

?y?

B

y?

??∞

0t−

i

?y?,τ?˙ ω?y?,τ?dτ

?

dy?.

(68)

Equation (68) establishes that the energy dissipated

within the idealized band can be written as a curvilinear

integralalongitslength.Theintegrand of (68)representsthe

energy dissipated per unit of area, which in terms of fracture

mechanics, is the fracture energy Gf:

?∞

If Gf is assumed to be a material property independent

of the spatial position of the point from (67) and (69) we

obtain,

?

By applying Green’s theorem and taking into account

(60), we can write the energy dissipated within the band as,

?

Ω∗

The local form of (71) is

Gf

?y??=

0t−

i

?y?,τ?˙ ω?y?,τ?dτ.

(69)

W∗=

Γ∗+∪Γ∗−Gfφ?x?, y??dy?= Gf

?

Γ∗φ?x?, y??dy?. (70)

W∗= Gf

∂φ

∂x?dΩ∗=

?

Ω∗gfdΩ∗.

(71)

gf = Gf

∂φ

∂x?=Gf

l∗,(72)

where

l∗?x?, y??=

?∂φ

∂x?

?−1

.

(73)

The parameter l∗plays the role of relating the specific

energy(perunitofvolume)andthefractureenergy(perunit

of area). l∗is also identified as the characteristic length or

crack band width used in existing cracking models [52]. For

the unidimensional case and using the proposed Hermitian

stress-strain softening law, the specific energy can be written

as

gf =σ0?f

2

, (74)

where σ0is the material strength and the degraded stress is

given by

σd= σ(1 −d(?)),

(75)

where the damage evolution law d(?) is given in terms of the

strain as follows:

d(?) = 1 −?0

?

?1+κ2(?)(2κ(?) − 3)?,

(76)

with

κ(?) =

? − ?0

?f− ?0.

(77)

Using (74) the failure strain ?fcan be written in terms of

the fracture energy and the characteristic length as it follows:

?f=2Gf

σ0l∗.

(78)

Page 11

International Journal of Aerospace Engineering 11

ζ

Node 8

(ϕ1,ζ) = (1,0)

η

y’

Node 5

Node 1

x’, n

ξ

Node 2

Node 3

Node 6

θj

(ϕ2,ζ) = (0,0)

(ϕ3,ζ) = (0,0)

(ϕ4,ζ) = (1,0)

Node 7

Figure 2: Determination of the characteristic length for hexahe-

dron elements.

2.6.4. Determination of the Function φ and the Characteristic

Length l∗. In order to apply the theory presented in the

previous section to the discretized medium, consider a mesh

of C0continuous hexahedron solid finite elements (see

Figure 2).

A set of cracked elements is determined by using

failure criteria for detecting the crack initiation, the crack

orientationdependsonthefibredirection.Thecrackedplane

is defined here as a normal vector which is parallel to the

fibres for fibre failure and normal to the fibre direction for

matrix failure. The algorithm described in this section for

determining the characteristic length was proposed by Oliver

[57]andithastheadvantagesofcalculatingthecharacteristic

length for arbitrary crack directions and any finite element

geometry.

From (71) the function φ has to be continuous and

derivable, satisfying (57). A simple function defined in

the isoparametric coordinates ξ and η which fulfils these

requirements is

φ?ξ,η?=

nc ?

i=1

Ni

?ξ,η?φi,

(79)

where nc is the number of corner nodes of a virtual plane

located at the midplane of the element (nc= 4 for our case),

Niare the standard C0shape functions of an element of nc

virtual nodes in its midplane and φiis the value of the φ at

corner i. If the crack location inside the element is known, φi

takes the value +1 if the corner node i is ahead the crack, and

0 otherwise. The function defined by (79) fulfils the required

condition of continuity within elements and takes the values

+1 for the nodes on the boundaries ahead the crack and 0 for

the nodes on the boundaries behind the crack (see Figure 3).

In general, however, the exact crack location is not

known, and usually only some indication of the onset

of cracking and the crack directions is obtained at the

η

Φ2= 1

Φ1= 1

ξ

n

Φ3= 0

Φ4= 1

Γ(e)

Figure 3: Finite element band modelling.

integration points. The following algorithm proposed by

Oliver [57] has been used for determining the characteristic

length at each integration point j, as shown in Figure 4,

(1) A set of local cartesian axes x?, y?is defined at the

centre of the element, this being identified by the values of

the isoparametric coordinates (ξ = 0, η = 0 and ζ = 0).

The direction of the local axis x?is defined by the normal to

the fracture plane, which is the fibre angle for fibre failure

(θj= θf) and (θj= θf+900) for matrix failure.

⎧

⎩

(2) Values of φ at each corner node are established

according to their position with respect to the local axis x?,

y?(φi= 1 if x?

(3) The characteristic length, at the present integration

point j with isoparametric coordinates ξj and ηj and

cracking angle θjis obtained as

⎨

x?

y?

i

i

⎫

⎬

⎭=

⎡

⎣

⎢

cos

?

θj

?

?

sin

?

?

θj

?

?

−sin

θj

?

cos

θj

⎤

⎦

⎥

⎧

⎩

⎨

xi

yi

⎫

⎭

⎬

(80)

i≥ 0, otherwise φi= 0).

l∗?

ξj,ηj

?

=

?∂φ(ξj,ηj)

∂x?

⎛

i=1

?−1

=

⎝

nc ?

⎡

⎣∂Ni

+∂Ni

?

ξj,ηj

∂x

?

cos

?

θj

?

?⎤

?

ξj,ηj

∂y

?

sin

?

θj

⎦φi

⎞

⎠

−1

(81)

where

⎧

⎪⎪⎪⎩

⎪⎪⎪⎨

∂N

∂x

∂N

∂y

⎫

⎪⎪⎪⎭

⎪⎪⎪⎬

= J−1

xy

⎧

⎪⎪⎪⎩

⎪⎪⎪⎨

∂N

∂ξ

∂N

∂η

⎫

⎪⎪⎪⎭

⎪⎪⎪⎬

, (82)

and Jxyis defined as the Jacobian matrix given by

Jxy=

⎡

⎢

⎢

⎢

⎣

∂x

∂ξ

∂x

∂η

∂y

∂ξ

∂y

∂η

⎤

⎥

⎥

⎥

⎦, (83)

Page 12

12 International Journal of Aerospace Engineering

η

y’

Φ2= 0

Φ1= 1

ξ

x’, n

Φ3= 0

Φ4= 1

θj

Figure 4: Computation of φ values at the virtual midplane of the

element.

where the partial derivatives with respect to the isoparamet-

ric coordinates are written as

∂x

∂ξ=1

4

?1 +η?x1−1

4(1+ξ)x1+1

4

?1 +η?x2−1

4(1 −ξ)x2−1

4

?1 −η?x3+1

4(1 −ξ)x3−1

4

?1 −η?x4,

∂x

∂η=1

4(1+ξ)x4,

∂y

∂ξ=1

4

?1 +η?y1−1

4(1+ξ)y1+1

4

?1+η?y2−1

4(1 − ξ)y2−1

4

?1 −η?y3+1

4(1 −ξ)y3−1

4

?1 −η?y4,

∂y

∂η=1

4(1+ξ)y4,

(84)

where the pairs (xi, yi) refers to the global coordinates of the

virtual nodes defining the midplane of the element.

For transverse compression failure a set of cracked

elements is determined by using the stress based criterion

defined by (28) and the cracked plane is defined by the

fracture angle θf. The function φ is given b

φ(ξ,ζ) =

nc ?

i=1

Ni(ξ,ζ)φi,

(85)

where nc is the number of corner nodes of a virtual plane

located at the midplane of the element according to Figure 5,

Niare the linear shape functions defined previously and nc

virtual nodes defining the virtual cracking midplane and φi

is the value of the φ at corner i.

x?, z?is an auxiliar coordinate system defined at the

centre of the element, this being identified by the values of

the isoparametric coordinates (ξ = 0, η = 0 and ζ = 0) with

the direction of the local axis x?is defined by the normal to

the fracture plane,

⎡

⎣

The values of φ at each corner node are established

accordingtotheirposition withrespecttothelocalaxisx?, z?

⎧

⎩

⎨

x?

z?

i

i

⎫

⎬

⎭=

⎢

cos

?

θj

?

?

sin

?

?

θj

?

?

−sin

θj

?

cos

θj

⎤

⎦

⎥

⎧

⎩

⎨

xi

zi

⎫

⎬

⎭.

(86)

(φi = 1 if x?

one described previously.

The characteristic length associated with transverse com-

pressionatthepresentintegrationpoint j withisoparametric

coordinates ξjand ζjand fracture angle θf is given by

i≥ 0, otherwise φi = 0) in a similar way as the

l∗?

ξj,ζj

?

=

?∂φ(ξj,ζj)

∂x?

⎛

i=1

?−1

=

⎝

nc ?

⎡

⎣∂Ni

+∂Ni(ξj,ζj)

?

∂x

ξj,ζj

?

cos

?

θf

?

?

∂z

sin(θf)

φi

?−1

(87)

where

⎧

⎪⎪⎩

⎪⎪⎨

∂N

∂x

∂N

∂z

⎫

⎪⎪⎭

⎪⎪⎬

= J−1

xz

⎧

⎪⎪⎪⎩

⎪⎪⎪⎨

∂N

∂ξ

∂N

∂ζ

⎫

⎪⎪⎪⎭

⎪⎪⎪⎬

, (88)

and Jxzis defined as the Jacobian matrix given by

Jxz=

⎡

⎢

⎢

⎣

∂x

∂ξ

∂x

∂ζ

∂z

∂ξ

∂z

∂ζ

⎤

⎥

⎥

⎦,(89)

where the partial derivatives with respect to the isoparamet-

ric coordinates are written as

∂x

∂ξ=1

4(1+ζ)x1−1

4(1+ζ)x2−1

4(1 −ζ)x3+1

4(1 −ζ)x4,

∂x

∂ζ

=1

4(1+ξ)x1+1

4(1 −ξ)x2−1

4(1 −ξ)x3−1

4(1+ξ)x4,

∂z

∂ξ=1

4(1+ζ)z1−1

4(1+ζ)z2−1

4(1 − ζ)z3+1

4(1 −ζ)z4,

∂z

∂ζ=1

4(1+ξ)z1+1

4(1 −ξ)z2−1

4(1 −ξ)z3−1

4(1+ξ)z4,

(90)

where the pairs (xi,zi) refer to the global coordinates of

the virtual nodes defining the midplane of the element.

In-plane shear cracking is strongly dependent on the fibre

orientarion within the element therefore, the characteristic

length associated with in-plane shear failure has been

assumed to be the same as the one defined for fibre failure or

failure in the warp direction. For out-of-plane shear failure

modes, cracks are assumed to be smeared over the thickness

of the element with a crack band defined between upper

and lower faces of the element which is equivalent to assume

θf = 900in (87),

?∂φ(ξj,ζj)

∂x?

l∗?

ξj,ζj

?

=

?−1

=

⎛

⎝

nc ?

i=1

?∂Ni(ξj,ζj)

∂z

?

φi

⎞

⎠

−1

.

(91)

Page 13

International Journal of Aerospace Engineering 13

ξ

Node 8

(ϕ1,η) = (1,0)

η

z’

Node 1

x’, n

ζ

Node 2

Node 3

Node 6

θf

Node 5

(ϕ2,η) = (0,0)

(ϕ3,η) = (0,0)

(ϕ4,η) = (1,0)

Node 7

Figure 5: Computation of the characteristic length for transverse

compression.

3.NumericalImplementation

This section presents details about the numerical imple-

mentation of the proposed failure model into ABAQUS

FE code. The code formulation is based on the updated

Lagrangian formulation which is used in conjunction with

thecentraldifferencetimeintegrationschemeforintegrating

the resultant set of nonlinear dynamic equations. The

method assumes a linear interpolation for velocities between

two subsequent time steps and no stiffness matrix inversions

are required during the analysis. The explicit method is

conditionally stable for nonlinear dynamic problems and the

stability for its explicit operator is based on a critical value of

the smallest time increment for a dilatational wave to cross

any element in the mesh. The numerical implementation

steps are listed as follows:

(1) Stresses and strain-rates are transformed from the

element axes to material axes:

?

˙ ?n+1

l

σd

l

?n

= T

= T ˙ ?n+1

?

σd

g

?n,

,

g

(92)

where the indices l and g refer to the material (local) and

element(global)coordinatesystems,respectivelyandT isthe

transformationmatrixdefinedby (2).Thesuperscriptsnand

n+1 refer to the previous and current time, respectively:

(2) Compute constitutive matrix C from (3) for trial

stresses.

(3) Based on the current time step compute strain

increments and update the elastic stresses and strains

σn+1= σn+CΔ?,

?n+1= ?n+Δ?.

(4) Compute failure index for fibre failure (σn+1

?σn+1

Store (?t

1

) = 1.

(93)

1

> 0):

Ft

11

?=σn+1

1

Xt

≥ 1.

(94)

1,0)n+1when Ft

1(σn+1

(5) Compute failure index for fibre kinking (σn+1

1

< 0):

Fc

1

?σn+1

1(σn+1

1

?=

???σn+1

) = 1.

1

Xc

???

≥ 1.

(95)

Store (?c

(6) Compute IFF index for tensile/shear matrix cracking

(σn+1

2

> 0):

?σn+1

Yt

1,0)n+1when Fc

1

Ft

2

?σn+1

2

,τn+1

23,τn+1

12

?=

2

?2

+

?τn+1

S23

23

?2

+

?τn+1

S12

12

?2

≥ 1.

(96)

Store (?t

(7) Compute IFF index for matrix crushing and

intralaminar shear cracking ( σn+1

?

SA

nn

m,0)n+1and (σt

m,0)n+1when Ft

2(σn+1

2

,τn+1

23,τn+1

12) = 1.

2

< 0):

?2

Fc

2

?

σn+1

nt ,σn+1

nl

?

=

σn+1

nt

23+μntσn+1

+

?

σn+1

nl

S12+μnlσn+1

nn

?2

≥ 1.

(97)

Store (?c

1.

(8) Compute failure index for inplane shear cracking:

m,0)n+1, (σc

m,0)n+1and (θf)n+1when Fc

2(σn+1

nt ,σn+1

nl) =

F12

?τn+1

12

?=

???τn+1

12

S12

???

≥ 1.

(98)

Store (γ12,0)n+1, (γi

(9) Compute characteristic lengths for each failure mode

using objectivity algorithm:

In-plane failure modes

?∂φ(ξj,ηj)

∂x?

⎛

i=1

?

∂y

12,0)n+1when F12(τn+1

12) = 1.

l∗?

ξj,ηj

?

=

?−1

=

⎝

nc ?

⎡

⎣∂Ni

+∂Ni

?

ξj,ηj

∂x

?

cos

?

θj

?

ξj,ηj

?

sin(θj)

⎤

⎦φi

⎞

⎠

−1

(99)

θj= θfibrefor fibre failure in tension and compression, θj=

θfibre+ 900for matrix cracking in tension/shear, θj = θfibre

for in-plane shear failure.

Out-of-plane failure modes

?

∂x?

l∗?

ξj,ζj

?

=

⎛

⎝∂φξj,ζj

?

⎞

⎠

−1

=

⎛

⎝

nc ?

i=1

⎡

⎣∂Ni

?

ξj,ζj

∂z

?

⎤

⎦φi

⎞

⎠

(100)

−1

,

θj = θf, where θf is the fracture plane orientation when

Fc

= 1 for matrix crushing/intralaminar shear

failure modes.

(10) Compute final strains based on intralaminar frac-

ture toughness and characteristic length associated with each

failure mode.

2(σnt,σnl)

Page 14

14 International Journal of Aerospace Engineering

Fibre failure

?t

1,f=2Gt

fibre

Xtl∗,

?c

1,f=2Gc

fibre

Xcl∗,

(101)

IFF (Matrix cracking in tension/shear)

?t

m,f=

2

σt

m,0

⎡

⎣

?cos2(θ)

gtmc

?λ

+

?sin2(θ)

gsmc

?λ⎤

⎦

−1/λ

,

(102)

IFF (Matrix cracking in compression/shear)

?c

m,f=2Gc

matrix

m,0l∗

σc

=2GIIc

σc

m,0l∗,

(103)

in-plane shear failure

γ12,f =2Gshear

S12l∗.

(104)

(11) Update damageparameters d1, dt

failure criteria are met with,

m, dc

m, d12whenthe

d1=

⎧

⎪⎪⎪⎪⎩

⎧

⎪⎪⎪⎪⎩

⎧

⎪⎪⎪⎪⎩

⎧

⎪⎪⎪⎪⎩

⎪⎪⎪⎪⎨

(d1)n+1

if (d1)n+1> (d1)n,

?

otherwise,

Ft

1

σn+1

1

?

> 1 or Fc

1

?

σn+1

1

?

> 1,

(d1)n

dt

m=

⎪⎪⎪⎪⎨

?dtm

?n+1

if?dtm

Ft

2

?n+1>?dtm

σn+1

2

,τn+1

?n,

12

?

23,τn+1

?

> 1,

?dtm

⎪⎪⎪⎪⎨

?dc

⎪⎪⎪⎪⎨

(d12)n

?n

?n+1

otherwise,

dc

m=

?dc

m

if?dc

Fc

2

m

?n+1>?dc

σn+1

m

?n,

> 1,

?

nt ,σn+1

nl

?

m

?n

otherwise,

d12=

(d12)n+1

if (d12)n+1> (d12)n,

?

otherwise.

F12

τn+1

12

?

> 1,

(105)

(12) Stress update

⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

σd

1

σd

2

σd

3

τd

12

τd

23

τd

13

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

n+1

=

⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1 −d1)σ1

?1 −dtm

σdcm

?1 −dtm

?1 −dtm

?σdcm

2

3

?(1 −d12)τdcm

?τdcm

τdcm

13

12

23

⎫

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

.

(106)

U1= Up

U1= Up

U1= Up

U1= Up

U1= 0

U1= 0

U1= 0

U1= 0

Undeformed shape

Deformed shape

Figure 6: Boundary conditions for tensile loading.

Undeformed shape

Deformed shape

U1= Up

U1= Up

U1= Up

U1= Up

U1= 0

U1= 0

U1= 0

U1= 0

Figure 7: Boundary conditions for compression loading.

(13)Stresstransformationfromthematerial(local)tothe

element coordinate system (global)

?

σd

g

?n+1

= (T)−1?

σd

l

?n+1.

(107)

(14) Compute element nodal forces, add element hour-

glass control nodal forces, transform element nodal forces

to global coordinates, solve for accelerations via equation of

motion and update velocities and displacements.

Page 15

International Journal of Aerospace Engineering 15

Undeformed shape

Deformed shape

U1= U2= U3= 0

U1= U2

U1= Up

U2= 0

U3= 0

= U3= 0

U1= Up

U2= 0

U3= 0

U1= Up

U2= 0

U3= 0

U1= Up

U2= 0

U3= 0

U1= U2= U3= 0

U1= U2= U3= 0

Figure 8: Boundary conditions for in-plane shear loading.

Loading-unloading-reloading (tension + compression)

Loading in compression only

Loading in tension only

–0.5

–0.25

–0.08 –0.040.04 0.080

0

0.25

0.5

0.75

1

Normalised stress

Strain (m/m)

Figure 9: Element loaded-unloaded-reloaded in tension and

compression in the fibre direction.

4.SingleElement Validation

Numerical simulations at element level were carried out

to validate the proposed damage model. The numerical

tests consisted of exciting each failure mode individually

and verify if damage irreversibility conditions and energy

concepts are always satisfied for each failure mode. The

material properties used in model were taken from [58] and

theyaresummarisedinTables1,2and3.Thematerialsystem

–0.08–0.04 0.040.080

Damage evolution (compression only)

Damage evolution (tension only)

Combined damage evolution (tension + compression)

Strain (m/m)

0

0.25

0.5

0.75

1

Damage

Figure 10: Damage evolution law due to combined tension and

compression loading in the fibre direction.

consistsofaquasiunidirectionalcarbonUDtapesuppliedby

EUROCARBON, which has a plain weave pattern with T700

carbon fibres in the warp direction and a small fraction of

PPG glass fibres in weft direction to hold the carbon fibres

together embedded into an infusible PRIME 20 LV epoxy

resin system.

The element was loaded under displacement control

in each direction with prescribed displacements U1

Up and boundary conditions shown in Figures 6, 7 and

8. The stress in the fibre direction was normalised with

respect to the fibre tensile strength whilst the stress in the

matrix (transverse) direction was normalised with respect

to the matrix compression strength. The shear stress was

normalised with respect to the shear strength. The structural

responses for one element loaded in tension, compres-

sion and combined tension-compression under loading-

unloading-reloading conditions in the fibre and matrix

directions are shown in Figures 9 and 11, respectively. As

damage commences the stresses are gradually reduced to

zero. The material stiffness in each direction are also reduced

as damage cummulates and during unloading the damage

irreversibility condition is fully satisfied in order to avoid

material self-healing as shown in Figures 10, 12 and 14. The

nonlinear behaviour in shear including damage combined

with plasticity effects is shown in Figure 13.

=

5.Coupon Tests Validation

In this section the predictions obtained using the proposed

failure model are compared against experimental results