A ThreeDimensional Ply Failure Model for Composite Structures
ABSTRACT 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 stressbased, 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.

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ABSTRACT: This paper presents a numerical study on the impact resistance of composite shells laminates using an energy based failure model. The damage model formulation is based on a methodology that combines stress based, continuum damage mechanics (CDM) and fracture mechanics approaches within a unified procedure by using a smeared cracking formulation. The damage model has been implemented as a userdefined material model in ABAQUS FE code within shell elements. Experimental results obtained from previous works were used to validate the damage model. Finite element models were developed in order to investigate the pressure and curvature effects on the impact response of laminated composite shells.Composite Structures  COMPOS STRUCT. 01/2010; 93(1):142152.  SourceAvailable from: Volnei Tita[Show abstract] [Hide abstract]
ABSTRACT: Recent improvements in manufacturing processes and materials properties associated with excellent mechanical characteristics and low weight have made composite materials very attractive for application on civil aircraft structures. However, even new designs are still very conservative, because the composite failure phenomenon is very complex. Several failure criteria and theories have been developed to describe the damage process and how it evolves, but the solution of the problem is still open. Moreover, modern filament winding techniques have been used to produce a wide variety of structural shapes not only cylindrical parts, but also “flat” laminates. Therefore, this work presents the development of a damage model and its application to simulate the progressive failure of flat composite laminates made using a filament winding process. The damage model was implemented as a UMAT (User Material Subroutine), in ABAQUSTM Finite Element (FE) framework. Progressive failure analyses were carried out using FE simulation in order to simulate the failure of flat filament wound composite structures under different loading conditions. In addition, experimental tests were performed in order to identify parameters related to the material model, as well as to evaluate both the potential and the limitations of the model. The difference between numerical and the average experimental results in a four point bending setup is only 1.6 % at maximum load amplitude. Another important issue is that the model parameters are not so complicated to be identified. This characteristic makes this model very attractive to be applied in an industrial environment.Applied Composite Materials 10/2013; · 1.29 Impact Factor  SourceAvailable from: Ghazi A. F. R. AbuFarsakh[Show abstract] [Hide abstract]
ABSTRACT: Variation of shear stresses xy on failure behavior of different composite materials was investigated for two types of composites; BoronEpoxy Narmco5505 and CarbonEpoxy AS4 (35016). The effect of material nonlinearity on failure behavior of unidirectional fibrous composite laminates was considered for several fiberorientation angles; ϴ=15, 30, 45, 60 and 75 degrees. An energybased nonlinear material model was adopted to predict the mechanical properties of a unidirectional composite lamina. A compatible failure criterion for nonlinear composite materials was utilized which incorporates the material model into the failure criterion. The resulting failure envelopes were compared for all the cases. KEYWORDS: Shear stress, Failure envelope, Composite material, Nonlinear behavior, Fiberorientation.Jordan Journal of Civil Engineering. 09/2013; EBSCO, AULRICH(Vol. 7, No. 4).
Page 1
Hindawi Publishing Corporation
International Journal of Aerospace Engineering
Volume 2009, Article ID 486063, 22 pages
doi:10.1155/2009/486063
Research Article
AThreeDimensionalPly 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), CTAITAIEM, Prac ¸a Mal. Eduardo Gomes 50,
12228900 S˜ ao Jos´ e dos CamposSP, 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 stressbased, 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, fibrematrix 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, TsaiHill 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 threedimensional failure criteria for unidirec
tional composites. In his model four distinct failure modes
Page 2
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 inplane 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 TsaiWu failure criterion
and maximum stress criterion to model lowvelocity impact
damage in composites. In this approach the TsaiWu 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 inplane damage model for predicting the residual
strength of notched laminated composites. Three inplane
failure modes are considered: matrix cracking, fibrematrix
shearing and fibre breakage. Their model is currently avail
ableinthematerialmodellibraryoftheLSDYNA3Dexplicit
finite element code. In their model the shear stressshear
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
fibrematrix 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 linenose impactor. They used
the ChangChang failure criterion to detect the initiation of
matrixcrackinganddelamination.Goodagreementbetween
experimental and numerical results was achieved.
Davies and Zhang [12] studied the lowvelocity impact
damage in carbon/epoxy composite plates impacted by
an hemispherical impactor. The plates tested were quasi
isotropic having a symmetrical layup. Different impact
energy levels, thickness, dimensions and boundary condi
tions were considered. The finite element modelling was
carried out using plate elements.
The ChangChang 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 ChangChang 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 HashinRotem 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 outofplane 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 Xradiographs 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 mixedmode delamination. Comparisons
were made with experimental and closedform 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
strainbased 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.
Page 3
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 elasticplastic
and it has two stages. The first stage is the postyield and
prefailure 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 inplane 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 inplane shear
modulus.Alinearelasticdamagebehaviourwasassumedfor
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 damagedevelopment 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 inplane damage
response of fibre reinforced composite structures during
crash and impact events. The damage model was imple
mented for shell elements into the PAMCRASH 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 quasiisotropic 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 inplane
damage model based on continuum damage mechanics
(CDM) into LSDYNA3D 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 ChangChang 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 substructuring concept
where one integration point is used for each sublaminate
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. Highvelocity and lowvelocity 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 planestress failure model
proposed by Davila and Camanho [43] and it accounts for
shear nonlinearities effects. The model was implemented
into LSDYNA3D finite element code and the authors
obtained good correlation between numerical predictions
and experimental results for static standard inplane 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
Page 4
4 International Journal of Aerospace Engineering
viscoplastic constitutive law and combines a quadratic stress
based criterion with the MohrCoulomb failure criterion
to predict interfibrefailure (IFF) without knowing a priori
the orientation of the fracture plane. The new version of
the model also uses the Second PiolaKirchhoff stress tensor
which potentially avoids material deformation problems.
Highorder 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. 3D Orthotropic StressStrain Relationship. The ortho
tropic material law that relates second PiolaKirchhoff stress
S to the GreenSt. 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 GreenSt. 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 Zdirection 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 GreenSt. 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
termsofthesecondPiolaKirchhoffstresstensorSasfollows:
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 crosssection 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:
Page 5
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 throughthethickness 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, outofplane 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
Page 6
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 inplane 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 stressstrain 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 InterfibreFailure
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 MohrCoulomb 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
thethickness direction (direction3 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 (direction1 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. InPlane Shear Failure. The observed behaviour of glass
and carbon fibres laminates generally shows marked rate
dependence in matrixdominated shear failure modes and
for this reason a rate dependent constitutive model has been
used to model the inplane 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 stressstrain behaviour for inplane 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 inplane shear
tests. α is the strainrate enhancement given by the following
law,
12is the initial shear modulus and c1, c2are material
α = 1+e(˙ γ12/c3)
(43)
wherec3isanothermaterialconstantobtainedfromdynamic
inplane shear tests. By decomposing the total shearstrain
into inelastic γi
shearstrain 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)
Thefailureindexforinplaneshearfailureisbasedonthe
maximum stress criterion and it is given by:
F12(τ12) =τ12
S12
≥ 1.
(45)
The proposed damage evolution law for inplane 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 stressstrain 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)
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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
stressstrain 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.
Inplane shear cracking is strongly dependent on the fibre
orientarion within the element therefore, the characteristic
length associated with inplane shear failure has been
assumed to be the same as the one defined for fibre failure or
failure in the warp direction. For outofplane 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 strainrates 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:
Inplane 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 inplane shear failure.
Outofplane 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)
inplane 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 inplane shear loading.
Loadingunloadingreloading (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 loadedunloadedreloaded 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 tensioncompression under loading
unloadingreloading 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 selfhealing 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
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