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Camanho.pdf
X Reunión de Usuarios de ABAQUS
ADVANCES IN THE SIMULATION OF DAMAGE AND
FRACTURE OF COMPOSITE STRUCTURES
Pedro Ponces Camanho, [email protected]
DEMEGI, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias,
4200-465 Porto, Portugal, www.fe.up.pt
ABSTRACT
This paper presents recent developments in the numerical simulation of damage and
structural collapse of advanced composite structures. The constitutive models presented
are developed in the framework of Continuum Damage Mechanics and Fracture
Mechanics, and can predict the onset and propagation of the different damage
mechanisms occurring in composite materials. Important aspects of the numerical
implementation of the models, including mesh objectivity and limit element size, are
discussed. The models proposed are implemented using Abaqus UEL and UMAT
subroutines, and an example of the simulation of fracture in a composite structure is
presented.
1. INTRODUCTION
The aerospace industry is committed to improve the performance of aircraft whilst
reducing emissions and weight. Such a goal can be achieved by the increased use of
advanced composite materials in primary aircraft and spacecraft structures. The next
generation of civil aircraft (Boeing 787, Airbus A380 and A350), and of aircraft jet
engines (General Electric GEnx engines) represent an important technological challenge
for the effective manufacture and design of composite structures.
The procedure used by the aerospace industry in the certification of composite
structures relies on a 'building-block' approach(1), where a large number of experimental
tests are performed at all stages of product development. The 'building-block' design
procedure results in high recurring and non-recurring costs. The use of advanced
analytical or numerical models for the prediction of the mechanical behaviour of
composite structures can replace some of the experimental tests and significantly reduce
costs.
Strength-based failure criteria are commonly used to predict failure in composite
materials. A large number of continuum-based criteria have been derived to relate
stresses and experimental measures of material strength to the onset of failure(2)-(5).
Failure criteria predict the onset of the several damage mechanisms occurring in
composites and, depending on the material, geometry and loading conditions, may also
predict final structural collapse.
For composite structures that can accumulate damage before structural collapse, the use
of failure criteria is not sufficient to predict ultimate failure. Simplified models, such as
the ply discount method, can be used to predict ultimate failure, but cannot represent
with satisfactory accuracy the quasi-brittle failure characteristic of a laminate that
results from the accumulation of different damage mechanisms.
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This paper presents computational models developed to accurately represent the fracture
process of advanced composite structures, from damage initiation up to final collapse.
The models are applied in the simulation of interlaminar damage (delamination) and
intralaminar damage (fibre fracture and matrix cracking).
2. SIMULATION OF DAMAGE AND FRACTURE USING FINITE ELEMENTS
The simulation of damage and structural collapse using Finite Element models is
normally done using the concepts of Fracture Mechanics, Continuum Damage
Mechanics or a combination of Fracture Mechanics and Continuum Damage
Mechanics. The two fundamental aspects of the numerical representation of damage and
fracture using finite elements are the kinematic description of the fracture process zone
(or crack) and the associated constitutive models.
The three kinematic descriptions are normally used(6), strong discontinuity, two weak
discontinuities and no discontinuities, can be represented by the displacement ( u ) and
strain ( ε ) fields shown in Figure 1.
ε
u
a)
strong
discontinuity
x
x
ε
u
b)
two weak
discontinuities
x
u
ε
x
c)
no discontinuities:
band of localized
defomation
x
x
Figure 1: Kinematic descriptions of localized deformation (after ref. (6)).
The Finite Element Models developed to predict failure using Fracture Mechanics
represent the boundary value problem of a structure containing a pre-crack. Using
Linear Elastic Fracture Mechanics, crack propagation is predicted by comparing the
computed values of the stress intensity factors, or the components of the energy release
rate, with the corresponding critical values, taken as material properties. The numerical
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computation of the stress intensity factors, or of the energy release rate, can be
performed using different techniques: collapsed finite elements, J-integral, and the
virtual crack closure technique (VCCT). Stability of crack growth can be easily assessed
by computing the values of the energy release rate (G) at different crack lengths (a) and
calculating the derivative of the energy release rate with respect to the crack length, i.e.
if ∂G / ∂a < 0 there is stable crack growth.
The use of Continuum Damage Mechanics in the prediction of failure does not require
any pre-existing crack, or the pre-knowledge of the location of crack initiation, and
normally uses a kinematic description that does not include discontinuities in the
displacement and strain fields (Figure 1c)). However, the use of Continuum Damage
Mechanics based on standard strain-softening constitutive models results in additional
difficulties. At the onset of localization the problem becomes ill-posed, the solution
bifurcates, and as a consequence the numerical solution depends upon the mesh
refinement. The bifurcation of the solution can be illustrated using a simple example of
a bar monotonically loaded in tension, shown in Figure 2, and using a simple damage
model defined as:
⎧ Eε , ε ≤ ε i
ε (ε − ε i )
⎪
σ ( ε ) = ⎨ (1 − d ) Eε , ε i < ε ≤ ε f , d = f
ε (ε f − ε i )
⎪ 0, ε > ε
f
⎩
(1)
σ
ΩΒ
F
ΩΑ
ΩΑ
σ
LS
L
εi
Figure 2: Bar under tensile load.
εf
ε
It is clear that the material properties, or the geometry of a real bar, cannot be exactly
uniform. Assuming that the strength of a small region of the bar, Ω B , is smaller than the
strength of the remaining portion of the bar, Ω A , damage localizes in the region Ω B .
The material in Ω B softens and, in order to satisfy the equilibrium conditions, the
material in Ω B must unload elastically. Using the equilibrium and compatibility
equations it is easy to show that the structural response during damage propagation,
established in terms of the time derivative of the applied load and displacement, F and
δ respectively, is calculated as:
F =
EAδ
⎛εf ⎞
L − LS ⎜ ⎟
⎝ εi ⎠
(2)
where A is the cross-section of the bar. Therefore, the post-peak structural response is a
function of the length of the localized zone, i.e., dF d δ = f ( LS ) . This effect is clearly
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shown in the Figure 3 where three different lengths for the localized zone of a 100mm
long bar with a square cross section are used: 10mm, 20mm and 50mm.
Figure 3: Bifurcation of the solution.
Supposing now that the problem under consideration is implemented in a finite element
code using a number N of 2-node linear beam elements, and assuming that the model is
able to capture the localized solution, the length of the localized zone is computed as:
LS = L / N . The post-peak solution is therefore strongly dependent of the number of
elements N, approaching the initial (linear-elastic) solution as the number of elements
tends to infinity. Furthermore, the energy dissipated in the localized zone, calculated as
ε
∫ V ∫ 0 σ d ε dV = 1 2σε f ALS , depends on the element length LS . At the limit, when the
f
element size tends to zero the computational model predicts failure without any energy
being dissipated, a physically unacceptable result.
There are two main procedures to assure the objectivity of the numerical solution:
methods acting at the level of the constitutive model, or the definition of the material
response as a function of a characteristic element length in the computational model.
The most general solution to assure objectivity is to act at the level of the constitutive
model, implementing non-local, gradient, viscous, or micropolar continuum models.
Non-local models replace a given variable s ( x ) by a non-local parameter obtained by
weighting the variable over a spatial neighbourhood(7): s ( x ) = ∫ V α ( x, ξ )s (ξ ) d ξ , where
α ( x, ξ ) is a weight function and ξ is a characteristic material length. Gradient models,
corresponding to the differential counterpart of integral non-local models, enforce highorder gradients in the constitutive models, for example as: σ = σ ( ε , ε , xx ) . Viscous
regularization is one technique used in Abaqus(8) to assure the objectivity of the
numerical solution, and it consists in using regularized damage variables, d v , in the
constitutive model. The regularized damage variables are calculated from the rate
equation:
1
d v = ( d − d v )
η
4
(3)
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Advances in the simulation of damage and fracture in composite structures
where η is the viscosity parameter controlling the rate at which the regularized damage
variable d v approachs the real damage variable d .
An alternative and simpler method to assure the objectivity of the numerical model
consists in adjusting the post-peak material response using a characteristic element
length. This technique, usually called the ‘crack-band model’(9), consists in assuring that
the computed dissipated energy due to a fracture process is constant and equal to the
product between the fracture toughness ( GC ) and the crack surface (A), and solving the
resulting equation for the ultimate strain:
∫ ∫
V
εf
0
σ d ε dV = GC A ⇒ ε f =
2GC
Ls Eε i
(4)
Using (4) in (2) it is clear that the post-peak response is objective and does not depend
on the element size LS . An additional material parameter, the fracture toughness, is
required, and the constitutive model is a function of a characteristic element length.
Methods based on a combination of Continuum Damage Mechanics and Fracture
Mechanics use kinematic descriptions with discontinuities in both the displacement and
strain fields, Figure 1a). The constitutive model is defined using the formalism of
Continuum Damage Mechanics and crack growth laws derived in the context of
Fracture Mechanics. The numerical implementation of this class of methods results in
cohesive finite elements that are discussed in the next section.
3 NUMERICAL SIMULATION OF DAMAGE IN ADVANCED COMPOSITES
The complex damage mechanisms occurring in advanced composite materials result in
additional difficulties in the numerical simulation of failure. The presence of two
constituents, fibre and matrix, and the extreme anisotropy in both stiffness and strength
properties result in damage mechanisms at different levels. The damage mechanisms
can be divided in intralaminar and interlaminar damage. As shown in Figure 4
intralaminar damage mechanisms correspond to fibre fracture and matrix cracking,
whereas interlaminar damage mechanisms correspond to the interfacial separation of the
plies (delamination).
Fibre fracture
Matrix transverse crack
Delamination
Figure 4: Damage mechanisms in laminated composites.
3.1 Interlaminar damage (delamination)
Delamination consists in the separation of the plies of a composite laminate. This
damage mechanism can occur as a result of impact loading, and it leads to a significant
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reduction in the compressive load-carrying capacity of a composite structure. The stress
gradients that occur near geometric discontinuities such as ply drop-offs, stiffener
terminations and flanges, bonded and bolted joints, and access holes promote
delamination initiation, trigger intraply damage mechanisms, and may cause a
significant loss of structural integrity. The numerical simulation of delamination is
normally performed using the virtual crack closure technique (VCCT) or cohesive
elements.
Virtual crack closure technique (VCCT)
The virtual crack closure technique proposed by Rybicki and Kanninen(10) has been used
to predict delamination growth in composite materials. This technique is based on the
assumption that when a crack extends by a small amount, the energy dissipated in the
process, ∆U , is equal to the work W required to close the crack to its original length,
i.e., W = ∆U = U ( a + ∆a ) − U ( a ) .
δΑΒ
X
A
A
B
δΑΒ
Y
B
y
l
x
b) crack length (a+l)
a) crack length a
Figure 5: VVCT method.
Figure 5 illustrates the steps required to calculate the energy release rates using the
VCCT. In Figure 5a) the crack tip is coincident with the nodes A and B, and it is
subjected to the nodal force FAB = FABX x+ FABY y . In Figure 5b) the crack tip has been
extended a length l, the nodes A and B are no longer coincident, and their relative
Y
displacement is given by: δ = δ ABX x+ δ AB
y . The work required to close the crack back to
Y
its original position is: W = 1 2 ( FABX δ ABX + FABY δ AB
) = ∆U . Therefore, the energy release rate
can be computed from the nodal forces and relative displacements obtained from a
Finite Element Model as:
∆U
1
1 X X
1 Y Y
X
Y
Y
≈
FABX δ AB
+ FAB
∴ GI =
FABδ AB and GII =
FABδ AB
δ AB
(
)
∆a → 0 b∆a
2bl
2bl
2bl
G = lim
(5)
where b is the width of the specimen.
The approach can be computationally effective when sufficiently refined meshes are
used, and when all the elements at the crack tip have the same dimensions in the crack
growth direction. Under these conditions, the energy release rates can be obtained from
only one analysis.
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Care must be taken in selecting the element size at the crack tip when using the VCCT
to simulate delamination. Raju et al.(11) have shown that the individual components of
the energy release rate do not converge when the ratio of the size of delamination tip
element to the ply thickness decreases. This effect does not occur for the total energy
release rate, and it is due to the oscillatory part of the stress singularity occurring in
cracks between dissimilar media. It was shown that when using delamination tip
elements with a size of one-quarter to one-half of the ply thickness, the individual
components obtained agreed well with the results from a model that had a resin-rich
layer incorporated between the plies.
Although providing valuable information concerning onset and stability of delamination
growth, the use of VCCT to simulate delamination growth requires the definition of an
initial delamination. For certain geometries and load cases, the location of the
delamination front might be difficult to determine a priori.
Cohesive elements
One approach to overcome the limitations of the VCCT is the use of interfacial cohesive
elements placed between the composite material layers. Interfacial cohesive elements
combine the strength of materials formulation for crack initiation with Fracture
Mechanics for crack propagation. The concept of cohesive elements is based on a
Dugdale-Barenblatt type cohesive zone(12) and was first applied to the analysis of
concrete cracking by Hillerborg et al. (13) .
Cohesive formulations relate displacement discontinuities, ∆ i , to the traction vector at
the process zone, τ j , and represent a process (softening) zone located ahead of a crack
tip (Figure 6).
y
Γ
1 2 3
x
τ
2
1
Cohesive
zone
4 5
Applied load
3
4
5
∆
Figure 6: Cohesive zone ahead of crack tip.
The formulation of a cohesive zone model is based on two main constituents:
kinematics of the interfacial surface and constitutive model. The calculation of the
displacement discontinuities and separation of loading modes is based on standard
mathematical procedures(14) and will not be elaborated upon here.
A constitutive law relating the cohesive tractions, τ j , to the displacement jump, ∆ i , is
required for modelling the behaviour of the material in the process zone. The
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constitutive laws may be formally written as τ j = Γ ( ∆i ) . A fundamental aspect in the
formulation of the constitutive model is the fact that the energy dissipated at crack
propagation must be equal to the fracture toughness, i.e., the following relation must be
satisfied:
∫
∆if
0
(6)
τ i d ∆ i = GC
Figure 7 illustrates several constitutive models that have been proposed(15): linear
elastic-perfectly plastic, linear elastic-linear softening, linear elastic-progressive
softening, linear elastic-regressive softening.
σ
Perfectly plastic (pp)
Linear softening (lin)
σc
Progressive softening (pro)
Regressive softening (reg)
Needleman (Ne)
Gc
0
pp
pro
lin
Ne
reg
Figure 7: Constitutive strain softening equations.
In order to simulate the cohesive zone process ahead of the crack tip represented in
Figure 6, a linear softening behaviour is usually implemented. A high initial elastic
stiffness avoids the introduction of artificial compliance in the model before damage
onset.
For pure Mode I, II or III loading, after the interfacial normal or shear tractions attain
their respective interlaminar tensile or shear strengths, the stiffness is gradually reduced
to zero. The area under the stress-relative displacement curves is the respective (Mode I,
II or III) fracture energy:
∫
∫
∫
∆1 f
0
∆2 f
0
∆3 f
0
τ 1 ( ∆1 )d ∆1 = GIC
τ 2 ( ∆ 2 )d ∆ 2 = GIIC
(7)
τ 3 ( ∆ 3 )d ∆ 3 = GIIIC
The properties required to define the interfacial behaviour are the initial elastic stiffness,
the corresponding fracture energies, GIC, GIIC and GIIIC and the corresponding
interlaminar tensile or shear strengths.
For mixed-mode loading, the formulation of the constitutive model is far more complex
because it is necessary to define a damage activation criterion taking into account all
components of the traction tensor, a damage evolution function based on the interaction
of the components of the energy release rate, and to avoid spurious energy generation
under variable mode-ratio.
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Under mixed-mode loading the constitutive model should be formulated using the
formalism of Damage Mechanics combined with a mixed-mode failure criterion for
crack propagation established from Fracture Mechanics concepts(16). A constitutive
model previously proposed and able to predict the onset and propagation of
delamination under variable mixed-mode conditions is summarized in Table 1.
Table 1: Constitutive model for interlaminar damage(14).
One important requirement for the use of cohesive elements in the simulation of fracture
is to have at least three cohesive elements in the process (softening) zone. This
requirement normally results in very refined meshes ahead of a crack tip. A procedure
recently proposed(16) to use coarse meshes in the simulation of delamination consists in
lowering the interfacial allowables, artificially increasing the process zone, but still
assuring the correct computation of the energy dissipated.
3.2 Intralaminar damage
The intralaminar damage mechanisms occur at different orientations, corresponding to
fracture planes with general orientations with respect to the fibre and ply thickness
directions (Figure 4). These complexities make the representation of intralaminar
damage using a kinematic description based on strong discontinuities (Figure 1a)) a
formidable task. The intralaminar damage mechanisms are usually represented as a band
of localized deformation (Figure 1c)) using constitutive models formulated in the
context of Continuum Damage Mechanics.
An accurate Continuum Damage Model should be based on damage activation functions
(failure criteria) able to represent the different damage mechanisms, and the
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corresponding computational implementation should avoid the mesh-dependence
problems outlined in section 2.
Damage activation functions (failure criteria)
There are several failure criteria that have been proposed to predict the onset of matrix
cracking and fibre fracture, some of which are implemented in Abaqus(8). However, few
criteria can represent several relevant aspects of the failure process of laminated
composites, e.g. the increase on apparent shear strength when applying moderate values
of transverse compression, or the detrimental effect of the in-plane shear stresses in
failure by fibre kinking.
The recently proposed LaRC04 failure criteria(5) can predict with accuracy intralaminar
damage in composite laminates, and it can be easily coded as an Abaqus UVARM user
subroutine. Figure 8 show the comparison between the predictions of LaRC04 failure
criteria and published experimental data(5).
Figure 8: Comparison between LaRC04 failure criteria and experimental data.
Continuum damage model
Using LaRC04 failure criteria(5) as damage activation functions, FM , it is possible to
formulate a Continuum Damage Model to predict the propagation of the several (M)
damage mechanisms occurring at intralaminar level. Each damage activation function
predicts one type of damage mechanism using the following equations:
( )
FM := φM σ t − rMt ≤ 0
(8)
where rMt are internal variables (equal to 1 at time t = 0 ), and the functions φM (σ t )
correspond to LaRC04 failure criteria. When a damage activation function is satisfied,
FM = 0 , the associated damage variable, d M , is different than zero, and the ply
compliance tensor is affected by the presence of damage. Using the model proposed in
ref. (18) the compliance matrix of a damaged ply is defined as:
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⎡
1
⎢
⎢ (1 − d1 ) E1
⎢
υ
H = ⎢ − 12
E2
⎢
⎢
⎢
0
⎢⎣
Advances in the simulation of damage and fracture in composite structures
−
υ12
E2
1
(1 − d 2 ) E2
0
⎤
⎥
⎥
⎥
⎥
0
⎥
⎥
1
⎥
(1 − d6 ) G12 ⎥⎦
0
(9)
where d1 is the damage variable associated with fibre fracture and d 2 , d6 are the
damage variables associated with matrix cracking.
Further to the damage activation functions and damaged compliance tensor, it is
necessary to define the evolution laws for the damage variables, d M . The damage
evolution laws need to assure that the computed energy dissipated is independent of the
refinement of the mesh.
A complete definition of a Continuum Damage Model for the simulation of intralaminar
damage can be found in reference (18). The corresponding algorithm for the integration
of the damage constitutive model implemented in an Abaqus UMAT subroutine is
shown in Table 2.
Table 2: Algorithm for the integration of intralaminar damage model(18).
The Continuum Damage Models developed to simulate localized intralaminar damage
use strain softening constitutive models. In order to avoid mesh dependent solutions, it
is necessary to regularize the energy dissipated for each damage mechanism using a
modification of the crack band model outlined in section 2. As a consequence of the
application of the crack band model there is a maximum element size that can be used in
the numerical model which avoids a physically unacceptable snap-back of the material
response. The maximum element size depends on the material properties and failure
mode, and can be calculated using closed-form equations(18).
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4. APPLICATIONS
This section presents an example of the application of advanced computational methods
in the simulation of damage onset and final failure of a composite structure.
4.1 Skin-stiffner debonding
Most composite components in aerospace structures are made of panels with co-cured
or adhesively bonded frames and stiffeners. Testing of stiffened panels has shown that
bond failure at the tip of the stiffener flange is a common failure mode. Test specimens
consisting of a stringer flange bonded onto a skin have been developed by Krueger et
al.(19) to study skin/stiffener debonding. The configuration of the specimens is shown in
Figure 9.
Figure 9: Skin-stiffener test specimens.
The specimens are 203 mm-long and 25.4 mm-wide. Both skin and flange were made
from IM6/3501-6 graphite/epoxy pre-preg tape with a nominal ply thickness of 0.188
mm. The skin lay-up consisting of 14 plies was (0º/45º/90º/-45º/45º/-45º/0º)s and the
flange lay-up consisting of 10 plies was (45º/90º/-45º/0º/90º)s.
The properties of the unidirectional graphite/epoxy and the properties of the interface
are shown in Tables 3 and 4, respectively.
Table 3: Composite ply properties(20).
Table 4: Interface properties(20).
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The complete model consists of 1,002 three-dimensional 8-node Abaqus C3D8I
laminated solid elements that allow non-symmetric lay-up, and 15,212 degrees of
freedom. Cohesive elements with the formulation outlined in section 3 were
implemented using an Abaqus UEL subroutine. The cohesive elements are used to
simulate the interface between the skin and the stiffener. This model does not contain
any pre-existing delaminations.
Deformed plots of the finite element model immediately before and after flange
separation are shown in Figure 10.
Figure 10: Debonding propagation (after ref. (20)).
It can be observed that only the refined end of the flange separates. It is worth noticing
that the debond growth is not symmetric across the width: the debond initiates on the
left corner of the flange shown on the bottom left of Figure 10 due to the lack of
symmetry introduced by the terminated plies at the flange tapered ends.
Figure 11 shows the load-extensometer measurement relation obtained in four
experiments and the corresponding numerical prediction.
Figure 11: Load-displacement relation (after ref. (14)).
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It can be observed that good accuracy in the prediction of the debond loads is obtained.
The predicted stiffness of the specimen is also in good agreement with the experimental
data.
5. CONCLUSIONS
The economic design of composite structures relies upon the availability of reliable and
general computational methods that can predict the mechanical behaviour of such
structures.
Reliable and validated computational methods to predict the stiffness or buckling loads
of composite structures are available. However, the prediction of fracture of composite
structures is a far more complex task. This paper presented several computational
models that can be used to predict the onset of damage and final failure of laminated
composite structures. The computational models are based on Fracture Mechanics or in
Damage Mechanics using different kinematic and constitutive models for interlaminar
and intralaminar damage.
Some of the computational methods required to accurately simulate damage in
composite materials, such as cohesive elements, have been implemented in commercial
Finite Element codes(8). Cohesive elements provide accurate prediction for both
delamination onset and growth, and do not require any user intervention during the
analysis.
The prediction of the onset of fibre fracture or transverse matrix cracking should be
based on novel failure criteria that account for relevant aspects of the mechanical
behaviour of a composite material. When predicting both the onset of intralaminar
damage and final failure the failure criteria should be used in a continuum damage
model able to represent the redistribution of load occurring in a quasi-brittle failure.
The computational methods described in this paper should be combined in order to
predict the interaction of intra- and interlaminar damage mechanisms. The simulation of
this interaction is a key aspect for the accurate representation of the response of
composite materials for several load cases, such as in low-velocity impact loads. Future
work will address the simulation of the interaction between the different damage
mechanisms, as well as damage models for other composite materials, such as woven or
non-crimp fabrics.
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6. REFERENCES
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2002.
(2) Hinton, M. J., Kaddour, A. S., and Soden, P. D., A comparison of the predictive
capabilities of current failure theories for composite laminates, judged against
experimental evidence. Composites Science and Technology, 62, 1725-1797, 2002.
(3) Puck, A. and Schürmann, H. Failure analysis of FRP laminates by means of
physically based phenomenological models. Composites Science and Technology, 58,
1045-1067, 1998.
(4) Hashin, Z. Failure criteria for unidirectional fiber composites. Journal of Applied
Mechanics, 47, 329-334, 1980.
(5) Pinho, S.T. , Dávila, C.G. , Camanho, P.P., Iannucci, L., and Robinson, P., Failure
models and criteria for FRP under in-plane or three-dimensional stress states including
shear non-linearity, NASA Technical Memorandum 213530, National Aeronautics and
Space Administration, U.S.A., 2005.
(6) Jirásek, M., Modeling of localized damage and fracture in quasibrittle materials,
Lecture Notes in Physics 568, eds. P.A. Vermeer et al., Springer, Berlin, 17-29, 2001.
(7) Bažant, Z. P. and Jirásek, M. Nonlocal integral formulations of plasticity and
damage: survey of progress. Journal of Engineering Mechanics. 128(11), 1119-1149,
2002.
(8) Abaqus Inc., ABAQUS 6.5 User's Manual, Pawtucket, U.S.A., 2005.
(9) Bažant, Z. P. and Oh, B. H., Crack band theory for fracture of concrete. Matériaux et
Constructions, 16(93), 155-177, 1983;
(10) Rybicki, E.F. and Kanninen, M.F., A finite element calculation of stress intensity
factors by a modified crack closure integral, Engineering Fracture Mechanics, 9, 931938, 1977.
(11) Raju, I. S., Crews, J. H., and Aminpour, M.A., Convergence of strain energy
release rate components for edge-delaminated composite laminates, Engineering
Fracture Mechanics 30(3), 383-396, 1988.
(12) Dudgale, D.S., Yielding of steel sheets containing slits, Journal of Mechanics and
Physics of Solids, 8, 100-104, 1960.
(13) Hilleborg, A., Modeer, M., and Petersson, P. E., Analysis of crack formation and
crack growth in concrete by fracture mechanics and finite elements, Cement and
Concrete Research, 6, 773-782, 1976.
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Advances in the simulation of damage and fracture in composite structures
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simulation of delamination in advanced composites under variable-mode loading,
Mechanics of Materials, in press, 2005.
(15) Camanho, P.P., Dávila C.G., and Ambur, D.R., Numerical simulation of
delamination growth in composite materials, NASA-Technical Paper 211041, National
Aeronautics and Space Administration, U.S.A., 2001
(16) Turon, A., Dávila, C.G., Camanho, P.P., and Costa, J., An engineering solution for
using coarse meshes in the simulation of delamination with cohesive zone models,
NASA Technical Memorandum 213547, National Aeronautics and Space
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(17) Camanho, P.P., Dávila, C.G., and Moura, M.F., Numerical simulation of mixedmode progressive delamination in composite materials, Journal of Composite Materials,
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(18) Maimí, P., Camanho, P.P., Mayugo, J.A., and Dávila, C.G., A Thermodynamically
consistent damage model for advanced composites, NASA-Technical Paper, in
preparation, 2005.
(19) Krueger, R., Cvitkovich, M.K., O'Brien, T.K., and Minguet, P.J., Testing and
analysis of composite skin/stringer debonding under multi-axial loading, Journal of
Composite Materials, 34, 1263-1300, 2000.
(20) Camanho, P.P., Dávila, C.G., and Pinho, S.T., Fracture analysis of composite cocured structural joints using decohesion elements, Fatigue and Fracture of Engineering
Materials and Structures, 27, 745-757, 2004.
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