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COMPUTATIONAL MECHANICS New Trends and Applications S. Idelsohn, E. Oñate and E. Dvorkin (Eds.) ©CIMNE, Barcelona, Spain 1998

A COMPUTATIONAL METHOD FOR DAMAGE INTENSITY PREDICTION IN A LAMINATED COMPOSITE STRUCTURE Pierre Ladevèze, Olivier Allix, Bernard Douchin and David Lévêque Laboratoire de Mécanique et Technologie (LMT - Cachan) ENS de Cachan / CNRS / Université Paris 6 61, Avenue du Président Wilson - 94235 Cachan Cedex - France E-mail : [email protected]; Web site : http://www.lmt.ens-cachan.fr

Keywords : Composite laminates, Meso-modeling, Damage, Identification tests, Non-linear computation Abstract : The present study describes the basic principles of a general damage computational approach for laminates and demonstrates this approach's prediction capabilities in simulating the complete fracture phenomenon in the case of several tests, both with and without delamination. In this paper, the Damage Mechanics approach is applied in order to solve delamination problems, such as the propagation of an initiated crack, the onset of delamination near edges, and the growth of delamination around holes. For the M55J/M18 carbon/epoxy material, numerical simulations are presented and then compared with experimental results.

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Pierre Ladevèze, Olivier Allix, Bernard Douchin, and David Lévêque

1. INTRODUCTION This study concerns a general damage computational approach for laminated structures which is able to predict, at any time and at any point, the "intensities" of the various damage mechanisms up to fracture. This paper presents the current "state-of-the-art" and highlights the prediction capabilities, as demonstrated by means of comparison with numerous experimental results. An initial step, which has been conducted in other studies, is to define what we term a "laminate meso-model". At the meso scale, characterized by the thickness of a single ply, the laminated structure can be described as a stacking sequence of homogeneous layers throughout the thickness and interlaminar interfaces. The main damage mechanisms then consist of : the breaking of fibers, the micro-cracking of matrices and the debonding of adjacent layers, i.e. delamination. The single-layer model includes both damage and inelasticity [Ladevèze 1992]1, [Ladevèze, Le Dantec 1992]2 and [Allix, Ladevèze, Vittecoq 1994]3. The interlaminar interface is described by a two-dimensional mechanical model which ensures traction and displacement transfer from one ply to another. Its mechanical behavior depends on the angle between the fibers of two adjacent layers [Allix, Ladevèze 1992, 1996]4,5. It is well-known that fracture simulation with a continuum damage model leads to severe theoretical and numerical difficulties. A second step, which has also been performed, involves overcoming these difficulties. For laminates and, more generally, for composites, we propose the concept of the meso-model, wherein the state of damage is uniform within each meso-constituent. For laminates, it is uniform throughout the thickness of each single layer; as a complement to this conceptual set-up, continuum damage models with delay effects have been introduced in [Ladevèze 1992]6. In this approach, two models are to be identified : the single-layer model, and the interface model. The appropriate tests to apply consist of : tension, bending and delamination. Each composite specimen, which contains several layers and interfaces, is computed in order to derive the material quantities intrinsic either to the single layer [Ladevèze, Le Dantec 1992]2, [Allix, Ladevèze, Vittecoq 1994]3 or to the interlaminar interface [Allix, Ladevèze 1996]5, [Allix, Corigliano, Ladevèze 1995]7 and [Allix, Ladevèze, Lévêque, Perret 1997]8. The procedure proposed herein is rather straightforward and has already been applied to various materials. Several comparisons with experimental results have been carried out in order to demonstrate both the possibilities and limitations of our proposed computational damage mechanics approach for laminates. Standard tests of delamination propagation (DCB, ENF and MMF) or initiation (edge delamination) as well as tests of holed-plate specimens in tension are to be considered herein. More detail can be founded in [Ladevèze 1992, 1995]6,9 and [Ladevèze and al. 1998]10. Here, we will present the "state-of-the-art" and the new advences of this damage computational approach for laminated structures. Special emphasis will be placed on the basic aspects of the finite element simulations of both interlaminar and intralaminar damage. In particular, the advantage of using a Damage Mechanics approach for initiation prediction as well as for the interpretation of standard Fracture Mechanics tests, in connection with experiments, is also discussed.

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Pierre Ladevèze, Olivier Allix, Bernard Douchin, and David Lévêque

2. MESO-MODELING CONCEPT Delamination often appears as the result of the interaction between different damage mechanisms, such as fiber breaking, transverse micro-cracking and the debonding of adjacent layers, which itself constitutes delamination [Highsmith, Reifsnider 1982]11, [Talreja 1985]12, [Whithney 1989]13 and [Crossman, Wang 1982]14. Our aim herein is to combine damage mechanics and delamination by including all of these damage mechanisms within the delamination analysis. In order to perform this combination, a damage meso-model, which allows predicting both delamination initiation and propagation far more accurately, has been developed. By virtue of the proposed approach, both initiation and propagation have been integrated into a single model. An initial step, which has been conducted in other studies, is to model the laminate as a stacking sequence of non-linear layers and non-linear interlaminar interfaces (Figure 1). At the layer level, the inner damage mechanisms are taken into account by means of internal damage variables. These damage variables are prescribed as uniform throughout the thickness of each ply, which serves to define what we have termed a "damage meso-model" [Ladevèze 1992]6. This step plays a crucial role in mesh-independent damage prediction. The singlelayer model and its characterization, including damage and inelasticity, were previously developed in other studies [Ladevèze, Le Dantec 1992]2 and [Allix, Ladevèze, Vittecoq 1994]3.

AAAAAAAAAAAAAAA AAAAAAA AAAAAAAA AAAAAAA Single-layer

Interface

Figure 1 : Laminate meso-modeling

The interlaminar interface is a two-dimensional entity which ensures traction and displacement transfer from one ply to another. Its mechanical behavior depends upon the angle between the fibers of two adjacent layers. Its primary attractive feature is to allow modeling the varying level of progressive degradation to the interlaminar connection [Allix, Ladevèze 1992, 1996]4,5. One consequence of this meso-model is that only two types of cracks are being considered : delamination cracks and orthogonal cracks, which extend across the entire thickness of the layer. 3. SINGLE-LAYER MODELING The damaged material's strain energy proposed below is written for the case of the plane stress assumption [Ladevèze, Le Dantec 1992]2. In what follows, subscripts 1, 2 and 3 will designate the fiber direction, the transverse direction inside the layer and the normal direction, respectively. The unilateral aspect of cracking in the transverse direction is taken into account by splitting the transverse energy into a "tension" energy and a "compression" energy. The energy for the damaged single layer is then obtained as follows :

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Pierre Ladevèze, Olivier Allix, Bernard Douchin, and David Lévêque

o

cp

ED,

=

V210 ν12, < σ 11 > 2 ζ < −σ 11 > + 1 [ + -( 0 + o ) σ11σ22 ] 2(1 − d F ) E10 E10 E2 E1,

+

1 2

[

< σ 22 > 2 (1 − d ' ) E20

+

< −σ 22 > 2 + E10

< σ 212 > 2 (1 − d ' )G120

(1)

]

where ζ is a material function that describes the non-linear compressive behavior in the fiber direction [Allix, Ladevèze, Vittecoq 1994]3. dF, d and d' are three scalar internal damage variables which remain constant within the thickness of each layer. d describes the shear damage and d' the transverse cracking. The thermodynamic forces associated with the mechanical dissipation are classically defined as : Yd =

> ∂ e  = σ : cst ∂d 2G120 (1 − d ' ) 2

(2)

> ∂ e  = Yd' = σ : cst ∂d ' 2 E2,o(1-d')2 2,+ < s11 > 2 + V 210 ∂ 1 e  >

where + represents the positive part of X and > denotes the mean value within the thickness. For static loadings, the damage evolution laws can be formally written as follows : d = Ad (Ydτ, Yd'τ , τ = t ) t

(3)

d' = Ad' (Ydτ, Yd'τ , τ = t ) t where the operators Ad and Ad' are material functions which need to be identified. The evolution of dF corresponds to the standard brittle fracture mechanism of the carbon fibers in tension. Further details, in particular with respect to the modeling of inelastic strains, can be found in [Ladevèze, Le Dantec 1992]2 and [Ladevèze 1992]6. It should be recalled that in order to perform a complete analysis for all cases, damage models with delay effects are introduced in the in-plane direction. Among the few tests employed in the identification process of the single-layer model, the tension test on a [±45]4s specimen served to enhance the inelastic and damageable behavior. In particular, the operator Ad is identified as a linear function of the square root of Yd. Figure 2 displays the evolution of the mean tension stress as a function of the longitudinal and transverse strains. A 3D FE simulation is compared with experimental results for several

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Pierre Ladevèze, Olivier Allix, Bernard Douchin, and David Lévêque

specimens. The material's non-linear behavior is well described by the computation up to complete rupture. 140

Applied stress (MPa)

120 100 80 60

Test 1 Test 2 Test 3 Test 4 Test 5 3D Simulation

40 20 0 -2

-1.5

-1

-0.5

0 Strain (%)

0.5

1

1.5

2

Figure 2 : Comparison between 3D FE simulation and tension tests on M55J/M18 [±45]4s laminates

4. INTERLAMINAR INTERFACE MODELING 4.1 Damageable behavior modeling The effect of the deterioration of the interlaminar connection on the connection's mechanical behavior is taken into account by means of three internal damage variables. The energy per unit area proposed in [Allix, Ladevèze,1992]4 is : 2 < σ 33 > 2 < σ 32 > 2 < σ 31 > 2 1 < −σ 33 > [ ] Ed = + 0 + + 2 k 30 k 3 (1 − d 3 ) k 20 (1 − d 2 ) k10 (1 − d 1 )

(4)

where ki,0 is an interlaminar stiffness value and di the internal damage indicator associated with its Fracture Mechanics mode, while subscript i corresponds to an orthotropic direction of the interface, as defined by the bisector of the fiber directions of the two adjacent layers (see Figure 3). ply i+1

AAAAAAA AAAAAAA AAAAAAA AAAAAAA AAAAAAA

interface ply i

3 2 1

Figure 3 : Orthotropic directions of the interlaminar interface

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Pierre Ladevèze, Olivier Allix, Bernard Douchin, and David Lévêque

From a classical standpoint, the damage energy release rates, associated with the dissipated energy ω both by damage and by unit area, are introduced as : 2 1 < σ 33 > Yd = 3 2 k 0 (1 − d ) 3 3

Yd1 =

(5)

2 1 < σ 32 > 2 k 20 (1 − d 2 )

2 1 < σ 31 > Yd2 = 2 k10 (1 − d 1 )

and : . . . φ = Yd3d, 3 + Yd1d, 1 + Yd2d, 2

(6)

with φ = 0 for satisfying the Clausius-Duheim inequality. In what follows, an "isotropic" damage evolution law is established. In this model [Allix, Ladevèze 1996]5, the damage evolution law is assumed to be governed by an equivalent damage energy release rate of the following form : Y (t) = sup = τ t

[(

(Yd3)α + (γ1Yd1)α + (γ2Yd2)α )

∧τ,

1/α

]

(7)

The evolution of the damage indicators is thus assumed to be strongly coupled. γ1, γ2 and α are three material parameters. A damage evolution law is then defined by choosing a material function ω, such that : d3 = d1 = d2 = ω(Y) if d < 1

(8)

d3 = d1 = d2 = 1 otherwise One simple case, used for application purposes, is :

ω(Y) = [

n < Y − Y0 > + n ] n + 1 Yc − Y0

(9)

in which a critical value Yc and a threshold value Yo have been introduced. High values for the n case correspond to a rather brittle interface. To summarize, the damage evolution law is defined by means of the six intrinsic material parameters : Yc, Yo, γ1, γ2, α and n. With respect to the onset of delamination, the significant parameters are Yo, n and the interface stiffnesses. It will be shown in the next section that Yc, γ1, γ2 and α are all related to the critical energy release rates. 4.2 Links with Fracture Mechanics One simple method for identifying the parameters related to delamination growth is to compare Damage Mechanics with Linear Elastic Fracture Mechanics. This was performed by comparing the mechanical dissipation yielded by the two approaches [Allix, Ladevèze 1996]5; the results of this comparison are presented below.

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Pierre Ladevèze, Olivier Allix, Bernard Douchin, and David Lévêque

In the case of pure-mode situations, when the critical energy release rate reaches its stabilized value at the propagation denoted by Gc,p , we obtain : (10) Y Y Gc,p I = Yc ; Gc,p II = c ; Gc,p III = c γ1 γ2 For a mixed-mode loading situation, we simply derive a standard LEFM model :  G  α  G  α  G α  pI  +  pII  +  pIII  = 1  GcI   GcII   GcIII 

(11)

wherein α governs the shape of the failure locus in the mixed-mode case. 4.3 Identification methodology Several standard Fracture Mechanics tests have been chosen to identify the interface damage model. These tests, conducted in an earlier work [Allix, Lévêque, Perret 1998]15, are the pure-mode I DCB (Double-Cantilever Beam) Test, the pure-mode II ENF (End-Notched Flexure) Test, and two mixed-mode tests : the MMF (Mixed-Mode Flexure) Test and the CLS (Cracked-Lap Shear) Test. Each specimen tested was a [(+θ/-θ)4s/(-θ/+θ)4s] laminate with θ = 0°, 22.5° or 45°, according to the three kinds of ±θ interlaminar interfaces being investigated. An anti-adhesive film was inserted at the mid-plane in order to initiate cracking. The critical energy release rate Gc is classically obtained by deriving the compliance of the specimen, as is usually carried out within the concept of Linear Elastic Fracture Mechanics. Nevertheless, in the case of carbon / epoxy laminates, the main assumptions of LEFM are not always satisfied, even in the simple case of a DCB specimen. Such is true, in particular, in the case of : nonunidirectional stacking sequences [Robinson, Song 1992]16, and R-curve-like phenomena [De Charentenay et al. 1984]17. In the former case, inner-layer damage mechanisms may be activated, which leads to an apparent energy release rate that's greater than the local interfacial energy release rate. In this case, it becomes necessary to perform a non-linear "re-analysis" for evaluating the damage state inside the layers [Allix, Ladevèze, Lévêque, Perret 1997]8. The critical energy release rates at propagation are thereby corrected with the part of the energy dissipated inside the layers. From these corrected rates and from the relationships existing between Fracture Mechanics and Damage Mechanics (10-11), we deduce the values of the critical energies Yc, the coupling coefficient γ1 and the parameter α. The identification results have been reported in Table 1. The interface parameters do seem to be independent of θ for all ±θ interfaces with θ ? 0°. It should also be pointed out that the 0°/0° interface appears to be something artificial. However, such an "artificial" interface can be introduced, for example, to describe a delamination crack in a thick layer. The other parameters associated with the delamination onset, namely Yo, n and the interface stiffnesses, are indirectly identified by comparing the tension strain at the onset of delamination between the experimental results of the Edge Delamination Tension tests and the results produced by a specialized software dedicated to free-edge effects [Daudeville, Ladevèze 1993]18 and [Allix, Lévêque, Perret 1997]19. More information on this identification methodology can be found in [Lévêque 1998]20.

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Pierre Ladevèze, Olivier Allix, Bernard Douchin, and David Lévêque

Interface

Yc (N mm-1)

γ1

α

0°/0° ±22.5° ±45°

0.11 ± 0.01 0.17 ± 0.01 0.19 ± 0.01

0.37 ± 0.15 0.36 ± 0.17 0.44 ± 0.16

1.59 1.12 1.19

Table 1 : Interface model parameters

5. NON-LINEAR SIMULATION OF DELAMINATION TESTS 5.1 Standard Fracture Mechanics tests From the Fracture Mechanics tests used for identification (Section 4.3), finite element predictions have been generated with the help of "ENDO-STRAT-EF", a three-dimensional non-linear finite element code that includes the meso-model [Gornet, Hochard, Ladevèze, Perret 1997]21. In this work, the same finite element mesh was used for all interlaminar fracture specimens. This mesh consists of 678 elements for each of the two arms (16 layers/arm) and 226 special elements for the interface. The mean thickness of a single ply is on the order of 0.1 mm; hence, one element in the ply thickness is chosen for the computation. Here, we will only show the prediction for the DCB, ENF and MMF tests conducted on unidirectional beams. For a DCB, ENF or MMF test, Figures 4-6 depict the following : a) the specimen geometry with loading and boundary conditions (the percentage of mode I on the global loading mode is also indicated); and b) the comparisons between experimental load-displacement curves and finite elementpredicted values. Experimental results and finite element predictions exhibit a strong correlation. In particular, the value of the length a of the debonding area is found to be quite close. Predicted values Experimental results

D.C.B. M55J/M18 [0,0]

50

51 mm 40

P

58 mm 65 mm 67 mm

30

δ

74 mm 20 P

a

10

a) DCB specimen (100% mode I) 0 -10 -1

0

1 2 3 Displacement (mm)

4

b) P-δ curves as a function of a Figure 4 : Comparison between experimental results on a DCB test and 3D F.E. predictions

8

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Pierre Ladevèze, Olivier Allix, Bernard Douchin, and David Lévêque

Experimental results Predicted values

E.N.F. M55J/M18 [0,0]

300 68 mm 250

P, δ

77 mm 200

a L

L

a) ENF specimen (0% mode I)

150 100 50 0 -50 -0,5

0

0,5 1 1,5 Displacement (mm)

2

2,5

b) P-δ curve as a function of a Figure 5 : Comparison between experimental results on an ENF test and 3D F.E. predictions

Predicted values Experimental results

M.M.F. M55J/M18

200

P, δ

a=53,96

a

a=59,90

a=65,58 a=70,96

150

L

a=45,38/L=90

L

a) MMF specimen (57% mode I)

100

50

a=78,15

0

-50 -0,5

0

0,5

1

1,5

2

2,5

Displacement (mm)

b) P-δ curves as a function of a Figure 6 : Comparison between experimental results on an MMF test and 3D F.E. predictions

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Pierre Ladevèze, Olivier Allix, Bernard Douchin, and David Lévêque

5.2 Edge Delamination Tension tests The experimental study of the initiation of delamination often requires EDT (Edge Delamination Tension) specimens [O’Brien 1982]22. Fracture Mechanics is not well-adapted for the analysis of such a test since the energy release rate vanishes at zero crack length. In addition, delamination, especially at its onset, appears to result from an intricate interaction between inner layer damage mechanisms and the deterioration of the interlaminar interface itself. The meso-modeling concept is thus useful in analyzing such a situation. In order to emphasize the value of Damage Mechanics in the prediction of delamination onset, let us consider the case in which damage phenomena are located in both layers and interfaces. A [03/±452/90]s EDT specimen under tension has been simulated. In the test, the delamination starts at the mid-plane (a 0°/0° interface oriented at 90°) at nearly 0.2% of longitudinal strain. After the delamination onset, the load still increases up to final rupture, due to the brittle fracture of fibers, while the delaminated area progressively expands from the free edges towards the central part of the specimen. Figure 7 shows the X-rayed damage map near the edges for such a specimen at about 0.4% of applied strain. The damage consists of both the delamination of the central interface (a 0°/0° interface) and transverse cracking limited by the delamination front. The onset of delamination has been correctly predicted and the comparison of delaminated area between computation (Figure 8) and the X-ray photograph (Figure 7) has proved to be a rather good one. In such a specimen, the damage occurring inside the layers can interact with delamination. It is shown in Figure 9 that the shear damage is not insignificant, especially near the free edges. This example demonstrates the necessity of including all of the damage mechanisms into the delamination analysis, even for quite simple specimens.

Figure 7 : X-ray photograph of a [03/±452/90]s EDT specimen at 0.41% of applied strain

0.0 0.4 0.8 1.0 Figure 8 : d3 damage indicator computed at the central interface (90°/90°) at 0.41% of longitudinal strain

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Pierre Ladevèze, Olivier Allix, Bernard Douchin, and David Lévêque

0.00 0

0.10

3

0.11

+45

2

-45

0.12 0.14

2

0.16

90

Figure 9 : Shear damage indicator: (d) computed inside the layers at the upper part of a [03/±452/90]s specimen

5.3 Holed plates in tension More reliable delamination tests would consist of those on laminated plates with a circular hole. In fact, for such specimens, the delamination crack initiation is reproducible and crack growth is often stable within a certain range. Also of interest in this test is the valuable information provided in terms of both the shape and size of the delaminated area, as revealed by means of X-ray photography. The difficulty herein is that, due to the complexity of the state of stresses, this test's interpretation requires complex computations. For such tests, a specialized software "DSDM" (Delamination Simulation by Damage Mechanics) has previously been developed [Allix 1992]23 with the intralaminar damage being incorporated into the analysis. In what follows, we will present an initial comparison between the experimental observations in the [03/±452/90]s laminate loaded in tension and the corresponding computation. Figure 10 displays the evolution of the X-rayed damage map near the hole for an increasing applied load. The initial damage, appearing at 55% of rupture (Figure 10a), is transverse cracking in 90°-plies near the hole, as well as matrix cracking in the 0°-plies both tangent to the hole and in the fiber direction, referred to as "splitting". Delamination alone begins at about 80% of rupture (Figure 10b). Just before rupture (Figure 10c), the delaminated area is always found to be located between those splittings developed in the 0°direction with a length of about two hole diameters. Micrographs performed have revealed (Figure 11) that the damage is well-advanced in several ways : splittings, transverse cracking not only in the 90°-plies but also in the ±45°-plies, multiple delamination at the 0°/+45°, ±45° (the most damaged) and -45°/90° interfaces. From the computational results, the splitting can be observed as shear damage in the 0°-layer (see Figure 12a). In fact, once the first 0°-fibers near the hole have cracked, the local load is then transferred by shear into the matrix at the adjacent fibers. The delaminated area computed in the ±45° interface is shown in Figure 12b as an example (the delaminated area corresponds to d3 = 1). Similarly, the other interfaces – except for the mid-plane – are found to be less delaminated.

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Pierre Ladevèze, Olivier Allix, Bernard Douchin, and David Lévêque

a) 55% of rupture load

b) 86% of rupture load

c) 99% of rupture load Figure 10 : X-ray photographs (x1.5) of a [03/±452/90]s-holed specimen (rupture load : 430 MPa)

Figure 11 : Micrograph in a section tangent to the hole of a [03/±452/90]s 92% of the rupture load

a) d indicator in the 0° layers

b) d3 indicator at the ±45° interface

specimen at

“d” value scale

Figure 12 : Damage maps computed in a [03/±452/90]s-holed specimen at the rupture load

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Pierre Ladevèze, Olivier Allix, Bernard Douchin, and David Lévêque

In order to achieve good comparative results, we have not, for the time being, used the Yc parameter identified previously (Table 1) which has led to the prediction of no delamination. Various explanations of this phenomenon are currently under review and progress is expected in the near future. 6. CONCLUSION A meso-model for laminated structures has been developed and described. In this model, the resistance to delamination can be characterized by the use of several material parameters. It has been shown that experimental results and finite element predictions exhibit rather strong correlation in standard Fracture Mechanics tests as well as in edge delamination tests. Moreover, an initial comparison between computation and a holed-specimen test in tension has revealed the advantage of applying such an approach, which is able to describe both innerlayer damage mechanisms and delamination growth. An extension to dynamics, using the proposed model, is in progress [Allix, Deü, 1997]24. Nonetheless, calculations performed with such a meso-model do lead to very significant computation times. One current challenge is to devise a more effective computational strategy and, in particular, to use parallel computers. REFERENCES [1] P. Ladevèze, “Towards a Fracture Theory”, Proceedings of the Third International Conference on Computational Plasticity, Part II, D.R.J. Owen, E. Oñate and E. Hinton Ed., Pineridge Press, Cambridge U.K., 1369-1400 (1992). [2] P. Ladevèze and E. Le Dantec, “Damage modeling of the elementary ply for laminated composites”, Composite Science and Technology, 43, 257-267 (1992). [3] O. Allix, P. Ladevèze and E. Vittecoq, “Modelling and identification of the mechanical behaviour of composite laminates in compression”, Composite Science and Technology, 51, 35-42 (1994). [4] O. Allix and P. Ladevèze, “Interlaminar interface modelling for the prediction of laminate delamination”, Composite Structures, 22, 235-242 (1992). [5] O. Allix and P. Ladevèze, “Damage mechanics of interfacial media : Basic aspects, identification and application to delamination”, Damage and Interfacial Debonding in Composites, Studies in Applied Mechanics, 44, Eds. Allen D. and Voyiadjis G., Elsevier, 167-88 (1996). [6] P. Ladevèze, “A damage computational method for composite structures”, J. Computer and Structure, 44 (1/2), 79-87 (1992). [7] O. Allix, A. Corigliano and P. Ladevèze, “Damage analysis of interlaminar fracture specimens”, Composite Structures, 31, 61-74 (1995). [8] O. Allix, P. Ladevèze, D. Lévêque and L. Perret, “Identification and validation of an interface damage model for delamination prediction”, Computational Plasticity, Eds. Owen D.R.J., Oñate E. and Hinton E., Barcelona, 1139-1147 (1997). [9] P. Ladevèze, “A damage Computational approach for Composites : Basic Aspects and Micromechanical Relations”, Computational Mechanics, 17, n° 1-2, 142-150 (1995). [10] P. Ladevèze, O. Allix, L. Gornet, D. Lévêque and L. Perret, “A computational damage mechanics approach of laminates : identification and comparison with experimental results”, Damage Mechanics in Engineering Materials, Eds. G. Z. Voyiajis, JW. W. Ju and J-L. Chaboche, Elsevier. [11] A.L. Highsmith and K.L. Reifsnider, “Stiffness reduction mechanism in composite material”, ASTM-STP 775, Damage in Composite Materials, A.S.T.M., 103-117 (1982).

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[12] R. Talreja, “Transverse cracking and stiffness reduction in composite laminates”, Journal of Composite Materials, 19, 355-375 (1985). [13] J.M. Whithney, “Experimental characterization of delamination fracture”, Interlaminar response of composite materials, Comp. Mat. Series, 5, Pagano N.J. Ed., 111-239 (1989). [14] F.W. Crossman and A.S.D. Wang, “The dependence of transverse cracking and delamination on ply thickness in graphite/epoxy laminates”, Damage in Composite Materials, ASTM-STP 775, K.L. Reifsnider Ed., 118-139 (1982). [15] O. Allix, D. Lévêque and L. Perret, “Interlaminar interface model identification and forecast of delamination in composite laminates”, Composite Science & Technology, Special Issue : 10th French National Colloquium on Composite Materials, to be published (1998). [16] P. Robinson and D.Q. Song, “A modified DCB specimen for mode I testing of multidirectional laminates”, Composite Science and Technology, 26, 1554-1577 (1992). [17] F.-X. De Charentenay, J.M. Harry, Y.J. Prel and M.L. Benzeggagh, “Characterizing the Effect of Delamination Defect by Mode I Delamination Test”, Effect of Defects in Composite Materials, ASTM-STP 836, 84-103 (1984). [18] L. Daudeville and P. Ladevèze, “A Damage Mechanics Tool for Laminate Delamination”, Journal of Composite Structures, 25, 547-555 (1993). [19] O. Allix, D. Lévêque and L. Perret, “On the identification of an interface damage model for the prediction of delamination initiation and growth”, First Int. Conf. on Damage and Failure of Interfaces, DFI-1, Sept. 21-24, 1997, Vienna, to be published. [20] D. Lévêque, “Analyse de la tenue au délaminage des composites stratifiés : identification d’un modèle d’interface interlaminaire”, PHD Thesis, École Normale Supérieure de Cachan, January 5, 1998. [21] L. Gornet, C. Hochard, P. Ladevèze and L. Perret, “Examples of delamination prediction by a damage computational approach”, First Int. Conf. on Damage and Failure of Interfaces, Vienna, Sept. 21-24, 1997, to be published. [22] T. K. O'Brien, “Characterisation of Delamination Onset and Growth in a Composite Laminate”, Damage in Composite Materials, Reifsnider K.L. Ed., ASTM-STP 775, 140-167 (1982). [23] O. Allix, “Damage analysis of delamination around a hole”, New Advances in Computational Structural Mechanics, P. Ladevèze and O.C. Zienkiewicz Eds., Elsevier Science Publishers B. V., 411-421 (1992). [24] O. Allix and J.-F. Deü, “Delay-damage modeling for fracture prediction of laminated composites under dynamic loading”, Engineering Transactions, 45, 29-46 (1997).

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