III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering

C.A. Mota Soares et.al. (eds.) Lisbon, Portugal, 5–8 June 2006

ADAPTIVE SIMULATION OF COHESIVE INTERFACE DEBONDING FOR CRASH- AND IMPACT ANALYSES

M. Nossek1, M. Sauer1, K. Thoma1

1 Fraunhofer Institute for High-Speed Dynamics -Ernst-Mach-Institut-

Eckerstraße 4 D-79104 Freiburg

Germany

nossek@emi.fhg.de sauer@emi.fhg.de

Keywords: Initial Rigid Cohesive Zone Model, Mesh Adaptation, Dynamic Crack Propaga-tion.

Abstract. In many applications, e.g. crushing of CFRP structures in automotive applications, modelling of interface debonding is of crucial importance. Although the amount of energy ab-sorbed by the delamination process itself might be negligible, it changes the integrity of a cross section and might trigger specific failure modes of entire parts.

On way of modelling debonding is the use of cohesive zone models. Such models are currently being implemented in some commercial finite element codes in the form of either particular element formulations or special contact algorithms. A traction-separation-relation describes the crack opening behaviour. It comprises an initial elastic behaviour and an energy based damage formulation. In such models, the initial rigid connection before crack growth starts is not reproduced. Particularly when wave propagation perpendicular to possible crack areas shall be investigated, initial elastic models have some drawbacks. In this paper, an alternative adaptive initial rigid cohesive zone model is presented for the simulation of delamination be-tween material boundaries in three-dimensional bodies. The model contains on one hand al-gorithms for initiation and crack propagation, on the other hand procedures to handle the mesh adaptation when crack propagation has been identified.

The properties of the implemented model are analysed by comparison of numerical results with results of the commercial software ABAQUS, where initial elastic cohesive elements were used. The test case is an in-plane impact on a laminated plate.

M. Nossek, M. Sauer, K. Thoma

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1 INTRODUCTION

Because of their light weight capabilities, structures made of Carbon-Fiber-Reinforced-Polymers (CFRP) are increasingly used e.g. in aeronautic or automotive industry. It is shown in [10] that these quasi-brittle layered materials can absorb more energy per unit mass under low velocity impact than many metallic materials.

The difficulties in modeling the mechanical behavior of these materials are the different types of failure. On one hand, there are different types of in-plane damage (fiber failure, ma-trix cracking or crushing) in every single layer possible, on the other hand, separations (de-laminations) of connected layers can take place. Particularly under low velocity impact delamination determinates the energy adsorption capability of the structure. The energy which is dissipated through the delamination process itself might be small, but the change in cross section reduces the structural strength and might trigger specific failure modes of entire parts [10]. Thus for a physically realistic modeling of the energy dissipation of layered structures delaminations have to be described adequately.

For simulation of crack propagation, element formulations or contact algorithms based on a cohesive zone model with an initial elastic stiffness are more and more frequently used. These special models are used where possible cracks can take place (e.g. by physical weak interfaces or through experimental observation). This can lead to a great effort in mesh gen-eration, and due to the required initial stiffness might change the mechanical response of the whole structure.

To overcome the disadvantages of these models, so called initial-rigid cohesive finite ele-ments can be used, which can be combined with an adaptive remeshing.

This paper presents an initial-rigid cohesive model and its implementation including the required adaptive remeshing tool for the simulation of delamination in three-dimensional lay-ered structures. The model will be compared with existing initial-elastic interface formula-tions by simulating an in-plane impact on a CFRP-plate.

2 BASIC EQUATIONS The fundamental mathematical relations for describing structural dynamical problems are

the conversation equations for mass, energy and momentum.

fvvDtDe

fDtDv

vDtD

+∇−=

+⋅∇−=

⋅∇−=

:

1

ρσ

σρ

ρρ

where

forcesvolumetricfstressenergyevelocityvdensity

timet

σ

ρ (1)

The equations in (1) describe a combined initial – boundary – value problem for which the exact solution can be approximated by numerical discretization. In this work finite element (FE) approximation with an updated Langrangian formulation is used. Thus the conservation of mass is ensured by the formulation and for low speed of loading analyzed here, the effects on energy are negligible. The conversation of momentum is formulated in weak form [4].

∫∫∫ΓΩΩ

Γ+Ω∂

∂−=⎥

⎦

⎤⎢⎣

⎡Ω

τ

ττσρ dNdx

NadNN II

JII (2)

M. Nossek, M. Sauer, K. Thoma

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Eq. (2) is solved in integrative mater over the element domain Ω and on the body sur-rounding surface Γ. The left side of eq. (2) contains the accelerations of the nodes aJ and the consistent mass matrix [..] which is calculated from the element density ρ and the nodal shape functions Ni (i=I, J). As we use explicit time integration (in a so-called hydrocode), a lumped mass matrix is advantageous, which is a diagonal matrix and a subdivision of the element mass to all nodes. The left term on the right side of eq. (2) denotes the internal forces, which result from the integration of the element stress distribution σ over the deformed element do-main. The deformed element is represented by the derivation of the shape functions. The sec-ond term of eq. (2) denotes the external loading on the surface surrounding the body. In eq. (2) body forces are neglected.

With the internal nodal forces calculated in the first term on the right side of eq. (2) and the lumped mass matrix, the conversation of momentum can solved by using the 1. Newton law in eq. (3).

amF ⋅= mFa = (3)

The acceleration of each node is computed from the total internal force of all connected elements, divided by the node mass. With this acceleration the new position of the node at the end of the timestep is calculated and this is the initial configuration for the next calculation loop.

With an extension of this procedure and an automatic mesh adaptation this numerical method is extended in the following, such that crack initiation on material boundaries in lay-ered structures can be described.

3 COHESIVE ZONE MODEL APPROACH

In fracture mechanics, three different modes of crack opening are considered.

Figure 1: Different types of crack opening modes.

In figure 1 a Mode-I crack is generated by normal traction within the body and Mode-II and –III are shear generated failure modes.

One method for analyzing the crack propagation is the cohesives zone model. The origin of this method goes back to Dugdale [5] or Barenblatt [8]. It is assumed that a small zone in front of a crack tip exists where cohesive damage (e.g. softening plasticity, void growth or micro cracking) takes place. All these micromechanical failure mechanisms are described by a relation between the traction on the crack surfaces and their relative displacement, see figure 2.

Mode-I Mode-II Mode-III

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Figure 2: Cohesive model.

For crack propagation in the body damage is initiated by an interfacial strength and the damage evolution is described by reducing the traction acting on the crack surface by increas-ing separation. The crack is fully opened when the area under the traction-separation relation is equal to the fracture toughness.

For numerical simulations cohesive zone models are implemented in many research and commercial finite element codes, e.g. in special element formulations (cohesive elements [1], [7], [12], [16], [18]) or contact algorithms [17]. All these models are based on an initially elastic behavior, a stress based failure identification and an energy based damage formulation.

These kinds of models are the state of the art for modeling delamination processes or adhe-sive debonding and require a separate discretisation of possible crack zones. Particularly in layered structures with many potential delamination zones between every layer, the elastic behavior of the crack zone with the required initial elasticity can strongly change the effective stiffness of the whole system. For a simple one-dimensional problem with elastic bulk mate-rial and zero thickness initial elastic interfaces it is shown in [15] that the effective modulus is influenced by the interface stiffness with eq. (4)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+−=

EhkEE

ieff

1

11 , (4)

where E is the elastic modulus of the bulk material, ki is the initial elastic stiffness of the interface and h is the distance between two cohesive planes. The interface stiffness ki has to be chosen high enough such that the second term tends to zero. On the other hand, a high in-terface stiffness requires a very small time step when an explicit integration scheme is applied.

Furthermore, an artificial initial elastic interface stiffness does not properly describe the wave propagation through the thickness. Thus this kind of cohesive zone model is not gener-ally applicable in analysis, especially if wave propagation through the thickness is important.

As alternative, “initial-rigid” cohesive models can be used, which require an adaptive mesh changing when a crack propagates. Initial rigid cohesive models are used e.g. to model dy-namic crack propagation or cracks between grains in ceramics (see [3], [6], [9]).

4 A NEW INITIAL RIGID COHESIVE ZONE MODEL In this chapter we present a new initial-rigid cohesive zone model and the implementation

in the explicit dynamic FE-code SOPHIA [13]. The CFRP structures that shall be analyzed here consist of thin layers where cracks can originate between neighbored layers. Usually, these structures are modeled with shell elements. However, the local stress state with possible

cohesive zone crack tip

separation

cohesive traction

M. Nossek, M. Sauer, K. Thoma

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singularities in front of the crack tip can not be adequately described with these elements. Therefore we use a discretisation with solid elements.

In the following, we only consider cracks of Mode-I and Mode-II. In structures with a very small laminate thickness and large in-plane dimensions the Mode-III crack opening type does not need to be analyzed separately, it can be described by a Mode-II crack opening. Still, for the identification of the crack mechanical mode a unique direction definition is required. This and other required initializations are described in the following.

First, in the initialization routine material boundaries within the model are identified by marking every node which is connected to elements with different material identification cards as potential delamination node. For the identification of a unique side of the crack the connected elements are assigned to a numerical “top” or “down” side. Generally, every node can have its own crack material properties (delamination strength, critical energy release rate). In practice, all nodes in one delamination plane have identical material properties. In the sec-ond step of the initialization procedure all potential delamination planes on all surrounding surfaces (facets) of each element are detected. The facets of the 8-node volumetric elements used here have four delamination nodes. Facets on the same material boundary are identified, the normal direction of this plane (a unit vector) is added (for elements on the top side) or subtracted (for elements on the down side) to every node connected to this plane. As result of this summation every node has its own Mode-I direction normal vector nr . Furthermore, a quarter of the facet area is added to the reference delamination area which is summed up at every node.

Figure 3: Initialization process.

After the initialization process, all following algorithms are integrated in the incremental solution process. In every time step the Mode-I delamination direction and the area associated to every node is updated.

Within one time step, during the integration of the element stresses, the resulting internal forces for delamination nodes are summed up separately for the numerical “top” and “down” side. From these forces and the associated area an effective stress vector (traction) on the ma-terial interface is calculated, which is used for the identification of crack initiation. Therefore, this stress vector is subdivided into its normal (Mode-I) and a tangential (Mode-II) part by taking the scalar product with the Mode-I delamination direction nr .

nr

nr

nr

nr

nr

(a) initial

numerical model

(c) identificated

delamination no-

(b) transparent illustration

(d) defining Mode-I

delamination direc-tion and associated

M. Nossek, M. Sauer, K. Thoma

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Figure 4: Forces acting on material boundaries.

For initiation of delamination under pure Mode-I or Mode-II loading (figure 4, (a)) the de-lamination stress can be directly compared to Mode-I or Mode-II strength. For a general load-ing on the delamination nodes a quadratic interaction criterion, eq. (5) proposed by [11], is used.

12

0

2

0 ≥⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

II

II

I

I

σσ

σσ

(5)

The brackets • denote that only positive stress for Mode-I is accountd for. 0Iσ and 0

IIσ are the single crack mode strength values.

If eq. (5) is complied, an adaptation of the FE-mesh is executed. This means that a new node is generated with the same information as the delamination node, the connectivity of both nodes and connected elements is rebuilt, the nodal mass is adjusted and the contact sur-faces for the existing contact algorithm are extended.

After splitting a node, a damage model based on fracture energy is used, which gradually reduces the cohesive strength with increasing crack opening.

For pure Mode-I loading the constitutive traction separation relation is depicted in figure (5).

Mat_ID 1

Mat_ID 2

nrnr

Mat_ID 1

Mat_ID 2

nr

tr

nr

tr

M. Nossek, M. Sauer, K. Thoma

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-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6

Figure 5: Constitutive relation for Mode-I crack opening.

With increasing Mode-I crack opening, the cohesive force decreases, such that the area un-der the traction-separation relation correlates to the Mode-I critical energy release rate GIc. For the linear damage model the final opening Mode-I separation can be calculated with eq. (6).

∫=≤f

dGGc

δ

δδσ0

)( 0

2

I

IcfI

Gσ

δ⋅

= (6)

The final and the current Mode-I separation are used to evaluate the damage constitutive relation in eq. (7).

0

maxmax

max

maxmaxmax

),max(

0

,1

0,

IcohI

fIIIf

I

I

III

I

d

ddd

otherwise

damage

unloadingd

d

σσ

δδδδδ

δδδδ

⋅=

=

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

<<−

≤≤⋅

=

(7)

The cohesive force for the current separation δI can be calculated from the cohesive stress and the associated area of the delamination node. For a partially opened crack a linear unload-ing behavior is defined, as shown in figure 5. Furthermore a contact algorithm (eq. (8)) is used to prevent interpenetrations under compression.

( ) dredt Fn

txmF

rrr−=

∆∆

= 2 , dt

dtred mm

mmm+⋅

= (8)

Iδ

0I

σ

IcG

fIδ

Iσ

M. Nossek, M. Sauer, K. Thoma

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Eq. (8) is used to calculate forces for a given Mode-I interpenetration x∆ , the mass of the node on the “top” (t) and “down” (d) side and the current time step length t∆ . The force acts on the Mode-I delamination direction nr and ensures the interpenetration free position of the both nodes on the end of the time step.

The constitutive relation under pure Mode-II loading is shown in figure 6. An increasing Mode-II crack opening decreases the cohesive traction.

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-1 .6 -1 .5 -1 .4 -1 .3 -1 .2 -1 .1 -1 -0 .9 -0 .8 -0 .7 -0 .6 -0 .5 -0 .4 -0 .3 -0 .2 -0 .1 0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1 1 .1 1 .2 1 .3 1 .4 1 .5 1 .6

Figure 6: Constitutive relation for Mode-II crack opening.

In contrast to a Mode-I delamination, the delamination direction for Mode-II t

r is not

clearly defined. We use the Mode-I delamination direction nr and the relative node separation vector δ

r, see eq. 9, to get the direction for Mode-II.

( ) nnt rrrr××= δ (9)

However, the node separation direction calculated with eq. (9) is very sensitive to small movements of the nodes, especially when the crack opening is very small. Therefore, we use eq. (10) to get a more stable direction for Mode-II by interpolating between the direction of Mode-II forces (which are stable right after node split) and the node separation vector (which is stable in later stages).

( ) ( )ααδδ −⋅+⋅= 1tFIIII

rrr; 10 ≤=≤ f

II

II

δδ

α (10)

Here ( )FIIδr

means the Mode-II direction calculated from the node acting force, tr

the Mode-II direction calculated from the relative node separation (eq. (9)), and α is a weighting factor. The constitutive relation for a Mode-II crack opening is summarized in eq. (11).

IIδ

0IIσ

IIcG

fIIδ

Iσ

fIIδ

0IIσ

M. Nossek, M. Sauer, K. Thoma

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0maxmax

max

maxmaxmax

00

),max(

0

,1

0,

2)(

IIcohII

fIIIIIIf

II

II

IIIIII

II

II

IIcfIIc

d

ddd

otherwise

damage

unloadingd

d

GdGGf

σσ

δδδδδ

δδδδ

σδδδσ

δ

⋅=

=

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

<<−

≤≤⋅

=

⋅=⇒=≤ ∫

(11)

With the constitutive relations of eq. (7) and eq. (11) a pure Mode-I and pure Mode-II crack damage can be described. For a general case with a mixture of Mode-I and Mode-II crack opening the interaction formulation of Benzeggagh and Kenane (B-K) [14] in eq. (12) is used.

( )η

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+=III

IIIcIIcIcc GG

GGGGG (12)

With eq. (12) the fracture toughness of every Mixed-Mode state Gc can be calculated. Here GIc and GIIc are the Mode-I and Mode-II critical energy release rates, the quotient of GI and GII describes the current Mixed-Mode ratio and η is a material dependent interaction coeffi-cient. By applying the B-K Mixed-Mode formulation, the final separation is defined in eq. (13).

( )

( ) ( )20220

2000

2

2

1

,

0,1

2

III

IIIm

I

II

fII

IIcIIcIcom

fm

otherwise

GGG

σβσβσσσ

δδβ

δ

δβ

βσδ

η

⋅+

+⋅⋅=

=

⎪⎩

⎪⎨

⎧>

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

⋅−+=

(13)

Under compression, the final separation is the Mode-II final separation. Under positive Mode-I separation the final separation can be calculated from the Mixed-Mode ratio β, the Mixed-Mode dependent delamination initiation strength 0

mσ and the material dependent inter-action coefficient η. This value of the Mixed-Mode final separation is the input parameter for a general constitutive relation in eq. (14).

M. Nossek, M. Sauer, K. Thoma

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δσσ

δδδ

δδδδ

δδδδ

r⋅⋅=

=

+=

⎪⎪⎩

⎪⎪⎨

⎧

>−

≤⋅=

0

maxmax

22

max

maxmaxmax

),max(

,1

,

mcoh

IIIm

mmfm

m

mmm

m

d

ddd

damage

unloadingdd

(14)

Here, d is an “isotropic” damage parameter for both Mode-I and Mode-II separation. This damage formulation reduces the cohesive traction if the Mixed-Mode crack opening separa-tion δm increases. On unloading, the cohesive traction reduces from the current maximum value dmax to zero. The cohesive traction is acting on the direction of the separation, which is identified by the normalized crack opening vector δ

r. Figure 7 shows graphically the soften-

ing response for an interaction coefficient η=2.284.

Figure 7: Constitutive surface.

In figure 7, the effective stress for Mixed-Mode ratio dependent separations is represented. The Mixed-Mode ration is marked in the δI – δII plane. The associated effective stress is given by the intersection of the normal to the δI – δII and the surface. Lines marked the constitutive relations under pure Mode-I and Mode-II separation. Unloading is represented by a straight return to the origin.

IδIIδ

mδ

fmδ

fIδ

fIIδ

Mode-I

Mode-II

omσ

cohmσ

M. Nossek, M. Sauer, K. Thoma

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5 SIMULATION OF PLANE IMPACT ON LAMINATED PLATES

In [10], different types of specimens made of layered CFRP were tested. Delamination was identified as crucial failure mode for the energy adsorbing capability under impact. One of these specimens was a plane plate consisting of 12 0°-orienated layers, as illustrated in figure 8 left. The geometrical dimensions were 70x40x4 (mm). For the experimental research, the plates were sharpened on the side of impact to trigger a repeatable failure mechanism. The test equipment and a typical failed specimen with visible delamination cracks and the typical y-shaped splitting are illustrated in figure 8.

Figure 8: Test specimen, experimental configuration and a typical damaged specimen.

In the work presented here, the initiation and progress of delaminations is investigated with the new initial-rigid cohesive zone model (implemented in Sophia) and compared to results of simulations with ABAQUS [2] and the initial-elastic cohesive element (COH3D8) based on the formulation of Camanho and Dávila [16] available there.

Elastic properties

E1 99 GPa E2 9 GPa E3 6 GPa G12 9 GPa G23 9 GPa G31 5 GPa η 12 0.26 [-] η 23 0.28 [-] η13 0.44 [-]

Fracture properties σo

I 80 MPa σo

II 150 MPa GIc 580 J/m² GIIc 2000 J/m²

Table 1: Material properties of 0° sublaminate [10].

1

2

3

M. Nossek, M. Sauer, K. Thoma

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For the analysis of delamination initiation, all in plane failure modes are deactivated. Thus a comparison with experimental results is not possible, but this is the aim of future work. In table 1 the elastic and fracture mechanical properties [10] of a homogenized 0°-layer are given.

In the research code Sophia, a brick element mesh is used. Full integrated elements are used to avoid hourglassing effects, an averaging of the hydrostatic stress state (pressure-averaging) is applied to minimize locking effects. Additionally, fully integrated elements al-low a discretisation with only one element over a layer thickness (12 elements on laminate thickness) for a rough description of bending stiffness of a single layer. With brick elements, the three-dimensional stress state is represented adequately, and the measured orthotropic ma-terial properties can directly be used.

The model for the commercial code ABAQUS is built with shell elements that model the constitutive behavior of the single layers (12 layers) and cohesive zone elements (COH3D8) that describe the out of plane constitutive behavior between neighbored layers. In order to model the correct out of plane stiffness, the of out-of-plane material properties of direction 3 are used for the cohesive elements.

For the numerical simulations the same in-plane mesh density 2x2 (mm) is used. In figure 9, the numerical models are illustrated, the element types are listed and the model size is given.

Sophia ABAQUS

Element types

Full integrated 8-node contin-uum element with pressure

averaging

Layer: Shell (S4R)

Interfaces: Cohesive Elements (COH3D8)

Number of Elements

35x20x12 = 8400

Σ = 8400

Layer: 12x35x20 = 8400 Interface: 11x35x20=7700

Σ = 16100

DOF 29484 58968 Figure 9: Numerical models of test specimen.

M. Nossek, M. Sauer, K. Thoma

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As can be seen in figure 9, the number of elements in Sophia is about half the number of elements for ABAQUS. In the third column a detail of the ABAQUS numerical model is shown with the twelve shell layers and the interface elements. On the left edge some cohesive elements are hidden.

Two impact velocities have been simulated. In figure 10, boundary force (on the bottom of the specimen) versus displacement of the impacted body are shown for low (1m/s) and high (10m/s) loading speed. The simulations shows a rapidly decreasing force, which comes from a rapidly growing delaminated area until the boundary is reached and a subsequent stability like failure (buckling) of the remaining structure. In both simulations, delamination is initiated on the same material boundary next to the two layers in the central plane of the plate.

-20000

-10000

0

10000

20000

30000

40000

50000

0 0.05 0.1 0.15 0.2 0.25 0.3Displacement [mm]

Forc

e [N

]

Sophia - initial rigid interfaceAbaqus - initial elastic interfaceAbaqus: crack

propagation initiatedSophia: crack propagation initiated

-40000

-20000

0

20000

40000

60000

80000

100000

120000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Displacement [mm]

Forc

e [N

]

Sophia - initial rigid interfaceAbaqus - initial elastic interface

Figure 10: Simulation of plane impact on CFRP-plates, with only out of plane failure modes.

Despite of the different ways of modeling, the results of both simulations are very similar under low velocity. However, large differences concerning the deformation of the structure exist. The initiation of delamination in both simulations takes place on different states of de-formation, these points are marked in figure 10, left. In Sophia the crack grows until it reaches the boundary, in ABAQUS, the crack stops after three quarter of specimen height. Addition-ally the experimentally observed y-shaped splitting of the structure is not visible in ABAQUS. In Sophia, this behaviour is reproduced, and this also triggers the crack propagation until the bottom. In ABAQUS the initiation of delamination is not only on the side of the impact, but delamination also starts from the middle of the structure.

Under high velocity, this behaviour also occurs. Additionally, in ABAQUS the impactor gets contact to the outer delaminated parts because of the smaller y-splitting deformation. This leads to a strong increase of the force at a displacement of 0.4 mm. Therefore, the energy dissipated in the deformation process is much bigger.

The out-of-plane stress distribution within the structure is also very different for both mod-els. In figure 11 the distributions of the damage initiating variable and of the transverse nor-mal and shear stress are shown.

sm

impactv 1= sm

impactv 10=

M. Nossek, M. Sauer, K. Thoma

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Damage initiation variable

[-] – [0..1]

Transverse normal stress distribution

[N/mm²]

Transverse shear stress distribution

[N/mm²]

Initial elastic cohesive

zone model

(ABAQUS)

Initial rigid cohesive

zone model

(Sophia) 1.00.80.60.40.20.0

15.00.0

-15.0-30.0-45.0

140.0120.0100.0

80.060.040.020.00.0

Figure 11: Damage initiation variable, out-of-plane normal and shear stress for ABAQUS and Sophia before delamination starts.

In Sophia, a localized stress distribution is visible. For the transverse normal stress a small compression zone exists next to the place of impact and a subsequent zone of tension. The out-of-plane shear stress is localized on the material boundary next to the two layers in the middle of the structure. Because of the shape of the specimen and the in-plane loading, de-lamination is initiated under shear in Mode-II. In the simulations with the initial elastic cohe-sive zone element in ABAQUS, this localized stress distribution can not be seen, the out-of-plane stress distribution is nearly homogeneous. The reason for this is the decoupling of in-plane and out-of-plane constitutive behavior with inextensible shell elements and cohesive elements. The three-dimensional stress state is not described correctly.

In the simulation with the initial-rigid cohesive zone model the simulation of the out of plane stress distribution is physically correct, because brick elements are used and the trans-verse shear strain behavior is adequately represented. This can not be achieved with the shell/cohesive element model.

6 CONCLUSION In this paper a new initial rigid cohesive zone model was presented, which was used for the

simulation of crack propagation in layered structures. A crack is initiated by analyzing the

M. Nossek, M. Sauer, K. Thoma

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forces on nodes at material boundaries. The forces are taken from the stress integration of the surrounding solid elements. The crack propagation is described by an automatic mesh adapta-tion and an energy based damage formulation. The B-K [14] Mixed-Mode interaction formu-lation is used for interaction of Mode-I and Mode-II crack propagation.

The new model is implemented in the research code Sophia and was compared with a model with shells and initial-elastic cohesive elements in ABAQUS by simulating in-plane impact on laminated plates. Only out-of-plane failure mechanisms were activated to ensure comparability of the models. It was shown that the initial-rigid cohesive zone model gives more efficiency (smaller model size) and a better representation of the three-dimensional stress state which leads to a more physical delamination initiation. In future work the test case will be simulated with additionally in-plane failure mechanisms so that the results can be di-rectly compared with experimental results.

REFERENCES [1] ABAQUS v6.5, Theory manual – Cohesive elements, 2005.

[2] ABAQUS, Version 6.5, Documentation.

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