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DAMAGE AND DISCRETE CRACK PROPAGATION

MODELLING

–

SOME RESULTS AND CHALLENGES FOR 2D AND 3D

CONFIGURATIONS

P.O. Bouchard1

1 Centre de Mise en Forme des Matériaux (CEMEF), Ecole des Mines de Paris, FRANCE.

ABSTRACT

In this paper we present a technique to model damage and fracture mechanics using remeshing in 2D and 3D

configurations. We use the finite element software FORGE2® and FORGE3® which can deal with elastic,

elastoplastic and elastic-viscoplastic materials in large deformation. Coupled damage models (Lemaître and

Gurson) have been implemented to model the progressive mechanical degradation of the material during its

deformation. Once damage reaches a critical value, fracture has to be modelled. 2D remeshing technique and crack

propagation criteria are presented to model automatic discrete crack propagation for different configurations.

Extension to 3D modelling of fracture is also discussed.

1 INTRODUCTION

Damage and Fracture Mechanics have been studied for many years now. During the last 50 years,

numerous different damage models and crack propagation techniques have been introduced and

implemented in different finite element software.

On a macro-scale, damage is characterized by a progressive loss of rigidity in the mechanical

behaviour of a material. From a micro-scale point of view, damage can be described by the three

well-known stages of nucleation of micro-voids, growth of these voids and finally coalescence

leading to fracture. Among the numerous models presented in the literature, the evolution of

damage can be coupled to the mechanical properties of the material or not. Once damage reaches a

critical value, a crack becomes initiated and has to be modelled.

Numerical modelling of crack propagation has been a real challenge for many years now. The

initial and more natural approach was based on a discrete crack propagation modelling approach

using remeshing techniques. To avoid numerical difficulties due to geometric topological changes

and remeshing during crack propagation, numerous different techniques have been introduced:

embedded crack models, mesh-free techniques, XFEM and so on. Each approach has its own

advantages and drawbacks, and I shall not in this paper discuss or compare each technique to the

others. Depending on the application, one approach can give better results than the others.

Sometimes, for instance in high shearing processes such as blanking, it is even sufficient to use a

simple “kill-element” technique.

After a brief description of the finite element software used for this study (§2), we introduce the

notion of damage and its coupling with the material’s mechanical behaviour (§3). Finally, discrete

crack propagation modelling is presented and applied to 2D and 3D configurations examples (§4).

2 FORGE2® AND FORGE3®

FORGE2® and FORGE3® have been developed to model large deformation of elastic,

elastoplastic and elastic-viscoplastic materials. They are based on a mixed velocity-pressure

formulation. The so-called Mini-element is used in FORGE3® (Arnold [1]). It is based on linear

isoparametric tetrahedra and a bubble function is added at element level in order to satisfy the

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Author manuscript, published in "ICF11 - International Conference on Fracture, Turin : Italy (2005)"

Brezzi/Babuska condition. The space discretization based on this element associated to the

incremental formulation of the virtual work principle lead to a set of discretized non-linear

equations. The well-known iterative Newton-Raphson linearization method is used. A one step

Euler scheme enables to compute the solution at time t+δt when the solution at time t is known.

FORGE2® and FORGE3® can both handle multimaterial structures using a nodal incremental

form of the penalty technique (Pichelin [2]). Besides this, an automatic adaptive remesher enables

to deal with large deformation without losing accuracy. The well-known Delaunay Triangulation

is used in 2D and a topological remeshing technique enables to deal with automatic 3D remeshing

(Gruau [3]).

3 DAMAGE MODELLING

Continuum damage mechanics is a constitutive theory that describes the progressive loss of

material integrity due to the propagation and coalescence of microcracks, microvoids, and similar

defects. Beyond a certain value of strain, void nucleation and void growth appear in the material:

this phenomenon, called damage, allows to model the ductile fracture of materials. When these

voids reach a critical size, they coalesce and give raise to instabilities or cracks propagation.

Damage models are generally based on the study of void growth, using different parameters such

as triaxiality, maximal principal stress, plastic strain and so on.

The first models to be used were uncoupled damage models, which means that the damage law

does not influence the mechanical properties of the material. The damage parameter is computed

using an integral of a strain and stress function, and its distribution can be computed in a post-

processing step. This approach is easy to introduce in a numerical software, but is quite unrealistic

because the damage evolution does not influence the material properties.

In order to better represent the evolution of damage in materials, coupled damage models have

been proposed. In this approach, damage and mechanical properties are directly linked and the

material fracture is modeled by a progressive decrease of the global response of the structure.

Contrary to the uncoupled approach, coupled damage models are quite difficult to introduce in

numerical software, but are closer to the physical phenomenon of micromechanical fracture of

ductile materials.

Some of these models use the notion of effective stress which represents the actual stress

transmitted by the bulk material between the microdefects. Another frequently used approach

consists in introducing a damage variable fv which represents the volumetric fraction of voids in

the material. The parameter fv is then used in the constitutive laws of the material and interacts

with the others state variables. The damage model of Tvergaard-Needleman, based on the model

introduced by Gurson3 belongs to these approaches.

The well-known Lemaître model [4] and Gurson-Tvergaard-Needleman model [5, 6] have been

implemented in FORGE3®. Details on these models can be found in [7].

Once damage reaches a critical value, fracture has to be taken into account. An easy way to

represent fracture is the so-called “kill element” technique. When the damage parameter reaches a

critical value inside an element, the element mechanical contribution to the stiffness matrix is set

to zero. Coupled with adaptive remeshing, this technique enables to model fracture easily in 3D

configurations. In Figure 1, a Lemaître damage model with adaptive remeshing and kill element

enables to model a blanking process with accuracy.

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Figure 1. Blanking simulation: isovalues of damage during fracture

However this technique involves a loss of volume during the simulation and the stress singularity

at the crack tip is not properly represented. In some cases, when the crack path has to be correctly

modelled, a more accurate technique is preferable.

4 CRACK PROPAGATION MODELLING

Our approach is based on discrete crack propagation, and we use automatic remeshing. Crack

initiation is based both on critical stress or critical damage. Once a crack is initiated the

propagation direction is computed using one of the following criteria: maximum circumferential

stress criterion, minimum strain energy density criterion or maximum strain energy release rate

criterion (Bouchard [8, 9]). In this last criterion, the strain energy release rate is computed using

the Gθ method which appears to be both efficient and accurate (Bouchard [10]). However, in high

shearing configurations, when mode II becomes predominant, such criteria are no more available.

In such cases, a maximum shear stress criterion has to be used (Bouchard [11]).

Once the crack propagation direction has been predicted, a new outline representing the crack

advance is added to the previous outline of the part. Then, a remeshing stage – based on the

Delaunay triangulation - is performed and the crack opens naturally due to tensile stresses applied

on its edges (Figure 2.):

Figure 2. propagation of a crack in a 2D mesh

The example in Figure 3, performed experimentally by Sumi, shows the growth of a crack starting

from a fillet in a structural member. The crack propagation depends on the welding residual

stresses and the bending stiffness of the structure. For the sake of simplicity, residual stresses are

not taken into account. The bending stiffness of the structure is modified by varying the size of the

bottom I-beam presented in figure 2 from 15mm to 315mm with an intermediate value of 115mm.

A linear elastic plane strain simulation is performed with a Young modulus and a Poisson ratio of

E=200GPa, and ν=0.3 respectively. The initial crack length is a0=5mm, and the maximum

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circumferential stress criterion is used to compute the crack path. Numerical simulations in figure

3 show the important influence of the bottom I-beam rigidity on the crack path:

Figure 3. Crack growth from a fillet and influence of the support beam rigidity on the crack path.

The following example concerns a pre-cracked part with an inclusion. In [3] we have studied the

propagation of a crack in a planar part with an off-center hole. It has been shown that the crack is

attracted by the hole since it creates a stress drop in this region.

In the present example, we replace the hole by an inclusion, and we study the influence of this

inclusion on the crack path. A rectangular part is pre-cracked and submitted to a tensile test. This

part contains an inclusion which may be more rigid or less rigid than the matrix. If Ematrix is the

Young modulus associated with the matrix, and Eincl the one associated with the inclusion, we

define R as the ratio : R=Ematrix/Eincl.

The numerical simulation is performed in plane strain, and the maximum strain energy release rate

is used. This example shows the ability of the software to deal with multimaterial applications.

Figure 4.a shows that for a soft inclusion – the inclusion is less rigid than the matrix – the crack is

still attracted by the inclusion. The crack reorientation is however less pronounced than the one

obtained with a hole.

Conversely, if the inclusion is more rigid (see figure 4.b), the crack is repulsed.

F=1

N

1

000

m

m

362

.

5 m

m

800

m

m

7

5 m

m

ρ

= 20mm

75 mm

5 mm

110 mm

Rigid beam (hI=315mm) Medium beam (hI=115mm) Flexible beam

h

I

= 15mm

,

115mm and

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Figure 4. Crack propagation in a part with a) a soft inclusion R=10, b) a hard inclusion R=0.1

In 3D configurations, discrete crack propagation modelling is far more complex from a topological

point of view since we have to deal with complex non planar surfaces representing crack faces.

Finite element software that manages to model 3D discrete crack propagation are extremely rare

(Carter [12], Schöllmann [13]). Our approach is still based on a modification of the part topology

coupled with surface and volume remeshing. Once the new crack front has been localized, the

surface mesh of the part is modified in order to insert new surface elements representing the new

faces of the crack advance. If necessary, a topological improvement of the surface mesh can be

performed at this stage. A volume remeshing of the new geometry is then performed. The 3D

topological remeshing technique used here is both efficient and well adapted for such complex

geometries [3]. Figure 5 shows a crack propagating in the same 3D T-shape structure presented in

figure 3. This example is purely intended for numerical feasibility purposes, since the crack

propagation direction is determined by the user in this example. However, it demonstrates the

ability of the 3D topological remesher to deal with complex 3D modifications such as crack

propagation.

Figure 5. 3D crack growth from a fillet

5 CONCLUSION

Recent advances in finite element software enable to model 3D complex forming processes or

structural analysis applications. In such simulations the use of accurate damage models and of

fracture modelling is sometimes necessary. Coupled damage models enable to couple the

evolution of damage with the mechanical behaviour of materials. Once a critical damage value is

reached, fracture has to be modelled explicitly. Our approach is based on discrete crack

propagation and automatic remeshing. In 2D, different crack propagation criteria have been

implemented and enable to model automatically crack propagation with accuracy. In 3D

Soft inclusion

a) Hard inclusion

b)

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configurations, discrete crack propagation is more complex. In some cases, a simple kill element

technique is accurate enough to model fracture. When the crack path becomes important, 3D

discrete crack propagation is needed. We have extended our 2D developments to 3D

configurations to model crack propagation. The first results are promising and have to be

generalized to more complex geometries.

6 REFERENCES

[1] Arnold, D.N., Brezzi, F. and Fortin, M., A stable finite element for Stokes equations,

Calcolo, 21, 337-344, 1984.

[2] Pichelin, E., Mocellin, K., Fourment, L. and Chenot, J.L., An application of a master-slave

algorithm for solving 3D contact problems between deformable bodies in forming processes,

European Journal of Finite Elements, 10, n°8, 857-880, 2001.

[3] Gruau, C. and Coupez, T., 3D tetrahedral, unstructured and anisotropic mesh generation

with adaptation to natural and multidomain metric to appear in Computational Methods in

Applied Mechanics and Engineering.

[4] Lemaître, J., A continuum damage mechanics model for ductile fracture, J. Engrg. Mat.

Techn., 107, 83-89, 1985.

[5] Gurson, A., Continuum theory of ductile rupture by void nucleation and growth : Part I-

Yield criteria and flow rules of porous ductile media, J. Engrg. Mat. Techn., 99, 2-15, 1977.

[6] Tvergaard, V. Material Failure by void growth to coalescence, Adv. in Appl. Mech., 27, 83-

151, 1990.

[7] Bouchard, P.O., Signorelli, J., Boussetta, R. and Fourment, L. Damage and Adaptive

Remeshing applied to 3D modeling of blanking and Milling, Computational Plasticity VII

(COMPLAS), Barcelona, 2003.

[8] Bouchard, P.O., Bay, F., Chastel, Y. and Tovena, I., Crack propagation using an advanced

remeshing technique, Comp. Meth. in Appl. Mech. and Engng, 189, 723-742, 2000.

[9] Bouchard, P.O., Bay, F., Chastel, Y., Numerical modelling of crack propagation: automatic

remeshing and comparison of different criteria, Comput. Methods Appl. Mech. Engrg. 192,

3887-3908, 2003.

[10] Bouchard, P.O., Bay, F., Chastel, Y., Gtheta Method Applied to Crack Propagation

Modelling, Proceedings of the Fifth World Congress on Computational Mechanics

(WCCM V), Vienna, Austria, Editors: Mang, H.A.; Rammerstorfer, F.G.; Eberhardsteiner, J.,

Publisher: Vienna University of Technology, Austria, ISBN 3-9501554-0-6,

http://wccm.tuwien.ac.at, July 7-12, 2002.

[11] Bouchard P.O., Bay, F., Chenot, J.L. and Hudin, O., Modeling of Sheet Metal Cutting by

Coupling Damage and Crack Propagation Theories, Simulation of Materials Processing:

Theory, Methods and Applications, Mori (ed.), 1001-1006, 2001.

[12] Carter, B.J. Wawrzynek, P.A., Ingraffea, A.R., Automated 3D Crack Growth Simulation,

Gallagher Special Issue of Int. J. Num. Meth. Engng 47, 229-253, 2000.

[13] Schöllmann, M., Fulland, M. and Richard, H.A., Development of a new software for

adaptive crack growth simulations in 3D structures, Engineering Fracture Mechanics,

Volume 70, Issue 2, 249-268, 2003.

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