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# Model of Crack Propagation with the Extended Finite Element Method

Nicolas Chevaugeon, Nicolas Moes

► To cite this version:

Nicolas Chevaugeon, Nicolas Moes. Model of Crack Propagation with the Extended Finite Element Method. 8th World Congress on Computational Mechanics (WCCM8), 5th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2008, Jun 2008, Venise (Italie), Italy). [https://hal.science/hal-04519910](https://hal.science/hal-04519910)

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# Model of Crack Propagation with the Extended Finite Element Method

Nicolas Chevaugeon, Nicolas Moes

► To cite this version:

Nicolas Chevaugeon, Nicolas Moes. Model of Crack Propagation with the Extended Finite Element Method. 8th World Congress on Computational Mechanics (WCCM8), 5th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2008, Jun 2008, Venice, Italy. hal-04519910

HAL Id: hal-04519910

https://hal.science/hal-04519910v1

Submitted on 25 Mar 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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Distributed under a Creative Commons Attribution 4.0 International License
MODEL OF CRACK PROPAGATION WITH THE EXTENDED FINITE
ELEMENT METHOD

* N. Chevaugeon¹ and N. Moës¹

¹ Institut de Recherche en Génie Civil et Mécanique, UMR CNRS 6183, École Centrale de Nantes,
B.P. 92101, 44321 Nantes Cedex 3, France
{nicolas.chevaugeon, nicolas.moes}@ec-nantes.fr

**Key Words: X-FEM, level-set method, crack propagation**

ABSTRACT

In this paper we will first review the combination of the level-sets and Extended Finite Element Method (X-FEM) techniques for the computation of the displacement field in the context of linear elasticity around cracks. Then we will discuss algorithms to extract stress intensity factors and the use of this results for crack propagation algorithm in fatigue context. Finally we will introduce promising techniques for quasi-static crack propagation problems.

The X-FEM and the level-set method constitute a good modeling combination for the simulation of
crack problems [1]. The X-FEM allow to enrich the discrete representation of the displacement field
with discontinuous and asymptotic solutions near the crack lip, using a partition of unity technique.
The geometry of the crack is represented by two level-sets. The first level-set represent the distance
to the crack surface. The second level-set has iso-surfaces orthogonal to the iso-surfaces of the first
one. The intersection of the zero-level of the two level-sets represent the crack lip. These two level-sets
define a coordinate system on the domain that permit to compute at each point, geometrical distance,
angle and local basis related to the crack geometry. This data simplify the enrichment procedure of the
X-FEM applied to crack and in particular the enrichment with term of the asymptotic expansion of the
elastic field around the singularity. It has been shown that with the right enrichment strategy and proper
elementary integration, optimal convergence rate can be obtain with the X-FEM/level-set combination
([2], [3], [4]).

Based on $G_{\theta}$ method, an algorithm have been developed to compute $J$ integrals or extract stress in-
tensity factors that take full advantage of the coordinate system defined by the two level set. We show
that our implementation give stable and convergent results for three-dimensional cases, with curved
crack-lip and non-planar crack surface.

In the context of fatigue analysis, the stress intensity factors are used to compute direction of crack
propagation using one of the literature given criteria (maximum hoop stress, maximum energy release
....). These data computed along the crack lip, are then exploited to extend a propagation speed for
the level-sets and then update the level-sets representing the crack. The level set update procedure is
of crucial importance, since the level-sets are, as was noted before, used for the accurate enrichment
# MODEL OF CRACK PROPAGATION WITH THE EXTENDED FINITE ELEMENT METHOD

* N. Chevaugeon¹ and N. Moës¹

¹ Institut de Recherche en Génie Civil et Mécanique, UMR CNRS 6183, École Centrale de Nantes,
B.P. 92101, 44321 Nantes Cedex 3, France
{nicolas.chevaugeon, nicolas.moes}@ec-nantes.fr

**Key Words:** X-FEM, level-set method, crack propagation

## ABSTRACT

In this paper we will first review the combination of the level-sets and Extended Finite Element Method (X-FEM) techniques for the computation of the displacement field in the context of linear elasticity around cracks. Then we will discuss algorithms to extract stress intensity factors and the use of this results for crack propagation algorithm in fatigue context. Finally we will introduce promising techniques for quasi-static crack propagation problems.

The X-FEM and the level-set method constitute a good modeling combination for the simulation of crack problems [1]. The X-FEM allow to enrich the discrete representation of the displacement field with discontinuous and asymptotic solutions near the crack lip, using a partition of unity technique. The geometry of the crack is represented by two level-sets. The first level-set represent the distance to the crack surface. The second level-set has iso-surfaces orthogonal to the iso-surfaces of the first one. The intersection of the zero-level of the two level-sets represent the crack lip. These two level-sets define a coordinate system on the domain that permit to compute at each point, geometrical distance, angle and local basis related to the crack geometry. This data simplify the enrichment procedure of the X-FEM applied to crack and in particular the enrichment with term of the asymptotic expansion of the elastic field around the singularity. It has been shown that with the right enrichment strategy and proper elementary integration, optimal convergence rate can be obtain with the X-FEM/level-set combination ([2], [3], [4]).

Based on $G_\theta$ method, an algorithm have been developed to compute $J$ integrals or extract stress intensity factors that take full advantage of the coordinate system defined by the two level set. We show that our implementation give stable and convergent results for three-dimensional cases, with curved crack-lip and non-planar crack surface.

In the context of fatigue analysis, the stress intensity factors are used to compute direction of crack propagation using one of the literature given criteria (maximum hoop stress, maximum energy release ....). These data computed along the crack lip, are then exploited to extend a propagation speed for the level-sets and then update the level-sets representing the crack. The level set update procedure is of crucial importance, since the level-sets are, as was noted before, used for the accurate enrichment
and extraction procedure [5]. We will discuss different algorithms, in term of speed and accuracy. The method is then applied to three dimensional industrials cases to predict the crack propagation path. The results are then related to a number of load cycle using a Paris law.

We then present first step toward the extension of our model to problems of crack propagation under quasi-static loading, using an implicit approach. We recast the model as a minimum principle : for a given dissipation we seek the level-sets representing the crack that minimize the potential energy. The minimization is implemented using a iterative level set propagation technique. Increments of dissipation pilot the global algorithm. the proposed method should be able to reproduce load-displacement curves with snap back behavior typical of the Brazilian test for example.

## REFERENCES

[1] N. Sukumar, N. Moës, B. Moran and T. Belytschko. "Extended finite element method for three-dimensional crack modelling". *Int. J. Numer. Meth. Engng.*, Vol 48, 1549–1570, 2000.

[2] F. L. Stazi, E. Budyn, J. Chessa and T. Belytschko. "An extended finite element method with higher-order elements for curved cracks". *Comput. Mech.*, Vol 31, 38–48, 2003.

[3] P. Laborde, J. Pommier, Y. Renard and M. Salaün. "High-order extended finite element method for cracked domains". *Int. J. Numer. Meth. Engng.*, Vol 64, 354–381, 2005.

[4] E. Béchet, H. Minnebo, N. Moës, B. Burgardt. "Improved implementation and robustness study of the X-FEM for stress analysis around cracks". *Int. J. Numer. Meth. Engng.*, Vol 64, 1033–1056, 2005.

[5] M. Duflot. "A study of the representation of cracks with level sets". *Int. J. Numer. Meth. Engng.*, Vol 70, 1261–1302, 2006.
and extraction procedure [5]. We will discuss different algorithms, in term of speed and accuracy. The method is then applied to three dimensional industrials cases to predict the crack propagation path. The results are then related to a number of load cycle using a Paris law.

We then present first step toward the extension of our model to problems of crack propagation under quasi-static loading, using an implicit approach. We recast the model as a minimum principle : for a given dissipation we seek the level-sets representing the crack that minimize the potential energy. The minimization is implemented using a iterative level set propagation technique. Increments of dissipation pilot the global algorithm. the proposed method should be able to reproduce load-displacement curves with snap back behavior typical of the Brazilian test for example.

## REFERENCES

[1] N. Sukumar, N. Moës, B. Moran and T. Belytschko. "Extended finite element method for three-dimensional crack modelling". *Int. J. Numer. Meth. Engng.*, Vol *48*, 1549–1570, 2000.

[2] F. L. Stazi, E. Budyn, J. Chessa and T. Belytschko. "An extended finite element method with higher-order elements for curved cracks". *Comput. Mech.*, Vol *31*, 38–48, 2003.

[3] P. Laborde, J. Pommier, Y. Renard and M. Salaün. "High-order extended finite element method for cracked domains". *Int. J. Numer. Meth. Engng.*, Vol *64*, 354–381, 2005.

[4] E. Béchet, H. Minnebo, N. Moës, B. Burgardt. "Improved implementation and robustness study of the X-FEM for stress analysis around cracks". *Int. J. Numer. Meth. Engng.*, Vol *64*, 1033–1056, 2005.

[5] M. Duflot. "A study of the representation of cracks with level sets". *Int. J. Numer. Meth. Engng.*, Vol *70*, 1261–1302, 2006.