Comptes Rendus
2D axisymmetric X-FEM modeling of the Hertzian cone crack system
Comptes Rendus. Mécanique, Volume 341 (2013) no. 9-10, pp. 715-725.

Hertzian cone cracks are nowadays a scholarly case making it possible to understand fracture of materials. However, the simulation of this physical phenomenon is not trivial and most theoretical models lead to the prediction of cone crack angles different from those observed experimentally. In the past, finite-element models have been developed based on a re-meshing procedure to explain this difference successfully, but with some limitations due the algorithms used. In this paper, we propose to use the X-FEM method to model Hertzian cone crack propagation with a 2D axisymmetric approach. The effect of various numerical parameters, such as mesh size or time step, is investigated and it is shown that they do not have a great impact on the crack angle result. The analysis of the stress field induced leads us to understand the difference in terms of cone crack angle based on the pre-existing stress field and those experimentally observed. As a conclusion, X-FEM is very efficient to reproduce faithfully several characteristics of the Hertzian cone crack phenomenon in a very simple manner.

La génération de fissures hertziennes de forme conique par des contacts mécaniques apparaît souvent comme un cas dʼécole pour étudier la rupture des matériaux fragiles. Cependant, la simulation de ce phénomène physique nʼest pas triviale, car la plupart des modèles analytiques prédisent des angles de fissuration différents de ceux observés expérimentalement. Cette différence a pu être expliquée par des modélisations éléments finis reposant sur des algorithmes de remaillage très spécifiques, engendrant malheureusement un certain nombre de limitations pratiques. Dans cet article, nous proposons dʼutiliser la méthode des éléments finis étendus pour représenter ce phénomène. Lʼangle de propagation obtenu est en très bon accord avec les résultats de la littérature, et la différence observée avec les modèles analytiques peut sʼexpliquer par une modification du champ de contraintes au cours de la propagation. Nous montrons aussi que les résultats obtenus sont robustes, cʼest-à-dire quʼils ne dépendent pas ou très peu de paramètres numériques tels que la taille des éléments ou la longueur dʼextension de fissure. En conclusion, la technique X-FEM apparaît comme très efficace et suffisamment précise pour modéliser la rupture en cône de Hertz.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crme.2013.09.004
Keywords: Cone crack, Indentation, Brittle materials, X-FEM, Silicate glasses, Contact
Keywords: Fissuration en cône, Indentation, Matériaux fragiles, X-FEM, Verres silicatés, Contact mécanique

David Y. Tumbajoy-Spinel 1; Éric Feulvarch 2; Jean-Michel Bergheau 2; Guillaume Kermouche 1

1 École nationale supérieure des mines de Saint-Étienne, centre SMS, LGF UMR CNRS 5307, 158 cours Fauriel, 42023 Saint-Étienne cedex 2, France
2 Université de Lyon, ENISE, LTDS, UMR5513 CNRS, 58 rue Jean Parot, 42023 Saint-Etienne cedex 2, France
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David Y. Tumbajoy-Spinel; Éric Feulvarch; Jean-Michel Bergheau; Guillaume Kermouche. 2D axisymmetric X-FEM modeling of the Hertzian cone crack system. Comptes Rendus. Mécanique, Volume 341 (2013) no. 9-10, pp. 715-725. doi : 10.1016/j.crme.2013.09.004. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2013.09.004/

[1] C. Kocer; R.E. Collins Measurement of very slow crack growth in glass, J. Am. Ceram. Soc., Volume 84 (2006), pp. 2585-2593

[2] R. Mouginot; D. Maugis Fracture indentation beneath flat and spherical punches, J. Mater. Sci., Volume 20 (1985), pp. 4354-4376

[3] A.C. Fischer-Cripps Predicting Hertzian fracture, J. Am. Chem. Soc., Volume 76 (1997) no. 5, pp. 1096-1105

[4] C. Kocer An automated incremental finite element study of Hertzian cone crack growth, Finite Elem. Anal. Des., Volume 39 (2002), pp. 639-660

[5] P.D. Warren; D.A. Hills; D.N. Dai Mechanics of Hertzian cracking, Tribol. Int., Volume 28 (1995) no. 6, pp. 357-362

[6] J.J. Benbow Cone cracks in fused silica, Proc. Phys. Soc., Volume 75 (1960) no. 5, pp. 697-699

[7] F.C. Frank; B.R. Lawn On the theory of Hertzian fracture, Proc. R. Soc., Volume 299 (1967), pp. 291-306

[8] B.R. Lawn; T.R. Wilshaw; N.E.W. Hartley A computer simulation study of Hertzian cone crack growth, Int. J. Fract., Volume 10 (1974), pp. 1-16

[9] C. Kocer Using a Hertzian fracture system to measure crack growth data: A review, Int. J. Fract., Volume 121 (2003), pp. 111-132

[10] J.T. Hagan Cone cracks around Vickers indentations in fused silica glass, J. Mater. Sci., Volume 14 (1979), pp. 462-466

[11] K.L. Johnson; J.J. OʼConnor; A.C. Woodward The effect of the indenter elasticity on the Hertzian fracture of brittle materials, Proc. R. Soc. A, Volume 334 (1973), pp. 95-117

[12] K. Zeng; K. Breder; D.J. Rowcliffe The Hertzian stress field and formation of cone cracks – I. Theoretical approach, Acta Metall. Mater., Volume 40 (1992) no. 10, pp. 2595-2600

[13] J.L. Swedlow (1976), pp. 506-521 (ASTM Special Technical Publication, No. 601)

[14] A. Seweryn A non-local stress and strain energy release rate mixed mode fracture initiation and propagation criteria, Eng. Fract. Mech., Volume 59 (1998) no. 6, pp. 737-760

[15] C. Kocer; R.E. Collins The angles of Hertzian cone cracks, J. Am. Chem. Soc., Volume 81 (1998) no. 7, pp. 1736-1742

[16] N. Moës; J. Dolbow; T. Belytschko A finite-element method for crack growth without remeshing, Int. J. Numer. Methods Eng., Volume 46 (1999) no. 1, pp. 131-150

[17] E. Bechet; H. Minnebo; N. Moës; B. Burgardt Improved implementation and robustness study of the X-FEM for stress analysis around cracks, Int. J. Numer. Methods Eng., Volume 64 (2005) no. 8, pp. 1033-1056

[18] X. Tran; S. Geniaut Development and industrial applications of X-FEM axisymmetric model for fracture mechanics, Eng. Fract. Mech., Volume 82 (2012), pp. 135-157

[19] N. Moës; A. Gravouil; T. Belytschko Non-planar 3D crack growth by the extended finite element and level set. I. Mechanical model, Int. J. Numer. Methods Eng., Volume 53 (2002) no. 11, pp. 2549-2568

[20] M. Stolarska; D.L. Chopp; N. Moes; T. Belytschko Modelling crack growth by level sets in the extended finite element method, Int. J. Numer. Methods Eng., Volume 51 (2001), pp. 943-960

[21] E. Feulvarch; M. Fontaine; J.-M. Bergheau XFEM investigation of a crack path in residual stresses resulting from quenching, Finite Elem. Anal. Des., Volume 75 (2013), pp. 62-70

[22] N. Sukumar; N. Moës; T. Belytschko; B. Moran Extended finite element method for three dimensional crack modelling, Int. J. Numer. Methods Eng., Volume 48 (2000) no. 11, pp. 1549-1570

[23] N. Sukumar; D. Chopp; N. Moës; T. Belytschko Modeling holes and inclusions by level sets in the extended finite-element method, Comput. Methods Appl. Mech. Eng., Volume 190 (2001) no. 46, pp. 6183-6200

[24] J. Sethian Level Sets Methods and Fast Marching Methods, Cambridge University Press, 1999

[25] P. Burchard; L. Cheng; B. Merriman; S. Osher Motion of curves in three spatial dimensions using level set approach, J. Comput. Phys., Volume 170 (2001), pp. 720-741

[26] A. Gravouil; N. Moës; T. Belytschko Non-planar 3D crack growth by the extended finite element and level set. II. Level set update, Int. J. Numer. Methods Eng., Volume 53 (2002) no. 11, pp. 2569-2586

[27] E.H. Yoffe Stress fields of radial shear tractions applied to anelastic half-space, Philos. Mag. A, Volume 54 (1986), pp. 553-558

[28] G. Kermouche; E. Barthel; D. Vandembroucq; P. Dubujet Mechanical modelling of indentation-induced densification in amorphous silica, Acta Mater., Volume 56 (2008), pp. 3222-3228

[29] R. Lacroix; G. Kermouche; J. Teisseire; E. Barthel Plastic deformation and residual stresses in amorphous silica pillars under uniaxial loading, Acta Mater., Volume 60 (2012), pp. 5555-5566

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