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Research Papers

Gurson's Criterion and Its Derivation Revisited

[+] Author and Article Information
Jean-Baptiste Leblond

Institut Jean Le Rond d'Alembert,
UMR 7190,
Universite Pierre et Marie Curie (Paris VI),
Tour 65-55, 4 place Jussieu,
Paris 75005, France
e-mail: jbl@lmm.jussieu.fr

Léo Morin

Institut Jean Le Rond d'Alembert,
UMR 7190,
Universite Pierre et Marie Curie (Paris VI),
Tour 65-55, 4 place Jussieu,
Paris 75005, France
e-mail: leo.morin@ens-cachan.fr

In his thesis [1], Gurson proposed an explicit approximation of the yield surface S2, but it was not clear whether the corresponding reversibility domain was even convex for all possible values of the parameters, and he discarded the proposal in his final paper [2].

This of course assumes that the voids are initially spherical.

Because of this influence of Lode's angle, the numerical porosity rate does not vanish for an exactly zero triaxiality, but for a small one, the sign of which depends upon the Lode angle; since for Gurson's model this rate vanishes for an exactly zero triaxiality, this implies that for the numerical results, the ratio f·/f·Gurson behaves oddly for very small triaxialities. This behavior is not represented in Fig. 2 because it is of little interest, both f· and f·Gurson being very small anyway under such conditions.

GTN: Gurson–Tvergaard–Needleman.

Manuscript received October 23, 2013; final manuscript received November 25, 2013; accepted manuscript posted November 28, 2013; published online January 7, 2014. Editor: Yonggang Huang.

J. Appl. Mech 81(5), 051012 (Jan 07, 2014) (7 pages) Paper No: JAM-13-1441; doi: 10.1115/1.4026112 History: Received October 23, 2013; Revised November 25, 2013; Accepted November 28, 2013

This paper revisits Gurson's (Gurson, A., 1975, “Plastic Flow and Fracture Behavior of Ductile Materials Incorporating Void Nucleation, Growth, and Interaction,” Ph.D. thesis, Brown University, Rhode Island; Gurson, 1977, “Continuum Theory of Ductile Rupture by Void Nucleation and Growth: Part I—Yield Criteria and Flow Rules for Porous Ductile Media,” ASME J. Eng. Mater. Technol., 99, pp. 2–15) classical limit-analysis of a hollow sphere made of some ideal-plastic von Mises material and subjected to conditions of homogeneous boundary strain rate (Mandel (Mandel, J., 1964, “Contribution Theorique a l'Etude de l'Ecrouissage et des Lois d'Ecoulement Plastique,” Proceedings of the 11th International Congress on Applied Mechanics, Springer, New York, pp. 502–509) and Hill (Hill, R., 1967, “The Essential Structure of Constitutive Laws for Metal Composites and Polycrystals,” J. Mech. Phys. Solids, 15, pp. 79–95)). Special emphasis is placed on successive approximations of the overall dissipation, based on a Taylor expansion of one term appearing in the integral defining it. Gurson considered only the approximation based on the first-order expansion, leading to his well-known homogenized criterion; higher-order approximations are considered here. The most important result is that the correction brought by the second-order approximation to the first-order one is significant for the porosity rate, if not for the overall yield criterion. This bears notable consequences upon the prediction of ductile damage under certain conditions.

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References

Gurson, A., 1975, “Plastic Flow and Fracture Behavior of Ductile Materials Incorporating Void Nucleation, Growth, and Interaction,” Ph.D. thesis, Brown University, Providence, RI.
Gurson, A., 1977, “Continuum Theory of Ductile Rupture by Void Nucleation and Growth: Part I—Yield Criteria and Flow Rules for Porous Ductile Media,” ASME J. Eng. Mater. Technol., 99(1), pp. 2–15. [CrossRef]
Mandel, J., 1964, “Contribution Theorique a l'Etude de l'Ecrouissage et des Lois d'Ecoulement Plastique,” Proceedings of the 11th International Congress on Applied Mechanics, Springer, New York, pp. 502–509.
Hill, R., 1967, “The Essential Structure of Constitutive Laws for Metal Composites and Polycrystals,” J. Mech. Phys. Solids, 15, pp. 79–95. [CrossRef]
Rice, J., and Tracey, D., 1969, “On the Enlargement of Voids in Triaxial Stress Fields,” J. Mech. Phys. Solids, 17, pp. 201–217. [CrossRef]
Garajeu, M., 1995, “Contribution a l'Etude du Comportement Non Lineaire de Milieux Poreux Avec ou Sans Renfort,” Ph.D. thesis, Universite d'Aix-Marseille 2, Marseille, France.
Monchiet, V., Charkaluk, E., and Kondo, D., 2011, “A Micromechanics-Based Modification of the Gurson Criterion by Using Eshelby-Like Velocity Fields,” Eur. J. Mech. A/Solids, 30, pp. 940–949. [CrossRef]
Alves, J., Cazacu, O., and Revil-Baudard, B., 2013, “New Criterion Describing Combined Effects of Lode Angle and Sign of Pressure on Yielding and Void Evolution,” Proceedings of IDDRG 2013, Zurich, Switzerland, June 2–5, P.Hora, ed., ETH Zurich, Zurich, Switzerland, pp. 169–174.
Cazacu, O., Revil-Baudard, B., Lebensohn, R., and Garajeu, M., 2013, “On the Combined Effect of Pressure and Third Invariant on Yielding of Porous Solids With von Mises Matrix,” ASME J. Appl. Mech., 80(6), p. 064501. [CrossRef]
Huang, Y., 1991, “Accurate Dilatation Rates for Spherical Voids in Triaxial Stress Fields,” ASME J. Appl. Mech., 58(4), pp. 1084–1086. [CrossRef]
Benzerga, A., and Leblond, J., 2010, “Ductile Fracture by Void Growth to Coalescence,” Adv. Appl. Mech., 44, pp. 169–305.
Gologanu, M., Leblond, J., and Devaux, J., 1993, “Approximate Models for Ductile Metals Containing Non-Spherical Voids—Case of Axisymmetric Prolate Ellipsoidal Cavities,” J. Mech. Phys. Solids, 41, pp. 1723–1754. [CrossRef]
Madou, K., and Leblond, J., 2012, “A Gurson-Type Criterion for Porous Ductile Solids Containing Arbitrary Ellipsoidal Voids—II: Determination of Yield Criterion Parameters,” J. Mech. Phys. Solids, 60, pp. 1037–1058. [CrossRef]
Madou, K., and Leblond, J., 2013, “Numerical Studies of Porous Ductile Materials Containing Arbitrary Ellipsoidal Voids—I: Yield Surfaces of Representative Cells,” Eur. J. Mech. A/Solids, 42, pp. 480–489. [CrossRef]
Gologanu, M., 1997, “Etude de Quelques Problemes de Rupture Ductile des Metaux,” Ph.D. thesis, Universite Paris 6, Paris.
Tvergaard, V., and Needleman, A., 1984, “Analysis of Cup-Cone Fracture in a Round Tensile Bar,” Acta Metall., 32, pp. 157–169. [CrossRef]
Tvergaard, V., 1981, “lnfluence of Voids on Shear Band Instabilities Under Plane Strain Conditions,” Int. J. Fract., 17, pp. 389–407. [CrossRef]
Sovik, O., and Thaulow, C., 1997, “Growth of Spheroidal Voids in Elastic-Plastic Solids,” Fatigue Fract. Eng. Mater. Struct., 20, pp. 1731–1744. [CrossRef]
Pardoen, T., and Hutchinson, J., 2000, “An Extended Model for Void Growth and Coalescence,” J. Mech. Phys. Solids, 48, pp. 2467–2512. [CrossRef]
Molinari, A., and Mercier, S., 2001, “Micromechanical Modelling of Porous Materials Under Dynamic Loading,” J. Mech. Phys. Solids, 49, pp. 1497–1516. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Comparison of approximate criteria (f=0.01)

Grahic Jump Location
Fig. 2

Comparison of values of f·/f·Gurson: second-order approximation and exact (numerical) values (f=0.01)

Grahic Jump Location
Fig. 3

Comparison of values of f·/f·Gurson: second-order approximation and GTN model (with q=1.25)

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