Research Papers

On Void Shape and Distribution Effects on Void Coalescence

[+] Author and Article Information
Pierre-Olivier Barrioz, Benoît Tanguy

CEA Saclay,
Université Paris-Saclay,
DEN Service d'Études des Matériaux Irradiés,
Gif-sur-Yvette 91191, France

Jérémy Hure

CEA Saclay,
Université Paris-Saclay,
DEN Service d'Études des Matériaux Irradiés,
Gif-sur-Yvette 91191, France
e-mail: jeremy.hure@cea.fr

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received May 18, 2018; final manuscript received September 21, 2018; published online October 18, 2018. Assoc. Editor: A. Amine Benzerga.

J. Appl. Mech 86(1), 011006 (Oct 18, 2018) (9 pages) Paper No: JAM-18-1298; doi: 10.1115/1.4041548 History: Received May 18, 2018; Revised September 21, 2018

Void coalescence is known to be the last microscopic event of ductile fracture in metal alloys and corresponds to the localization of plastic flow in between voids. Limit-analysis has been used to provide coalescence criteria that have been subsequently recast into effective macroscopic yield criteria, leading to models for porous materials valid for high porosities. Such coalescence models have remained up to now restricted to cubic or hexagonal lattices of spheroidal voids. Based on the limit-analysis kinematic approach, a methodology is first proposed to get upper-bound estimates of coalescence stress for arbitrary void shapes and lattices. Semi-analytical coalescence criteria are derived for elliptic cylinder voids in elliptic cylinder unit cells for an isotropic matrix material, and validated through comparisons to numerical limit-analysis simulations. The physical application of these criteria for realistic void shapes and lattices is finally assessed numerically.

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Puttick, K. E. , 1959, “ Ductile Fracture in Metals,” Philos. Mag., 4(44), pp. 964–969. [CrossRef]
Pardoen, T. , and Hutchinson, J. W. , 2000, “ An Extended Model for Void Growth and Coalescence,” J. Mech. Phys. Solids, 48(12), pp. 2467–2512. [CrossRef]
Benzerga, A. A. , and Leblond, J.-B. , 2010, “ Ductile Fracture by Void Growth to Coalescence,” Adv. Appl. Mech., 44, pp. 169–305. [CrossRef]
Pineau, A. , Benzerga, A. A. , and Pardoen, T. , 2016, “ Failure of Metals I: Brittle and Ductile Fracture,” Acta. Mater., 107, pp. 424–483. [CrossRef]
Besson, J. , 2010, “ Continuum Models of Ductile Fracture: A Review,” Int. J. Damage Mech., 19(1), pp. 3–52. [CrossRef]
Gurson, A. , 1977, “ Continuum Theory of Ductile Rupture by Void Nucleation and Growth—Part I: Yield Criteria and Flow Rules for Porous Ductile Media,” J. Eng. Mat. Technol., 99(1), pp. 2–15. [CrossRef]
Rousselier, G. , 1981, “ Finite Deformation Constitutive Relations Including Ductile Fracture Damage,” Three-Dimensional Constitutive Relations and Ductile Fracture, North Holland, Amsterdam, The Netherlands.
Ponte Castañeda, P. , 1991, “ The Effective Mechanical Properties of Nonlinear Isotropic Composites,” J. Mech. Phys. Solids, 39(1), pp. 45–71. [CrossRef]
Gologanu, M. , Leblond, J.-B. , Perrin, G. , and Devaux, J. , 1997, “ Recent Extension of Gurson's Model for Porous Ductile Metals,” Continuum Micromechanics (CISM Lectures Series), Springer, Vienna, Austria, pp. 61–130.
Madou, K. , and Leblond, J.-B. , 2012, “ A Gurson-Type Criterion for Porous Ductile Solids Containing Arbitrary Ellipsoidal Voids—I: Limit-Analysis of Some Representative Cell,” J. Mech. Phys. Solids, 60(5), pp. 1020–1036. [CrossRef]
Madou, K. , and Leblond, J.-B. , 2012, “ A Gurson-Type Criterion for Porous Ductile Solids Containing Arbitrary Ellipsoidal Voids—II: Determination of Yield Criterion Parameters,” J. Mech. Phys. Solids, 60(5), pp. 1037–1058. [CrossRef]
Benzerga, A. A. , and Besson, J. , 2001, “ Plastic Potentials for Anisotropic Porous Solids,” Eur. J. Mech. A/Solids, 20(3), pp. 397–434. [CrossRef]
Han, X. , Besson, J. , Forest, S. , Tanguy, B. , and Bugat, S. , 2013, “ A Yield Function for Single Crystals Containing Voids,” Int. J. Solids Struct., 50(14–15), pp. 2115–2131. [CrossRef]
Paux, J. , Morin, L. , Brenner, R. , and Kondo, D. , 2015, “ An Approximate Yield Criterion for Porous Single Crystals,” Eur. J. Mech. A/Solids, 51, pp. 1–10. [CrossRef]
Morin, L. , Leblond, J. , and Kondo, D. , 2015, “ A Gurson-Type Criterion for Plastically Anisotropic Solids Containing Arbitrary Ellipsoidal Voids,” Int. J. Solids Struct., 77, pp. 86–101. [CrossRef]
Mbiakop, A. , Constantinescu, A. , and Danas, K. , 2015, “ An Analytical Model for Porous Single Crystals With Ellipsoidal Voids,” J. Mech. Phys. Solids, 84, pp. 436–467. [CrossRef]
Song, D. , and Ponte Castañeda, P. , 2017, “ A Finite-Strain Homogenization Model for Viscoplastic Porous Single Crystals—I: Theory,” J. Mech. Phys. Solids, 107, pp. 560–579. [CrossRef]
Thomason, P. F. , 1968, “ A Theory for Ductile Fracture by Internal Necking of Cavities,” J. Inst. Met., 96, p. 360.
Thomason, P. F. , 1985, “ Three-Dimensional Models for the Plastic Limit-Loads at Incipient Failure of the Intervoid Matrix in Ductile Porous Solids,” Acta. Metall., 33(6), pp. 1079–1085. [CrossRef]
Benzerga, A. A. , Besson, J. , and Pineau, A. , 2004, “ Anisotropic Ductile Failure—Part I: Experiments,” Acta Mater., 52(15), pp. 4623–4638. [CrossRef]
Benzerga, A. A. , 2002, “ Micromechanics of Coalescence in Ductile Fracture,” J. Mech. Phys. Solids, 50(6), pp. 1331–1362. [CrossRef]
Weck, A. , and Wilkinson, D. , 2008, “ Experimental Investigation of Void Coalescence in Metallic Sheets Containing Laser Drilled Holes,” Acta Mater., 56(8), pp. 1774–1784. [CrossRef]
Weck, A. , Wilkinson, D. , Maire, E. , and Toda, H. , 2008, “ Visualization by X-Ray Tomography of Void Growth and Coalescence Leading to Fracture in Model Materials,” Acta Mater., 56(12), pp. 2919–2928. [CrossRef]
Scheyvaerts, F. , Onck, P. , Tekoğlu, C. , and Pardoen, T. , 2011, “ The Growth and Coalescence of Ellipsoidal Voids in Plane Strain Under Combined Shear and Tension,” J. Mech. Phys. Solids, 59(2), pp. 373–397. [CrossRef]
Fabrègue, D. , and Pardoen, T. , 2008, “ A Constitutive Model for Elastoplastic Solids Containing Primary and Secondary Voids,” J. Mech. Phys. Solids, 56(3), pp. 719–741. [CrossRef]
Tekoğlu, C. , Leblond, J. , and Pardoen, T. , 2012, “ A Criterion for the Onset of Void Coalescence Under Combined Tension and Shear,” J. Mech. Phys. Solids, 60(7), pp. 1363–1381. [CrossRef]
Benzerga, A. A. , and Leblond, J.-B. , 2014, “ Effective Yield-Criterion Accounting for Microvoid Coalescence,” ASME J. Appl. Mech., 81(3), p. 031009. [CrossRef]
Morin, L. , Leblond, J.-B. , and Benzerga, A. A. , 2015, “ Coalescence of Voids by Internal Necking: Theoretical Estimates and Numerical Results,” J. Mech. Phys. Solids, 75, pp. 140–158. [CrossRef]
Torki, M. E. , Benzerga, A. A. , and Leblond, J.-B. , 2015, “ On Void Coalescence Under Combined Tension and Shear,” ASME J. Appl. Mech., 82(7), p. 071005. [CrossRef]
Torki, M. E. , Tekoğlu, C. , Leblond, J.-B. , and Benzerga, A. A. , 2017, “ Theoretical and Numerical Analysis of Void Coalescence in Porous Ductile Solids Under Arbitrary Loadings,” Int. J. Plast., 91, pp. 160–181. [CrossRef]
Hure, J. , and Barrioz, P. , 2016, “ Theoretical Estimates for Flat Voids Coalescence by Internal Necking,” Eur. J. Mech. A/Solids, 60, pp. 217–226. [CrossRef]
Keralavarma, S. , and Chockalingam, S. , 2016, “ A Criterion for Void Coalescence in Anisotropic Ductile Materials,” Int. J. Plast., 82, pp. 159–176. [CrossRef]
Morin, L. , 2015, “ Influence Des Effets de Forme et de Taille Des Cavités, et de L'anisotropie Plastique Sur la Rupture Ductile,” Ph.D. thesis, Université Pierre et Marie Curie, Paris, France.
Gallican, V. , and Hure, J. , 2017, “ Anisotropic Coalescence Criterion for Nanoporous Materials,” J. Mech. Phys. Solids, 108, pp. 30–48. [CrossRef]
Cao, T. , Mazière, M. , Danas, K. , and Besson, J. , 2015, “ A Model for Ductile Damage Prediction at Low Stress Triaxialities Incrorporating Void Shape Change and Void Rotation,” Int. J. Solids Struct., 63, pp. 240–263. [CrossRef]
Navas, V. , Bernacki, M. , and Bouchard, P. , 2018, “ Void Growth and Coalescence in a Three-Dimensional Non-Periodic Void Cluster,” Int. J. Solids Struct., 139–140, pp. 65–78. [CrossRef]
Miehe, C. , Apel, N. , and Lambrecht, M. , 2002, “ Anisotropic Additive Plasticity in the Logarithmic Strain Space: Modular Kinematic Formulation and Implementation Based on Incremental Minimization Principles for Standard Materials,” Comput. Methods Appl. Mech. Eng., 191(47–48), pp. 5383–5425. [CrossRef]
Koplik, J. , and Needleman, A. , 1988, “ Void Growth and Coalescence in Porous Plastic Solids,” Int. J. Solids Struct., 24(8), pp. 835–853. [CrossRef]
Tekoğlu, C. , 2015, “ Void Coalescence in Ductile Solids Containing Two Populations of Voids,” Eng. Fract. Mech., 147, pp. 418–430. [CrossRef]
Moulinec, H. , and Suquet, P. , 1998, “ A Numerical Method for Computing the Overall Response of Nonlinear Composites With Complex Microstructure,” Comput. Methods Appl. Mech. Eng., 157(1–2), pp. 69–94. [CrossRef]
Paux, J. , Brenner, R. , and Kondo, D. , 2018, “ Plastic Yield Criterion and Hardening of Porous Single Crsytals,” Int. J. Solids Struct., 132–133, pp. 80–95. [CrossRef]
Helfer, T. , Michel, B. , Proix, J.-M. , Salvo, M. , Sercombe, J. , and Casella, M. , 2015, “ Introducing the Open-Source Mfront Code Generator: Application to Mechanical Behaviors and Materials Knowledge Management Within the PLEIADES Fuel Element Modelling Platform,” Comput. Math. Appl., 70(5), pp. 994–1023. [CrossRef]


Grahic Jump Location
Fig. 1

Top: Slice views in the e¯2−e¯3 plane of the porous unit cells used in the simulations, with different anisotropies of voids distribution. Bottom: Evolution of macroscopic stress (through volume averaging) in the principal loading direction Σ11 (solid lines) and transverse macroscopic deformation gradients F22 (dotted lines) and F33 (dashed lines) as a function of F11.

Grahic Jump Location
Fig. 2

Reference cylindrical unit cell Ωref and deformed unit cell Ω. Coalescence deformation mode corresponds to localized plastic flow in a coalescence band Ωcoa associated with (almost) rigid body motion of the outer parts Ω\Ωcoa, thus to uniaxial straining conditions D=D33e¯3⊗e¯3.

Grahic Jump Location
Fig. 3

Elliptic cylinder unit cell with coaxial elliptic cylinder void considered in this study. The principal axes of the mechanical loading are assumed to be the same of the ones of the unit cell (and void).

Grahic Jump Location
Fig. 4

One-eighth of the typical periodic unit cell with cylindrical void used for FFT simulations to assess numerically coalescence stress. Three different constitutive equations are used: zero rigidity for the void, elasto-plastic von Mises plasticity, and fictive elastic material, respectively. Macroscopic strain is imposed: E=E33e¯3⊗e¯3. Macroscopic coalescence stress Σ33 is computed through volume averaging over the white and blue regions only.

Grahic Jump Location
Fig. 5

(a)–(e) Coalescence stress for elliptic cylinder unit cells with elliptic cylinder voids, as a function of the parameter α, for various values of W1 and χ1. Solid lines correspond to Eq. (35), squares to numerical results. (f)–(h) Comparisons of the analytical and numerical strain rate fields (arbitrary units). Numerical results are taken at an height z = h.

Grahic Jump Location
Fig. 6

Hexagonal-type lattices of ellipsoidal voids. Unit cells in red solid lines are used to perform numerical simulations, and the results are compared to predictions from the elliptic cells inscribed in the Voronoi cells. Unit cells used for FFT simulations are shown on right and left sides (see Fig. 4 caption for details about the colors).

Grahic Jump Location
Fig. 7

(a)–(d) Coalescence stress for hexagonal-type lattices of ellipsoidal voids as a function of the parameter α, for various values of W1 and χ1. Squares correspond to numerical results, solid lines to Eq. (35) considering the elliptic unit cell inscribed in the Voronoi cell of the void. (b) Evolution of the equivalent strain rate field taken at z = h/2 obtained with numerical simulations as the parameter α decreases. Arbitrary units.



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