. B. Fig, 1 ? Comparaison des courants obtenus par Gourdin (1989) et par notre modélisation 1D. Un bon accord jusqu'à t = 22 µs est observé. Au-delà, en raison des essais avec charge creuse de Gourdin, des différences notoires existent naturellement, 1989.

A. C. Traitement and . Images-d-'expansion-d-'anneaux-distances-vraies, Une résolution spatiale supérieure à celle de la caméra Imacon

. L. Sera-peut-Être-nécessaire, objectif de cette partie est essentiellement de montrer la faisabilité d'une telle procédure. (a) Image de l'anneau en expansion (b) Traitement 1 : extraction de l'anneau (c) Traitement 2 : évaluation des contours Fig. C.1 ? Traitement d'une image d'expansion d'anneau obtenue grâce à la caméra Imacon 200. Le premier traitement revient à extraire de l'image l'anneau

M. Bibliographie-altynova, X. Hu, and G. S. Daehn, Increased ductility in high velocity electromagnetic ring expansion, Met. and Mat. Trans. 27A, pp.1837-1844, 1996.

R. Becker, Ring fragmentation predictions using the Gurson model with material stability conditions as failure criteria, International Journal of Solids and Structures, vol.39, issue.13-14, pp.3555-3580, 2002.
DOI : 10.1016/S0020-7683(02)00170-1

G. Besnard, J. Lagrange, F. Hild, S. Roux, and C. Voltz, Characterization of Necking Phenomena in High-Speed Experiments by Using a Single Camera, EURASIP Journal on Image and Video Processing, 2010.
DOI : 10.1016/S0143-8166(98)00052-9

URL : https://hal.archives-ouvertes.fr/hal-00497354

J. Bishop and O. Strack, A statistical method for verifying mesh convergence in Monte Carlo simulations with application to fragmentation, International Journal for Numerical Methods in Engineering, vol.150, issue.11, pp.279-306, 2011.
DOI : 10.1007/978-1-4612-0871-6

P. Chassaing, Turbulences en mécanique des fluides. Collection Polytech, 2000.

P. C. Chou, J. Carleone, C. A. Tanzio, and R. D. Ciccarelli, Shaped charge jet breakup measurements and surface instability calculations, 1977.
DOI : 10.21236/ada040444

URL : http://www.dtic.mil/dtic/tr/fulltext/u2/a040444.pdf

A. Considère, Mémoire sur l'emploi du fer et de l'acier dans les constructions, Annales des Ponts et Chaussées, vol.9, issue.34, pp.574-775, 1885.

C. Denoual and F. Hild, A damage model for the dynamic fragmentation of brittle solids, Computer Methods in Applied Mechanics and Engineering, vol.183, issue.3-4, pp.247-258, 2000.
DOI : 10.1016/S0045-7825(99)00221-2

URL : https://hal.archives-ouvertes.fr/hal-00013966

C. Fressengeas and A. Molinari, Inertia and thermal effects on the localization of plastic flow, Acta Metallurgica, vol.33, issue.3, pp.387-396, 1985.
DOI : 10.1016/0001-6160(85)90081-1

C. Fressengeas and A. Molinari, Fragmentation of rapidly stretching sheets, Eur, 1994.

I. M. Fyfe and A. M. Rajendran, Dynamic pre-strain and inertia effects on the fracture of metals, Journal of the Mechanics and Physics of Solids, vol.28, issue.1, pp.17-26, 1980.
DOI : 10.1016/0022-5096(80)90009-5

D. Goto, R. Becker, T. Orzechowski, H. Springer, A. Sunwoo et al., Investigation of the fracture and fragmentation of explosively driven rings and cylinders, International Journal of Impact Engineering, vol.35, issue.12, pp.1547-1556, 2008.
DOI : 10.1016/j.ijimpeng.2008.07.081

N. Goubot, Application du test d'expansion d'anneau à l'étude du comportement dynamique du cuivre, 1994.

W. Gourdin, Metallurgical effects on the constitutive and fragmentation behavior of ofhc copper rings, Conference on Shock Compression of Condensed Matter, 1987.

W. H. Gourdin, Analysis and assessment of electromagnetic ring expansion as a high???strain???rate test, Journal of Applied Physics, vol.12, issue.2, pp.411-422, 1989.
DOI : 10.1007/BF00542818

D. E. Grady, Fragmentation of solids under impulsive stress loading, Journal of Geophysical Research: Solid Earth, vol.44, issue.B2, pp.1047-1054, 1981.
DOI : 10.1115/1.3424150

D. E. Grady, Fragmentation of Rings and Shells -The Legacy of N.F. Mott, 2006.

D. E. Grady and D. A. Benson, Fragmentation of metal rings by electromagnetic loading, Experimental Mechanics, vol.47, issue.B2, pp.393-400, 1983.
DOI : 10.1007/978-1-4613-3219-0_8

D. E. Grady and M. Olsen, A statistics and energy based theory of dynamic fragmentation, International Journal of Impact Engineering, vol.29, issue.1-10, pp.293-306, 2003.
DOI : 10.1016/j.ijimpeng.2003.09.026

P. Guduru, M. Bharathi, and L. Freund, The Influence of a Surface Coating on the High-rate Fragmentation of a Ductile Material, International Journal of Fracture, vol.23, issue.1-4, pp.89-108, 2006.
DOI : 10.1007/s10704-005-3066-0

P. Guduru and L. Freund, The dynamics of multiple neck formation and fragmentation in high rate extension of ductile materials, International Journal of Solids and Structures, vol.39, issue.21-22, pp.5615-5632, 2002.
DOI : 10.1016/S0020-7683(02)00367-0

J. Han and V. Tvergaard, Effect of inertia on the necking as a precursor to dynamic fracture, Eur. J. Mech. A/Solids, vol.14, pp.287-307, 1995.

R. Hill and J. W. Hutchinson, Bifurcation phenomena in the plane tension test, Journal of the Mechanics and Physics of Solids, vol.23, issue.4-5, 1975.
DOI : 10.1016/0022-5096(75)90027-7

T. Hiroe, K. Fujiwara, H. Hata, and H. Takahashi, Deformation and fragmentation behaviour of exploded metal cylinders and the effects of wall materials, configuration, explosive energy and initiated locations, International Journal of Impact Engineering, vol.35, issue.12, pp.1578-1586, 2008.
DOI : 10.1016/j.ijimpeng.2008.07.002

C. Hoggatt and R. F. Recht, Fracture Behavior of Tubular Bombs, Journal of Applied Physics, vol.58, issue.3, pp.1856-1862, 1968.
DOI : 10.1115/1.3629585

C. Hoggatt and R. F. Recht, Stress-strain data obtained at high rates using an expanding ring, Experimental Mechanics, vol.8, issue.1, pp.441-448, 1969.
DOI : 10.1007/BF02410405

M. Hopson, C. Scott, and R. Patel, Computational comparisons of homogeneous and statistical descriptions of AerMet100 steel subjected to high strain rate loading, International Journal of Impact Engineering, vol.38, issue.6, 2011.
DOI : 10.1016/j.ijimpeng.2010.10.016

J. W. Hutchinson and K. Neale, Influence of strain-rate sensitivity on necking under uniaxial tension, Acta Metallurgica, vol.25, issue.8, pp.839-846, 1977.
DOI : 10.1016/0001-6160(77)90168-7

V. Jeanclaude and C. Fressengeas, Dynamic Necking of Rods at High Strain Rates, Le Journal de Physique IV, vol.07, issue.C3, 1997.
DOI : 10.1051/jp4:19973119

URL : https://hal.archives-ouvertes.fr/jpa-00255404

J. N. Johnson, Dynamic fracture and spallation in ductile solids, Journal of Applied Physics, vol.99, issue.4, pp.2812-2825, 1981.
DOI : 10.1115/1.3443401

W. Johnson and R. Mehl, Reaction kinetics in processes of nucleation and growth, Trans. AIMME, vol.135, pp.414-458, 1939.

D. Jouve, Etude analytique de l'instabilité plastique de striction pour une plaque sollicitée en traction biaxiale, 2010.

R. R. Karpp and J. Simon, An estimate of the strength of a copper shaped charge jet and the effect of strength on the breakup of a stretching jet, 1976.

H. Knoepfel, Pulsed High Magnetic Fields, 1970.

F. Llorca and A. Juanicotena, Expanding Ring Test : Numerical Simulation - Application to the Analysis of Experimental Data, Le Journal de Physique IV, vol.07, issue.C3, pp.235-240, 1997.
DOI : 10.1051/jp4:1997342

URL : https://hal.archives-ouvertes.fr/jpa-00255499

S. Mercier, N. Granier, A. Molinari, F. Llorca, and F. Buy, Multiple necking during the dynamic expansion of hemispherical metallic shells, from experiments to modelling, Journal of the Mechanics and Physics of Solids, vol.58, issue.7, pp.955-982, 2010.
DOI : 10.1016/j.jmps.2010.05.001

S. Mercier and A. Molinari, Predictions of bifurcation and instabilities during dynamic extension, International Journal of Solids and Structures, vol.40, issue.8, pp.1995-2016, 2003.
DOI : 10.1016/S0020-7683(03)00020-9

S. Mercier and A. Molinari, Analysis of multiple necking in rings under rapid radial expansion, International Journal of Impact Engineering, vol.30, issue.4, pp.403-419, 2004.
DOI : 10.1016/S0734-743X(03)00063-0

J. Meulbroek, K. Ramesh, P. Swaminathan, and A. Lennon, CTH simulations of an expanding ring to study fragmentation, International Journal of Impact Engineering, vol.35, issue.12, pp.1661-1665, 2008.
DOI : 10.1016/j.ijimpeng.2008.07.066

W. Mock and W. H. Holt, Fragmentation behavior of Armco iron and HF???1 steel explosive???filled cylinders, Journal of Applied Physics, vol.189, issue.5, pp.2344-2351, 1983.
DOI : 10.1098/rspa.1947.0042

N. Mott, Fragmentation of Shell Cases, Proceedings of The Royal Society . Series A. pp, pp.300-308, 1947.
DOI : 10.1098/rspa.1947.0042

URL : http://rspa.royalsocietypublishing.org/content/royprsa/189/1018/300.full.pdf

F. Murnaghan, Finite Deformations of an Elastic Solid, American Journal of Mathematics, vol.59, issue.2, 1951.
DOI : 10.2307/2371405

P. R. Nayak, Random process model of rough surfaces. Transactions of the ASME, Journal of Lubrication Technology, pp.398-407, 1971.
DOI : 10.1115/1.3451608

F. I. Niordson, A unit for testing materials at high strain rates, Experimental Mechanics, vol.11, issue.1, pp.29-32, 1965.
DOI : 10.1007/BF02320901

F. Olive, A. Nicaud, J. Marilleau, and R. Loichot, Rupture behaviour of metals in explosive expansion, Inst. Phys. Conf. Ser, vol.47, pp.242-251, 1979.

A. Pandolfi, P. Krysl, and M. Ortiz, Finite element simulation of ring expansion and fragmentation, International Journal of Fracture, vol.95, issue.1/4, pp.279-297, 1999.
DOI : 10.1023/A:1018672922734

B. Persson, O. Albohr, U. Tartaglino, A. Volokitin, and E. Tosatti, On the nature of surface roughness with application to contact mechanics, sealing, rubber friction and adhesion, Journal of Physics: Condensed Matter, vol.17, issue.1, 2005.
DOI : 10.1088/0953-8984/17/1/R01

J. Petit, Materials under Extreme Loadings, pp.313-332, 2010.

J. Petit and J. Dequiedt, Constitutive relations for copper under shock wave loading: Twinning activation, Mechanics of Materials, vol.38, issue.3, pp.173-185, 2006.
DOI : 10.1016/j.mechmat.2005.06.005

J. Petit, V. Jeanclaude, and C. Fressengeas, Breakup of Copper shaped-charge jets: Experiment, numerical simulations, and analytical modeling, Journal of Applied Physics, vol.80, issue.12, 2005.
DOI : 10.1016/S0924-0136(98)00099-5

URL : https://hal.archives-ouvertes.fr/hal-00119280

D. Preston and D. Wallace, A model of the shear modulus, Solid State Communications, vol.81, issue.3, pp.277-281, 1992.
DOI : 10.1016/0038-1098(92)90514-A

J. Price, An acetylene cylinder explosion: A most probable cause analysis, Engineering Failure Analysis, vol.13, issue.4, 2006.
DOI : 10.1016/j.engfailanal.2005.04.014

J. Rodríguez-martínez, G. Vadillo, R. Zaera, and J. Fernández-sáez, On the complete extinction of selected imperfection wavelengths in dynamically expanded ductile rings, Mechanics of Materials, vol.60, pp.107-120, 2013.
DOI : 10.1016/j.mechmat.2013.01.009

A. Rusinek and R. Zaera, Finite element simulation of steel ring fragmentation under radial expansion, International Journal of Impact Engineering, vol.34, issue.4, pp.799-822, 2007.
DOI : 10.1016/j.ijimpeng.2006.01.003

URL : https://e-archivo.uc3m.es/bitstream/10016/7341/3/finite_zaera_ijie_2007_ps.pdf

V. B. Shenoy and L. B. Freund, Necking bifurcations during high strain rate extension, Journal of the Mechanics and Physics of Solids, vol.47, issue.11, pp.2209-2233, 1999.
DOI : 10.1016/S0022-5096(99)00031-9

P. Slate, M. Billings, and P. Fuller, The rupture behaviour of metals at high strain rates, J. Inst. Metals, vol.95, pp.244-251, 1967.

N. Sorensen and L. Freund, Unstable neck formation in a ductile ring subjected to impulsive radial loading, International Journal of Solids and Structures, vol.37, issue.16, pp.2265-2283, 2000.
DOI : 10.1016/S0020-7683(98)00315-1

G. Taylor, The fragmentation of tubular bomb Scientific papers of G.I. Taylor Vol III, pp.387-390, 1963.

D. Wesenberg and M. Sagartz, Dynamic fracture of 6061-T6 aluminium cylinders, 1977.
DOI : 10.1115/1.3424150

F. Zerilli and R. Armstrong, Dislocation???mechanics???based constitutive relations for material dynamics calculations, Journal of Applied Physics, vol.46, issue.5, pp.1816-1825, 1987.
DOI : 10.1063/1.1707363

URL : https://zenodo.org/record/1232906/files/article.pdf

H. Zhang and K. Ravi-chandar, On the dynamics of necking and fragmentation ??? I. Real-time and post-mortem observations in Al 6061-O, International Journal of Fracture, vol.21, issue.6, pp.183-217, 2006.
DOI : 10.1098/rspa.1950.0050

H. Zhang and K. Ravi-chandar, On the dynamics of necking and fragmentation???II. Effect of material properties, geometrical constraints and absolute size, International Journal of Fracture, vol.33, issue.1, pp.3-36, 2008.
DOI : 10.1007/978-1-4613-2880-3_6

F. Zhou, J. F. Molinari, and K. Ramesh, An elastic???visco-plastic analysis of ductile expanding ring, International Journal of Impact Engineering, vol.33, issue.1-12, pp.880-891, 2006.
DOI : 10.1016/j.ijimpeng.2006.09.070

F. Zhou, T. Wright, and K. Ramesh, The formation of multiple adiabatic shear bands, Journal of the Mechanics and Physics of Solids, vol.54, issue.7, pp.1376-1400, 2006.
DOI : 10.1016/j.jmps.2006.01.006