Simulations of necking during plane strain tensile tests

J. Phys. IV France 134 (2006) 429 434 C EDP Sciences, Les Ulis DOI: 10.1051/jp4:2006134066 Simulations of necking during plane strain tensile tests F. Dalle 1 1 CEA/DAM Île de France, BP. 12, 91680 Bruyères-le-Châtel, France Abstract. A hydrodynamic code is used to simulate plane strain tension experiments: extension of sheets and expansion of cylinders. The material is characterized by an elastic-plastic constitutive law. The stress state in the structure is first approximately uniform, but numerical fluctuations proper to the hydrodynamic code grow and give birth to a non homogeneous deformation state. The last step shows the concentration of plastic deformation in necks, prior to rupture. Simulations show very clearly that the wavelength of instabilities and the instant of their appearance depends on some physical values: traction speed, aspect ratio, parameters of the constitutive law. Simulations of plane strain expansion of rings are compared to experimental results of expansion of cylinders. Several fragments are created and their size is similar, which validates partly the simulations. 1. INTRODUCTION Necking is a well-known metallurgical phenomenon, the last step of a material behaviour prior to rupture [1]. In quasistatic uniaxial tension tests, localization of plastic strain usually appears at the center of the specimen. From a dynamical point of view, several workers have studied the rapid expansion of rings [2-4]. In these experiments, after a period of almost homogeneous expansion at a high strain rate, the rings develop multiple necks, some of which provide sites for final fracture. Grady and Benson [2] performed experiments on copper and aluminium rings and have reported both the fracture strain and the number of fragments/necks as a function of the velocity of expansion, both of which increase with the expansion velocity. The aim of this study is to describe and study the spontaneous initiation and growth of necking in a hydronamic code with an elastic plastic extension. The simulated influence of expansion velocity gives coherent results with experiments. Other important features (constitutive law, aspect ratio) are identified and discussed. 2. A SPONTANEOUS NECKING PHENOMENON The first specimen considered here is a rectangular strip which occupies the region 0 x L 0, e 0 /2 y e 0 /2 and extends indefinitely in the z direction (fig. 1). The coordinates (x,y) represent the Lagrangian coordinates of a material point. The face x = 0 of the specimen is subjected to a mirror condition all along the experiment. The boundary condition on face x = L 0 is specified as V x = V 0. The first test is carried out with a traction speed V 0 = 1m.s 1. y e 0 L 0 V 0 x Figure 1. Schematic diagram of the sheet extension test. Article published by EDP Sciences and available at http://www.edpsciences.org/jp4 or http://dx.doi.org/10.1051/jp4:2006134066

430 JOURNAL DE PHYSIQUE IV The material considered here is an elastic plastic tantalum described by the law given below (1) taken from Steinberg et al. [5]. The shear modulus and flow stress increase with plastic strain ε p and pressure P and decrease with temperature T. It obeys to Von Mises plasticity criterion. ( Y = Y 0 (1 + βε p ) n 1 + G P Max(P ;0) G 0 (ρ/ρ 0 ) 1/3 G ) T Max (T 300; 0) G 0 and Y 0 (1 + βε p ) n Y max The onset of necking in a tensile specimen is associated with a bifurcation from a state of homogeneous uniaxial stressing. This is why initial conditions are set into the specimen of fig. 1, which overcomes the complications of wave propagation effects [4]. At t = 0, elastic solution given by equations (1) is applied for a traction speed V 0 = 1m/s. (1) x V x (x, t = 0) = V 0 L V y (y, t = 0) = ν 1 ν V y 0 L (2) Results are given in figures 2 to 6 for t = 1 ms, 2 ms, 3.3 ms, 3.6 ms and 5.5 ms. Each of these steps is described by a cartography of plastic strain and profiles along the x-axis: plain lines for plastic strain ε p profiles, dotted lines for Von Mises flow stress σ eq and lines with cross symbols for elastic limit Y. First step (fig. 2) shows quasi homogeneity in the specimen, strains and stresses being constant along the x axis. Both profiles of σ eq and Y are superposed: the specimen deforms plastically. Then, at 2 ms, some perturbations lead to an inhomogeneous behaviour in the specimen (fig. 3). A pattern of bands at 45 deg. appears on the plastic strain cartography. Later (fig. 4), inhomogeneities have grown and give sinusoidal profiles of stress/strain along the x-axis. Plastic deformation starts to concentrate in two areas of the tensile specimen, but the whole structure still deforms plastically since Y and σ eq profiles are superposed. At 3.6 ms (fig. 5), the least solicited parts of the structure (in white on the cartography) are back in the elastic field: plastic strain localises in two necks (in black on the cartography). Finally, at 5.5 ms (fig. 6), only one neck has fully developed, next step would be the rupture of the specimen at this place. A spontaneous necking phenomenon proper to the code has been put into light and needs to be validated. Its reality will be discussed through identification of the physical dependences and through comparison with experimental data. Precisely, two elements are of special interest for validation: the period P of the sinusoidal instability as can be measured on figure 4, which gives the number of necks; and the instant at which necking starts t s. This instant is defined as the moment when part of the structure gets back into the elastic field. t = 1 µs 6.468 e -3 < ε p < 6.472 e -3 Figure 2. Cartography of plastic strain and x-profiles at 1 μs.

EURODYMAT 2006 431 t = 2 µs 1.771 e -2 < ε p < 1.775 e -2 Figure 3. Cartography of plastic strain and x-profiles at 2 μs. t = 3.3 µs 3.126 e -2 < ε p < 3.317 e -2 Figure 4. Cartography of plastic strain and x-profiles at 3.3 μs. t = 3.6 µs 3.155 e -2 < ε p < 4.245 e -2 Figure 5. Cartography of plastic strain and x-profiles at 3.6 μs. t = 5.5 µs 3.155 e -2 < ε p < 2.477 e -1 Figure 6. Cartography of plastic strain and x-profiles at 5.5 μs.

432 JOURNAL DE PHYSIQUE IV 3. IDENTIFICATION OF PHYSICAL DEPENDENCES The influence of several physical parameters is studied on both period P and instant t s. First figure 7a shows the variations of the estimated number of necks L s /P (L s is the total length of the strip at the instant t s ) and of the necking elongation El s = V 0 *t s with traction speed V 0. This parametric study with velocity shows clearly that the simulations agree well with the experimental trends mentioned in the introduction: the ductility (corresponding to the simulated El s ) and number of necks increase with an increase of the traction speed. Figures 7b and 7c show the variation of the instant t s with the traction speed V 0, with the initial aspect ratio L 0 /e 0, with the hardening exponent n and with the coefficient G T /G 0. Figure 7c shows the competitive influences of hardening (through the exponent n) and softening (coefficient G T /G 0 ) mechanisms on the instant t s. These results constitute the first step to find a necking criterion at least for the present bidimensional case. An analytical relationship is likely to be found in the near future to synthesize these results. a) b) c) Figure 7. a) Variations of the number of necks L s /P and of the necking elongation El s with traction speed V 0. b) Variations of instant t s with traction speed V 0 and aspect ratio L 0 /e 0. c) Variations of instant t s with hardening exponent n and coefficient G T /(1000*G 0 ). 4. EXPERIMENTAL VALIDATION OF NECKING SIMULATIONS This part is devoted to the validation of necking simulations based on some precise experimental data. Results of cylinder expansions which have been carried out by F. Olive et al [6] have been used. A cylinder of explosive, conically hollowed at its extremities, is initiated at its centre. A tube of Plexiglas closely surrounds the explosive, to prevent detonation products from emerging on the surface. A steel tube surrounds the Plexiglas tube and its expansion is observed by means of an ultra-rapid framing camera. After detonation, the shell is accelerated and then moves at a quasi constant speed of 980 steel plexiglas explosive detonators (1) (5) (4) (8) Figure 8. Expansion of cylinder: experimental results [7] [8].

EURODYMAT 2006 433 t = 49.4 µs, R = 91 mm 0.234 < ε p < 1.748 y plexiglas steel explosive x t = 68.5 µs, R = 110 mm 0.234 < ε p < 3.347 a) b) c) Figure 9. Expansion of ring: plane strain simulation. m.s 1. Four images have been taken out of the film at t 0 before detonation (1), at t 0 + 4Δt (4),t 0 + 5Δt (5) and t 0 + 8Δt (8). Necking starts between t 0 and t 0 + 4Δt but its detection is difficult before t 0 + 4Δt. Then the necks and fissure network tends to develop parallel to the cylinder axis. The motion of the shell is supposed to be purely radial, which allows us to simulate such an experiment by the plane strain expansion of a bi-dimensional ring. Figure 9a shows the design of the simulated assembly, only half of the structure has been calculated with suitable symmetric boundary condition at y = 0. The equation of state of the explosive has been adjusted to reach the expansion speed of 980 m.s 1, the detonation is activated at the center (0,0) and a circular wave propagates in the ring of Plexiglas and in the ring of steel. An elastic rigid-plastic constitutive law has been chosen for steel. The calculations are performed with 180 elements in the circumference of the steel ring and 10 meshes in the cross section. After the first shock the ring continues to expand driven only by its radial inertia. As expected, the necking phenomenon appears during expansion: figure 9b and 9c show the cartography of plastic deformation in the ring at 49.4 μs and 68.5 μs, corresponding to the picture (8) on figure 8. It is possible to compare the experimental fragment size, between 6 and 15.5 mm, and the simulated distance between the necks, about 5.7 to 10.7 mm. Thus, experiment and simulation are in good agreement. As far as the instant t s of necking initiation is concerned, the experimental measure is more difficult; we estimate that necking appears between picture (1) and picture (4) of the expansion film (fig. 8), that is to say between t = 17.5 and 34.2 μs. During the simulation, necking starts at t s = 27.9 μs which is in the experimental range. However, if we change for an SCG [5] constitutive law for steel, the agreement of the simulation with the experiment is worse, yet the range of fragment size remains reasonable. 5. CONCLUSIONS The hydrodynamic code used in this study reveals an inherent necking initiation and growth whose reality has been proved. The uniaxial study of the influence of physical parameters such as traction speed, aspect ratio, hardening and softening coefficients from the constitutive law is a first step in the elaboration of a necking criterion. The increase of ductility and decrease of fragment size with velocity is properly yielded by simulation. An experiment of cylinder expansion has been used to validate the number of necks obtained by simulation. A study of biaxial extension tests of sheets is planned in order to compare with the experimental results of sphere expansions. Acknowledgement The author is grateful to C. Denoual (CEA/DAM) for useful discussions.

434 JOURNAL DE PHYSIQUE IV References [1] D. Francois, A. Pineau and A. Zaoui, in Comportement mécanique des matériaux, Ed. Hermès Paris, ISBN 2-86601-314-X, ISBN 2-86601-348-4, 1995. [2] D.E. Grady and D.A. Benson, Exp. Mech., 12, 393 (1983). [3] M. Altynova, X. Hu and G.S. Daehn, Met. and Mat. Trans. A, Vol. 27A, 1837 (1996). [4] V.B. Shenoy and L.B. Freund, J. Mech. Phys. Solids, 2209 (1999). [5] D.J. Steinberg, S.G. Cochran and M.W. Guinan, J. Appl. Phys. 51 (3), 1498 (1980). [6] F. Olive, A. Nicaud, J. Marilleau and R. Loichot, Inst. Phys. Conf. Ser. No. 47 Chapter 2, 242 (1979). [7] J.N. Oeconomos, M. Carnot, M. Schmitt, Proceedings of Journées Détonique de Gramat, Tome 1, NC-5, 1988. [8] Carnot M., Rapport de stage de fin d études de l ENSMP, 1988.