Damage modeling for Taylor impact simulations

Transcription

1 J. Phys. IV France 134 (2006) C EDP Sciences, Les Ulis DOI: /jp4: Damage modeling for Taylor impact simulations C.E. Anderson Jr. 1, I.S. Chocron 1 and A.E. Nicholls 1 1 Engineering Dynamics Department, Southwest Research Institute, PO Drawer 28510, San Antonio, TX 78228, USA Abstract. G. I. Taylor showed that dynamic material properties could be deduced from the impact of a projectile against a rigid boundary. The Taylor anvil test became very useful with the advent of numerical simulations and has been used to infer and/or to validate material constitutive constants. A new experimental facility has been developed to conduct Taylor anvil impacts to support validation of constitutive constants used in simulations. Typically, numerical simulations are conducted assuming 2-D cylindrical symmetry, but such computations cannot hope to capture the damage observed in higher velocity experiments. A computational study was initiated to examine the ability to simulate damage and subsequent deformation of the Taylor specimens. Three-dimensional simulations, using the Johnson-Cook damage model, were conducted with the nonlinear Eulerian wavecode CTH. The results of the simulations are compared to experimental deformations of 6061-T6 aluminum specimens as a function of impact velocity, and conclusions regarding the ability to simulate fracture and reproduce the observed deformations are summarized. 1. INTRODUCTION G. I. Taylor recognized that the impact of a flat-ended projectile into a rigid surface permitted an estimate of a dynamic flow stress [1]. The experimental procedure has become known as the Taylor impact or Taylor anvil test. Although Ref. [1] is the usually quoted work, Taylor had been interested in the testing of material at high rates of loading for some time [2]. There is a figure in Ref. [2] that shows the results of mild-steel cylinders fired at armor steel plates, from which Taylor estimated the dynamic yield strength. Wilkins and Guinan [3] were the first to conduct numerical simulations of the Taylor impact test and compare these results to experiments. They tested several metals, and in particular, they have a relatively large number of tests for 1090 steel, tantalum, and 6061-T6 aluminum. The authors show that the final length of the cylinder can be reproduced fairly well by numerical simulations, over the range of impact velocities considered, using a constant flow stress for each material. But they also demonstrated that better agreement of the final shape is obtained by including the effects of work hardening. Experiments were conducted at our laboratory using 6061-T6 aluminum specimens [4]. The specimens were cm long with a 6.35-mm diameter (L/D = 5). The samples were precision ground to produce end faces that were flat and parallel to within mm. Considerable care was taken to insure that the specimens impactednormallyagainstthe anvil. Images of the specimens, as a function of impact velocity, are shown in Fig. 1. A description of the imaging system used to reconstruct the 3-D images of the specimens is given in Ref. [4]. Considerable damage (ductile fracturing) is observed in the specimens for impact velocities greater than 350 m/s. 2. NUMERICAL SIMULATIONS D Cylindrically Symmetric Simulations Numerical simulations were conducted and the results compared to the experimental data. Simulations were conducted using the multi-material, nonlinear Eulerian wavecode CTH [5]. CTH uses the van Leer algorithm for second order accurate advection that has been generalized to account for a non-uniform and finite grid, and multiple materials. CTH includes advanced constitutive models for simulating large deformations in many types of materials. Article published by EDP Sciences and available at or

2 332 JOURNAL DE PHYSIQUE IV Figure 1. Images of Taylor anvil specimens. The Mie-Grüneisen equation of state was used to represent the pressure-density-internal energy response of the materials. The Johnson-Cook constitutive model [6] was used to describe the flow stress as a function of strain, strain rate, and temperature, Eq. (1): σ eq = ( A + Bεp) n (1 + C ln ε )(1 T m ) ε = ε/ ε o ε o = s 1 T = T T (1) ref T ref = 300 K T melt T ref where ε is the equivalent plastic strain, ε is the dimensionless strain rate for ε o = s 1 and T is the homologous temperature. The five material constants are A, B, n, C, m. The expression in the first set of brackets gives the stress as a function of strain for ε = and T = 0. The expressions in the second and third sets of brackets represent the effects of strain rate and temperature, respectively. The basic form of the model is readily adaptable to most computer codes since it uses variables (ε, ε, T )thatare available in the codes. Constitutive parameters had been determined previously for 6061-T6 [7]. The yield and strainhardening behavior of the materials is described by A, B,andn. These constants were determined from quasi-static tension tests and quasi-static torsion tests. The strain rate constant, C, was determinedfrom quasi-static tension tests and high-strain-rate Hopkinson bar tension tests. There is a very small strain rate effect for this aluminum alloy. The thermal softening exponent, m, was determined from handbook data for elevated temperatures. The constants were determined to be: A = 324 MPa; B = 114 MPa; n = 0.42; C = 0.002; and m = T melt for 6061-T6 is 925 K. The 2-D cylindrically symmetric option of CTH was used for the initial simulations. The simulation results for the normalized length, and normalized diameter at the impact surface, are shown in Fig. 2, along with the experimental data. The dashed lines use the as-determined constitutive constants given in the preceding paragraph. In general, the simulated lengths are slightly shorter than measured experimentally; and because the length is a little too short, the predictions for the diameter are a little too large (since the volume is nominally conserved). The final length is very sensitive to the initial flow stress. The simulations were redone, but with a 5% increase in A (340 MPa). The results are shown as the solid lines in Fig. 2. Agreement between the simulations and the experimental data is somewhat better. Of course, the normalized diameters do not agree particularly well at the higher impact velocities where there is considerable fracturing of the specimens (see Fig. 1). What is perhaps somewhat surprising is that the computed length is not particularly affected by fracture. The radius of the 289-m/s impact specimen was measured as a function of distance from the impact end. Measurements were taken along different diameters; these results are plotted as the solid points in Fig. 3. The largest uncertainty in the deformed shape is in the transition ( neck region) from a nominally uniform cylinder to the large deformation at the impact end. The simulated results (using the modified value for the constitutive parameter A) are also shown. Agreementis quite good.

3 EURODYMAT CTH: A=324 MPa CTH: A=340 MPa CTH: 3D L f /L o CTH: A=324 MPa CTH: A=340 MPa CTH: 3D D f /D o a) Normalized length b) Normalized diameter Figure 2. Comparison of 2-D simulations with experimental data CTH 4.0 R (mm) 3.0 Original Radius Y (mm) Figure 3. Comparison of calculated profile to experiment for the 289-m/s impact D Simulations Convergence Study Typically, Taylor anvil simulations are conducted assuming 2-D cylindrical symmetry, but such computations cannot hope to capture the damage observed at the higher impact velocities. The 3-D option of CTH was used for the simulations with damage. Prior to invoking damage, a grid convergence study was conducted. Square (or cubic) zoning was used throughout the problem. In 2-D, results converged using 10 zones to resolve the initial radius (all 2-D results presented in this article were run with 20 zones across the initial radius). Somewhat surprising, it took the same zone size in 3-D to achieve convergence. Previous work showed that for certain types of problems, numerical convergence was achieved faster in 3-D than in 2-D [8]. For the Taylor anvil simulations, the final length and diameter were still slightly different using 10, 15, and 20 zones to resolve the initial radius, although the final dimensions were within the magnitude of the elastic oscillations at the end of impact. Most of the 3-D simulations were conducted using 10 zones to resolve the initial radius. A comparison of the 3-D results (using A = 340 MPa) with fully resolved 2-D results is shown in Fig. 2; the dotted lines denote the 3-D

4 334 JOURNAL DE PHYSIQUE IV results. The differences in the 2-D and 3-D results (approximately 2% at the highest impact velocities) are essentially the same as the differences between the two initial flow stresses the parameter A in Eq. (1) in 2-D simulations Johnson-Cook Damage Model The Johnson-Cook damage model [9] was used to investigate the response of the Taylor anvil specimens to damage. The damage model is given by Eq. (2): 1 D = Δε p ε f (2a) ε f = D 1 + D 2 e D 3σ σ = σ (2b) σ eq The parameter D represents the damage that is accumulated. D is a function of the incremental plastic strain Δε p and the equivalent strain to fracture ε f. The damage function explicitly accounts for the observation that material can undergo larger strains in compression than in tension; this is done through the exponential term in Eq. (2b) where σ is a function of the mean stress ( σ) and the equivalent stress (σ eq ). For example, σ has a value of +1/3 for a standard tensile test (before necking) and -1/3 for a standard compression test. When D =, fracture occurs; that is, the material no longer can support deviatoric stresses (e.g., tension and/or shear) and the material essentially behaves like a fluid. The damage constants for 6061-T6 are [7]: D 1 = 0, D 2 = 1.11, and D 3 = 1.5. For an uniaxial tensile test, these constants give a strain to failure of The initial simulations used the as-given damage constants, Case 1 (the dot-dashed line in Fig. 4), and the results for normalized length are compared to the 3-D simulations without damage. Numerically, the cylinder disintegrates and becomes way too short using the as-given damage constants. Also, damage accumulates at very low impact velocities, in contrast to experiment. This implies that the computed failure strains are unrealistically low. Results are not given for the diameter because of extensive failure in the simulations. A second approach was tried, Case 2 (long dashed line). Experimentally, it appears that ductile fracture is just beginning for the 289-m/s impact case. The simulation for an impact velocity of 289 m/s with damage suppressed shows equivalent plastic strains on the order of 1.8. D 1 was set to 1.60 (a little below the maximum values seen in the simulations), and D 2 = D 3 = 0. Now, damage cannot accumulate until the equivalent plastic strain reaches 1.60; thus, the low velocity results overlap the nodamage simulations. But at impact velocities greater than 300 m/s, there is again too much damage numerically, with the numerical specimen being too short and the diameter too large. Next, it was decided to try using all three damage parameters: D 1 = 1.60, D 2 = 1.11, and D 3 = 1.5. These results Case 3 are also shown in Fig. 4 as the short-dashed lines. Of course, where failure has not occurred, the results agree with the no-damage simulations. But at the higher impact velocities, the calculated normalized length and diameters agree very well with the experiments. Unfortunately, for a uniaxial tensile test, the damage model predicts that the strain to failure is over three times the measured failure strain Analysis of Results Notwithstanding the relatively good agreement of Case 3 to the experimental data, there are some major issues, one of which has already been mentioned (failure strain for a tensile test). An image of the 1 The model, as originally formulated, includes strain rate and temperature effects, but these terms are seldom used because of the lack of test data to determine constants.

5 EURODYMAT L f / L o D f / D o No Damage Case 1 Case 2 Case a) Normalized length b) Normalized diameter Figure 4. Comparison of 3-D simulations with experimental data for various damage constants. Figure 5. Simulation of 450-m/s impact with D 1 = 1.60, D 2 = D 3 = 0 (Case 2): a) 10 zones to resolve initial radius; b) 15 zones to resolve initial radius. computational results for the 450-m/s case is shown in Fig. 5(a) for Case 2. The same calculations, but using 15 zones to resolve the initial radius, instead of 10, is shown in Fig. 5(b). Here, the projectile is disintegrating, i.e., flying apart at the impact surface. 2 This will be discussed further in the paragraphs below. Equivalent plastic strain contours for the 289-m/s impact are shown in Fig. 6. This is the impact at which damage on the periphery (circumference) of the impact end appears to be just initiating. The strain is approximately 60% at the periphery, which is about the strain to failure for a uniaxial stress tensile test. But there are strains larger than 150% towards the centerline. In the simulations, damage (failure) is initiating in the interior of the mushroomed region, and not on the periphery. Thus, in Fig. 5, and for all the simulations, the numerical specimens are failing from the interior. With finer zoning, Fig. 5(b), the criterion for failure is achieved nearly simultaneously over a larger volume, with the effect that more of the projectile material fails. Since the material fails in the interior, and since the mushrooming has a net velocity radially during deformation, the numerical specimen flies apart, i.e., disintegrates. 2 For Case 3, where the three damage constants are nonzero, the mushroom is very regular and smooth. There is interior damage, but no ductile fracturing at the periphery as shown in Fig. 1, or as in Fig. 5.

6 336 JOURNAL DE PHYSIQUE IV A: 0.1 B: 0.2 C: 0.4 D: 0.6 E: 0.8 F: G: 1.5 H: 2.0 Figure 6. Equivalent plastic strain contours for the 289-m/s impact case (calculation done without damage). Thus, we conclude that an equivalent plastic strain damage model is not sufficient to characterize damage nucleation and growth for Taylor anvil impacts. It would appear, in addition to equivalent plastic strain, that the strain components may need to be monitored and included within the damage model. 3. SUMMARY A study has been conducted to examine the ability of numerical simulations to reproduce damage (ductile fracture) as a function of impact velocity for Taylor impact specimens. The specimens, made from 6061-T6 aluminum, begin to fracture around the circumference at the impact end of the specimen for an impact velocity of 289 m/s. As the impact velocity is increased further, the deformation and fracturing becomes more extensive. Three-dimensional numerical simulations were conducted using the Johnson-Cook constitutive model to describe the flow stress as a function of strain, strain rate, and temperature; the Johnson-Cook damage model was used to describe the response to damage. The Johnson-Cook damage model, which uses accumulated equivalent plastic strain as the damage variable, explicitly accounts for the observation that failure strain is a function of the state of stress. However, using the as-given damage constants considerably overpredicted damage. Agreement with experiment was obtained by requiring an initially large value for the failure strains (through the damage constant D 1 ), thereby suppressing the on-set of failure until larger equivalent plastic strain accumulated within the specimen. However, failure initiates numerically in the interior of the specimen (where the equivalent plastic strains are highest), and not on the periphery as observed in the experiments. Thus, it is concluded that using the equivalent plastic strain as the sole indicator of damage is not sufficient to describe the ductile damage observed in Taylor anvil experiments. Work is continuing as we investigate more appropriate computational damage models. References [1] Taylor, G.I., The use of flat-ended projectiles for determining dynamic yield stress. Part I. Theoretical considerations, Proc. R. Soc. A, 194, , [2] Taylor, G.I., Testing of materials at high rates of loading, J. Inst. of Civil Engineers, 26, , [3] Wilkins, M.L. and Guinan, M.W., Impact of cylinders on a rigid boundary, J. Appl. Phys., 44(3), , [4] Anderson, Jr. C.E., Nicholls, A.E., Chocron, I.S. and Ryckman, R.A., Taylor anvil impact, Shock Compression in Condensed Matter 2005, Baltimore, MD, 1 5 August 2005.

7 EURODYMAT [5] McGlaun, J.M., Thompson, S.L. and Elrick, M.G., CTH: A three-dimensional shock wave physics code, Int. J. Impact Engng., 10, , [6] Johnson, G.R. and Cook, W.H., A constitutive model and data for metals subjected to large strain, high strain rates and high temperatures, 7 th Int. Symp. Ballistics, The Hague, The Netherlands, , [7] Dannemann, K.A., Anderson, Jr., C.E. and Johnson, G.R., Modeling the ballistic impact performance of two aluminum alloys, in Modeling the Performance of Engineering Structural Materials II (D.R. Leseur and T.S. Srivatsan, Eds.), TMS, 63 75, [8] Littlefield, D.L. and Anderson, Jr., C.E., A study of zoning requirements for 2-D and 3-D longrod penetration, Shock Compression of Condensed Matter 1995 (S. C. Schmidt and W. C. Tao, Eds.), pp , AIP Press, Woodbury, NY, [9] Johnson, G.R. and Cook, W.H., Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures, Engng. Fract. Mech., 21(1), 31 48, 1985.