258–261, 274–277, 307–310, 311–314, 351–354, 359–362, 40 519–522, 607–610, 858–861, 862–865, 866–869, 909–912, 948 1045–1048, 1069–1072, 1073–1076, 1081–1084, 1097–1100, 1 1131–1134, 1187–1190, 1195–1198, 1199–1202, 1215–1218, 1 1261–1264, 1357–1360,1387–1390,1503–1506, 1527–1530,15 1539–1542 CP845, Shock Compression of Condensed Matter - 2005, edited by M. D. Furnish, M. Elert, T. P. Russell, and C. T. White © 2006 American Institute of Physics 0-7354-0341-4/06/$23.00

FOR ONLY THE ARTICLES ON PP. 165–168, 169–174, 183–1 JOHNSON-COOK STRENGTH MODEL CONSTANTS FOR 274–277, 307–310, 311–314, 351–354, 359–362, 409–412, 49 VASCOMAX 300 AND 607–610, 1080 STEELS 858–861, 862–865, 866–869, 909–912, 948–951, 101 1

1

J. D. Cinnamon , A. N. Palazotto , N. S.

1069–1072, 1073–1076, 1081–1084, 1097–1100, 1105–1108, 1 1187–1190, 1195–1198, 1199–1202, 1215–1218, 1233-1236, 1 2 Brar , Z. Kennan1 and D. Bajaj2, 1527–1530,1535–1538, an 1357–1360,1387–1390,1503–1506

1

Department of Aeronautics and Astronautics, Air Force Institute of Technology, WPAFB, OH 45433 2 THIS LINE BELOW HERE: University of Dayton Research Institute,USE Dayton, OHCREDIT 45469-0182 CP845, Shock Compression of Condensed Matter - 2005,

Abstract. High strength steels, VascoMax 300 and 1080, arebycharacterized edited M. D. Furnish, M. under Elert, T. tension P. Russell, at andstrain C. T. White rates of ~1/s, ~500/s, ~1000/s, and ~1500/s and at high temperatures using the quasi-static and split 2006 American Institute of Physics 0-7354-0341-4/06/$23.00 Hopkinson bar techniques. The data on 1080 steel exhibited a typical strain hardening response, whereas Vasco-Max 300 steel showed diminishing flow stress beyond yielding because of localized necking in gauge section of the tested specimens. The tension data are analyzed to determine the Johnson-Cook (J-C) strength model constants for the two steels. The flow stress values for VascoMax are adjusted to account for necking, and the corrected J-C model is developed. Keywords: Johnson-Cook, constitutive, 1080 steel, VascoMax 300. PACS: 62.20.Fe, 83.10.Gr, 83.60.La INTRODUCTION*

the user must input a constitutive model and define a new material. CTH does possess equation-ofstate (EOS) models for VascoMax 300 and Iron (which is arguably close enough to 1080 steel at high pressure and temperature). Therefore, the purpose of this work is to develop strain rate viscoplastic constitutive models for 1080 steel and VascoMax 300 for use in CTH. The Split Hopkinson Bar test was utilized to obtain the necessary data to construct Johnson-Cook constitutive models for the materials.

Current efforts to model the hypervelocity gouging phenomenon (rail damage developed at test speeds in access of 1.5 km/s that hinders the particular test being carried out) at the Holloman Air Force Base High Speed Test Track (HHSTT) have revolved around using the Eulerian shock wave physics code, CTH. To accomplish this, one needs to include very exact material characteristics in the high strain rate regime. While the numerical models have been successful in recreating the gouging interaction, they have not been based on the materials used at the HHSTT [1-4]. Elemental iron and VascoMax 250 are the closest materials in CTH to the 1080 steel and VascoMax 300 steel present in the gouging problem. In order to utilize materials that are not present in the CTH database,

SPLIT HOPKINSON BAR TEST OVERVIEW A typical Split Hopkinson Bar (SHB) test apparatus was used to test specimens of 1080 steel and VascoMax 300 (a review of the SHB test evaluation appears next). It should be noted that the SHB in itself is not unique but by considering the test it was found that VascoMax 300 has very different viscoplastic material response than has not been observed previously. The bars in this SHB apparatus are 0.5 inch diameter Inconel 718. The

* The views expressed in this work are those of the authors and do not reflect the official policy or position of the United States Air Force, the Department of Defense, or the U.S. Government. This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

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Eq. 5 from noting that σ = Eε = P / A , where E is the test material elastic modulus and A is the Hopkinson bar cross-sectional area.

striker bar was capable of generating stress pulses that created strain-rates in the test specimens of up to ~1500/sec. The stress pulse is assumed to be:

σ = ρ c0Vs

(1)

⎧u = t c ε dt = c t (ε − ε )dt ⎫ i r 0 ∫0 ⎪ 1 ∫0 0 1 ⎪ ⎨ ⎬ t t ⎪u2 = ∫ c0ε 2 dt = c0 ∫ ε t dt ⎪ 0 0 ⎩ ⎭ u1 − u2 c0 t εs = = ∫ (ε i − ε r − ε t ) dt L L 0

where ρ is the material density (7900 kg/m3), c0 is the material sound (elastic wave) speed, and Vs is the striker bar velocity. The striker bar velocity can be measured and the elastic wave speed can be found using (where E is the bar elastic modulus): c0 =

E

ρ

=

195 GPa = 4968 m / s 7900 kg / m3

(3)

(4)

⎧ P1 = EAε1 = EA(ε i + ε r ) ⎫ ⎨ ⎬ ⎩ P2 = EAε 2 = EAε t ⎭

(2)

The created compressive stress pulse travels through the incident bar, through a collar surrounding the test specimen, to the end of the transmitter bar. The free end reflects the pulse back as a tensile wave that arrives back at the specimen (where the collar now has no effect). Figure 1 illustrates the test apparatus and shows this arriving tensile wave as εi, the incident strain wave. The incident wave is partially reflected as εr and transmitted as εt. The strain gauges on the apparatus bars allow for the measurement of these strain pulses.

(5)

Assuming the forces are the same at both ends of the specimen, Eq. 5 implies that εi + εr = εt and therefore from Eq. 4:

εs =

c0 L

∫

t 0

(ε t − ε r − ε r − ε t )dt = −

2c0 L

∫

t 0

ε r dt

(6)

which is the specimen strain. This is available from the strain gauge measurements of εr. The force at the specimen ends must equal the force in the bars, which requires:

σs =

A A σ b = Eε t As As

(7)

where σb is the stress in the bar and As is the gauge cross-sectional area of the test specimen. The specimen strain-rate is obtained from Eq. 6 as:

εs = −

2c0 εr L

(8)

With these relationships a set of material data (stress and strain) can be gathered at varying strainrates and temperature. From this data, a constitutive model can be created.

Figure 1. SHB Test Apparatus Schematic.

Following the theory developed in [5-7], the values of specimen strain-rate and stress can be computed from these strain measurements. The displacements of the ends of the specimen in Figure 1 can be expressed in Eq. 3, where ε = ∂u / ∂x and σ = Eε . The average strain in the specimen can be found from Eq. 4, where L is the length of the specimen test section. The forces, P, at the ends of the specimen can be computed in

SHB TEST RESULTS A series of SHB tests were conducted on material specimens machined at the HHSTT to be identical to those materials in use in the field. The 1080 steel test results were typical of a strainhardening material. Figure 2 shows a typical stress-strain curve generated by the SHB. Consistent with the assumptions within the SHB

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relationships, the 1080 steel specimens showed no measurable necking in the specimens. Table 1 summarizes these tests results.

Figure 3. Typical VascoMax 300 stress-strain data. TABLE 2. Uncorrected VascoMax 300 Results. Figure 2. Typical 1080 steel stress-strain data.

Test No. Q1, Q2 3, 4 15, 16 19, 20 17, 18 1, 2 9, 28 10, 11 12, 13 6, 7 21, 23 24, 25 26, 27

TABLE 1. 1080 SHB Results. Test No. Q1, Q2 3, 4 11, 12 16, 17 18, 20 6, 7 13, 14 23, 24 22, 31 25, 26 33, 34 27, 28 36, 38

Temp (°F) 70 70 300 500 750 70 300 500 750 70 300 500 750

Strainrate (s-1) ~1 ~500 ~500 ~500 ~500 ~1000 ~1000 ~1000 ~1000 ~1500 ~1500 ~1500 ~1500

Stress (GPa) (ε~.06) 1.03, 1.06 1.23, 1.21 1.03, .99 .87, .90 1.05, .95 1.30, 1.24 .68, 1.07 .91, .60 .94, 1.05 1.18, 1.18 1.43, 1.09 .83, 1.42 .69, .96

Mean stress (GPa) 1.048 1.22 1.01 .89 1.00 1.27 .88 .75 .99 1.18 1.26 1.12 .82

As As σ original , ε adjusted = ε original Ameasured Ameasured

Strainrate (s-1) ~1 ~500 ~500 ~500 ~500 ~1000 ~1000 ~1000 ~1000 ~1500 ~1500 ~1500 ~1500

Stress (GPa) (ε~.06) 1.99, 1.92 1.88, 1.97 1.56, 1.50 1.50, 1.54 1.16, 1.12 2.00, 1.78 1.74, 1.80 1.50, 1.57 1.28, 1.21 1.91, 2.12 1.99, 1.95 1.81, 1.71 1.48, 1.39

Mean stress (GPa) 1.955 1.92 1.53 1.52 1.14 1.89 1.77 1.53 1.24 2.01 1.97 1.76 1.43

Table 3 summarized the results of modifying the stress and strain of the final reliable point in the recorded data. Because we do not have necking data over the strain range, we can only adjust the final value and allow a linear fit between that point and the selected yield point. Figure 4 shows a typical stress-strain curve from the modified data.

VascoMax 300, on the other hand, did not behave as a typical strain hardening material. Figure 3 shows a typical VascoMax 300 stressstrain profile. It exhibits little strain hardening before the material begins to fail. Additionally, the specimens experienced significant necking during the testing process. Table 2 summarizes the VascoMax test results. In order to account for the necking in the specimens, a stress and strain modification was made based on an assumption of incompressibility. Detailed measurements were taken of the post-test specimens to determine the necked cross-sectional area, Ameasured. The data was then corrected using: σ adjusted =

Temp (°F) 70 70 500 750 1000 70 500 750 1000 70 500 750 1000

(9) Figure 4. Typical modified VascoMax 300 curve.

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TABLE 3. Corrected VascoMax 300 Results. Test No. Q1, Q2 3, 4 15, 16 19, 20 17, 18 1, 2 9, 28 10, 11 12, 13 6, 7 21, 23 24, 25 26, 27

Temp (°F) 70 70 500 750 1000 70 500 750 1000 70 500 750 1000

Stress (GPa) (ε~.07) 2.53, 2.39 2.50, 2.54 1.90, 1.88 1.92, 2.00 1.44, 1.46 2.55, 2.62 1.95, 2.08 1.89, 1.80 1.55, 1.49 2.48, 2.56 2.03, 2.11 1.86, 1.85 1.75, 1.78

Strainrate (s-1) ~1 ~500 ~500 ~500 ~500 ~1000 ~1000 ~1000 ~1000 ~1500 ~1500 ~1500 ~1500

TABLE 4. Material Properties & J-C Coefficients. Mean stress (GPa) 2.46 2.52 1.89 1.96 1.45 2.59 2.02 1.84 1.52 2.52 2.07 1.86 1.76

Property E ν Tmelt A (σyield) B C m n

1080 Steel 202.8 GPa .27 1670 K 525 MPa 3.59 GPa .029 .7525 .6677

VascoMax 300 180.7 GPa .283 1685 K 2.17 GPa 9.4 GPa .0046 .7799 1.175

CONSTITUTIVE MODEL With the experimental data from the SHB, a constitutive model was developed. The JohnsonCook relationship [8] was chosen due to the relative simplicity in deriving the coefficients from the SHB data. This relationship states: σ = [ A + Bε ][1 + C ln ε ][1 − T ] n

*

*m

Figure 6. Stress v. Strain-Rate, VascoMax 300.

(10)

ε* = ε / ε0 and T * = (T − Troom ) /(Tmelt − Troom )

REFERENCES

The data from the SHB was used to determine the best fit of J-C coefficients (see Table 4).

1. 2.

CONCLUSIONS 3.

Johnson-Cook constitutive models were developed using the SHB data. The stress versus strain-rate diagrams (Figures 5 & 6) over various temperatures shows good agreement with experiment. These models will be useful in future CTH modeling of the HHSTT gouging problem.

4.

5. 6. 7. 8.

Figure 5. Stress v. Strain-Rate, 1080 steel.

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Laird, D. and A. Palazotto. “Effects of Temperature on the Process of Hypervelocity Gouging.” AIAA Journal, 41(11):2251-2260, 2003 Laird, D. and A. Palazotto. “Gouge development during hypervelocity sliding impact.” International Journal of Impact Engineering, 30:205-223, 2004 Szmerekovsky, A. G., A. N. Palazotto, and W. P. Baker. “Scaling Numerical Models for Hypervelocity Test Sled Slipper-Rail Impacts.” International Journal of Impact Engineering, (Manuscript 1754, accepted for publication), 2004 Szmerekovsky, Andrew G. “The Physical Understanding of the Use of Coatings to Mitigate Hypervelocity Gouging Considering Real Test Sled Dimensions AFIT/DS/ENY 04-06.” Ph.D. Dissertation, Air Force Institute of Technology, Sept 2004 Lindholm, U. S. “Some Experiments with the Split Hopkinson Pressure Bar.” Journal of the Mechanics of Physical Solids, 12:317-335, 1964 Nicholas, T., “Tensile testing of materials at high rates of strain.” Experimental Mechanics, 117-185, May 1981 Zukas, Jonas A., Theodore Nicholas, H. F. Swift, L. B. Greszczuk, and D. R. Curran. Impact Dynamics. Krieger Publishing Co., Malabar, FL, 1992 Johnson, Gordon R. and William H. Cook. “A Constitutive Model and Data for Metals Subjected to Large Strains, High Strain Rates, and High Temperatures.” Proceedings of the 7th International Symposium Ballistics, American Defense Preparation Organization, The Hague, Netherlands, 541-547, April 1983