Johnson CookstrengthmodelforVascomax

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JOHNSON-COOK STRENGTH MODEL CONSTANTS FOR

VASCOMAX 300 AND 1080 STEELS

J. D. Cinnamon1, A. N. Palazotto

1, N. S. Brar

2, Z. Kennan

1and D. Bajaj

2

1Department of Aeronautics and Astronautics, Air Force Institute of Technology, WPAFB, OH 45433

2University of Dayton Research Institute, Dayton, OH 45469-0182

Abstract. High strength steels, VascoMax 300 and 1080, are characterized under tension at strain

rates of ~1/s, ~500/s, ~1000/s, and ~1500/s and at high temperatures using the quasi-static and split

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*

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]. Elementaliron 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,

* 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.

the user must input a constitutive model and define

a new material. CTH does possess equation-of-state (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.

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 SHBapparatus are 0.5 inch diameter Inconel 718. The

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


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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:

0 sc V = (1)

where 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):

0 3

1954968 /

7900 /

E GPac m s

kg m= = = (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 i, the incident strainwave. The incident wave is partially reflected asrand transmitted as t. The strain gauges on theapparatus bars allow for the measurement of these

strain pulses.

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

/u x = and E = . The average strain in thespecimen 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

Eq. 5 from noting that /E P A = = , where E isthe test material elastic modulus and A is the

Hopkinson bar cross-sectional area.

1 0 1 00 0

2 0 2 00 0

( )t t

i r

t t

t

u c dt c dt

u c dt c dt

= =

= =

(3

01 2

0( )

t

s i r t

cu udt

L L

= = (4

1 1

2 2

( )i r

t

P EA EA

P EA EA

= = +

= =

(5

Assuming the forces are the same at both ends of

the specimen, Eq. 5 implies that i + r = t andtherefore from Eq. 4:

0 0

0 0

2( )

t t

s t r r t r

c cdt dt

L L = = (6

which is the specimen strain. This is available

from the strain gauge measurements of r. Theforce at the specimen ends must equal the force in

the bars, which requires:

s b t

s s

A AE

A A = = (7

where b is the stress in the bar and As is the gaugecross-sectional area of the test specimen. Thespecimen strain-rate is obtained from Eq. 6 as:

02s r

c

L = (8

With these relationships a set of material data

(stress and strain) can be gathered at varying strain-

rates and temperature. From this data, a

constitutive model can be created.

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 strain-

hardening 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 2. Typical 1080 steel stress-strain data.

TABLE 1. 1080 SHB Results.

Test

No.

Temp

(F)Strain-

rate (s-1)

Stress (GPa)

(~.06)Mean stress

(GPa)

Q1, Q2 70 ~1 1.03, 1.06 1.048

3, 4 70 ~500 1.23, 1.21 1.22

11, 12 300 ~500 1.03, .99 1.01

16, 17 500 ~500 .87, .90 .89

18, 20 750 ~500 1.05, .95 1.00

6, 7 70 ~1000 1.30, 1.24 1.27

13, 14 300 ~1000 .68, 1.07 .88

23, 24 500 ~1000 .91, .60 .75

22, 31 750 ~1000 .94, 1.05 .99

25, 26 70 ~1500 1.18, 1.18 1.18

33, 34 300 ~1500 1.43, 1.09 1.2627, 28 500 ~1500 .83, 1.42 1.12

36, 38 750 ~1500 .69, .96 .82

VascoMax 300, on the other hand, did not

behave as a typical strain hardening material.

Figure 3 shows a typical VascoMax 300 stress-

strain 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 strainmodification 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:

,s sadjusted original adjusted original

measured measured

A A

A A = = (9)

Figure 3. Typical VascoMax 300 stress-strain data.

TABLE 2. Uncorrected VascoMax 300 Results.

Test

No.

Temp

(F)Strain-

rate (s-1)

Stress (GPa)

(~.06)Mean stress

(GPa)

Q1, Q2 70 ~1 1.99, 1.92 1.9553, 4 70 ~500 1.88, 1.97 1.92

15, 16 500 ~500 1.56, 1.50 1.53

19, 20 750 ~500 1.50, 1.54 1.52

17, 18 1000 ~500 1.16, 1.12 1.14

1, 2 70 ~1000 2.00, 1.78 1.89

9, 28 500 ~1000 1.74, 1.80 1.77

10, 11 750 ~1000 1.50, 1.57 1.53

12, 13 1000 ~1000 1.28, 1.21 1.24

6, 7 70 ~1500 1.91, 2.12 2.01

21, 23 500 ~1500 1.99, 1.95 1.97

24, 25 750 ~1500 1.81, 1.71 1.76

26, 27 1000 ~1500 1.48, 1.39 1.43

Table 3 summarized the results of modifying thestress 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.

Figure 4. Typical modified VascoMax 300 curve.


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TABLE 3. Corrected VascoMax 300 Results.

Test

No.

Temp

(F)Strain-

rate (s-1)

Stress (GPa)

(~.07)Mean stress

(GPa)

Q1, Q2 70 ~1 2.53, 2.39 2.46

3, 4 70 ~500 2.50, 2.54 2.52

15, 16 500 ~500 1.90, 1.88 1.89

19, 20 750 ~500 1.92, 2.00 1.96

17, 18 1000 ~500 1.44, 1.46 1.451, 2 70 ~1000 2.55, 2.62 2.59

9, 28 500 ~1000 1.95, 2.08 2.02

10, 11 750 ~1000 1.89, 1.80 1.84

12, 13 1000 ~1000 1.55, 1.49 1.52

6, 7 70 ~1500 2.48, 2.56 2.52

21, 23 500 ~1500 2.03, 2.11 2.07

24, 25 750 ~1500 1.86, 1.85 1.86

26, 27 1000 ~1500 1.75, 1.78 1.76

CONSTITUTIVE MODEL

With the experimental data from the SHB, a

constitutive model was developed. The Johnson-Cook relationship [8] was chosen due to the

relative simplicity in deriving the coefficients from

the SHB data. This relationship states:

* *

* *

0

[ ][1 ln ][1 ]

/ ( ) /( )

n m

room melt room

A B C T

and T T T T T

= + +

= =

(10)

The data from the SHB was used to determine the

best fit of J-C coefficients (see Table 4).

CONCLUSIONS

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.

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

TABLE 4. Material Properties & J-C Coefficients.

Property 1080 Steel VascoMax 300

E 202.8 GPa 180.7 GPa

.27 .283Tmelt 1670 K 1685 K

A (yield) 525 MPa 2.17 GPaB 3.59 GPa 9.4 GPa

C .029 .0046m .7525 .7799

n .6677 1.175

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

REFERENCES

1. Laird, D. and A. Palazotto. Effects of Temperature onthe Process of Hypervelocity Gouging. AIAA Journal,

41(11):2251-2260, 2003

2. Laird, D. and A. Palazotto. Gouge development duringhypervelocity sliding impact. International Journal of

Impact Engineering, 30:205-223, 2004

3. 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

4. Szmerekovsky, Andrew G. The Physical Understandingof 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

5. Lindholm, U. S. Some Experiments with the SplitHopkinson Pressure Bar. Journal of the Mechanics of

Physical Solids, 12:317-335, 19646. Nicholas, T., Tensile testing of materials at high rates of

strain. Experimental Mechanics, 117-185, May 1981

7. Zukas, Jonas A., Theodore Nicholas, H. F. Swift, L. B.Greszczuk, and D. R. Curran. Impact Dynamics. Krieger

Publishing Co., Malabar, FL, 1992

8. Johnson, Gordon R. and William H. Cook. AConstitutive 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


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