Thermomechanical response of HSLA-65 steel plates: experiments and modeling

Mechanics of Materials 37 (2005) 379–405 www.elsevier.com/locate/mechmat

Thermomechanical response of HSLA-65 steel plates: experiments and modeling Sia Nemat-Nasser *, Wei-Guo Guo Department of Mechanical and Aerospace Engineering, Center of Excellence for Advanced Materials, University of California, 9500 Gilman Drive, San Diego, La Jolla, CA 92093-0416, USA Received 1 June 2003

Abstract To understand and model the thermomechanical response of high-strength low-alloy steel (HSLA-65), uniaxial compression tests are performed on cylindrical samples, using an Instron servohydraulic testing machine and UCSDÕs enhanced Hopkinson technique. True strains exceeding 60% are achieved in these tests, over the range of strain rates from 10
3/s to about 8500/s, and at initial temperatures from 77 to 1000 K. The microstructure of the undeformed and deformed samples is examined through optical microscopy. The experimental results show: (1) HSLA-65 steel displays good ductility and plasticity (strain > 60%) even at low temperatures (even at 77 K) and high strain rates; (2) at relatively high temperatures and low strain rates (especially below about 0.1/s), its strength is temperature-insensitive, indicating that the material has good high-temperature weldability; (3) slight dynamic strain aging (DSA) occurs at temperatures over 400 K and in the range of strain rates from 0.001/s to 3000/s, the maximum values of the stress shifting to higher temperatures with increasing strain rates; and (4) the microstructure of the material is not affected much by the changes in the strain rate and temperature. Finally, based on the mechanism of dislocation motion, and using our experimental data, the parameters of a physically-based model developed earlier for AL-6XN stainless steel [J. Mech. Phys. Solids 49 (2001) 1823] are estimated and the model predictions are compared with various experimental results, excluding the dynamic strain aging effects. Good agreement between the theoretical predictions and experimental results is obtained. In order to further verify the model independently of the experiments used for the evaluation of the model parameters, additional compression tests at a strain rate of 8500/s and various initial temperatures are performed, and the results are compared with the model predictions. Good correlation is observed. As an alternative to this model, the experimental data are also used to estimate the parameters in the Johnson–Cook model [Proceedings of the Seventh International Symposium on Ballistics, The Hague, The Netherlands, p. 541] and the resulting model predictions are compared with the experimental data, again

*

Corresponding author. Tel.: +1 858 5344914; fax: +1 858 5342727. E-mail address: [email protected] (S. Nemat-Nasser).

0167-6636/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2003.08.017

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excluding the dynamic strain-aging effects. These and related results suggest that the physically-based model has a better prediction capability over a broader strain rate and temperature range. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: HSLA steel; Strain rate; Aging; Modeling

1. Introduction The HSLA steels were developed in the 1960s by microalloying low-carbon steels with Nb, V, and Ti in the 0.01–0.1 wt.% range. The increased strength of HSLA steels is attributed to a combination of ferrite grain refinement and precipitation strengthening (Militzer et al., 2000). Steels used for high-strength structural applications are quenched and tempered, low- to medium-carbon, low-alloy steels. In the navyÕs specifications. This is the HY series of steels that are primarily low-alloy Ni– Cr–Mo–V steel with about 0.2% C. In this series of steels, carbon is the main martensitic strengthening element. With a carbon equivalent of around 0.80, the steels are difficult to weld and require costly pre-heating as well as post-weld heat treatments (Dhua et al., 2001). To further improve the steelÕs properties and to attain a higher strength, better impact toughness, and easier weldability, a new family of low-carbon, copper-bearing, high-strength, low-alloy steels has been developed over the last two decades, an example being the HSLA-80 steel. More recently, a new high strength, low-alloy steel, HSLA-65, has been produced, which has a 65 ksi minimum yield strength, and is suitable for use in the Naval surface vessels and submarines, with potential for application in commercial carriers. This steel will allow the use of thinner plates, leading to structural weight reduction. Preliminary calculations indicate that, if it is used in hull plates, it could provide equal or greater service life than the traditional high-strength steel, but with less weight. The same would be true for the hullÕs interior supporting structures. For example, NSS (Newport News Shipbuilding, Ship Structure Committee) estimates potential weight saving could be up to 1500 long tons per aircraft carrier. This steel also offers improved weldability, formability, and toughness,

resulting from the specified alloying elements with limitations on carbon, sulfur, and residual element content. As a traditional metal, steel has been the subject of extensive study in the past few decades, both experimentally and theoretically. However, to our knowledge, HSLA-65 steel has not been studied systematically over a broad range of strain rates and temperatures. The present paper reports the results of systematic experiments over strain rates ranging from 10
3/s to about 8500/s, and initial temperatures ranging from 77 to 1000 K. Using the results of these experiments, a physically-based model is developed and its predictions are compared with the experimental results. To further assess and compare the model results with other models, we have examined the Johnson– Cook model, and have compared its predictions with our experimental results.

2. Experimental procedure and results 2.1. Material and samples All tests are carried out on an HSLA-65 baseplate. The chemical composition of this structural steel baseplate, analyzed with CAP-017C (ICPAES) and ASTM E 1019-00 by IMR, Inc., is listed in Table 1. All samples have a 5 mm nominal diameter and 5 mm height. To reduce the end friction on the samples during the low- and high-strain deformation, the sample ends are first polished using waterproof silicon carbide paper, 1200 and 4000 grit, and then they are greased for low- and room-temperature tests. A molybdenum-powder lubricant is used for the high-temperature experiments. To examine the microstructure of the undeformed and deformed samples, the samples are

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Table 1 Major alloy content of HSLA-65 (wt.%) C

Mn

Cu

Si

Cr

Mo

V

Ti

Al

Nb

Ni

P

S

Balance

0.08

1.40

<0.01

0.24

0.01

0.02

0.07

0.01

0.03

0.04

<0.01

0.005

0.005

Fe

sectioned along the loading and transverse directions, and then polished and etched, as required by standard metallography. The etching reagent is the Nital: 98 ml alcohol and 2 ml HNO3. Fig. 1a–c shows the microstructure of an undeformed sample. Fig. 1a displays the microstructure normal to the rolling direction of the plate (top view). Fig.

1b is along the rolling direction (front view along rolling direction). Fig. 1c is parallel to the rolling direction through the plate thickness (side view). In these figures, white regions are ferrites and black regions are pearlites. It is known that ferrites have lower strength and hardness, but higher plasticity and toughness, whereas pearlites have

Fig. 1. (a) HSLA-65 plate; microstructure normal to the rolling direction of the plate (top view), (b) microstructure in the rolling direction of the plate (front view along rolling direction) and (c) microstructure transverse direction to the rolling direction of the plate (side view).

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reversed properties. Fig. 1a–c shows that the microstructure of this HSLA-65 plate is not the same in the two considered directions. Pearlites in Fig. 1b and c form black strips, while pearlites in Fig. 1a have a non-uniform distribution. Hence, the microstructure of the plate is not uniform. Since the material properties usually depend on the microstructure, we prepared 10 samples along various directions of the baseplate, and tested them under various loading conditions. Fig. 2 shows some of the results for indicated strain rates. The samples for this figure are identified in the following manner according to their orientation and location: TOP––Top part of the plate along the thickness; MID––Middle part of the plate along the thickness; and BOT––Bottom part of the plate along the thickness; Across––along transverse direction of the plate; Width––along the longitudinal direction of the plate. In Fig. 2, the stress in the thickness direction (marked as Top, Middle, and Bottom) is lower by about 80 MPa than that in the other two directions (marked as Width and Across) of the plate. The final sample shape in ‘‘Width’’ and ‘‘Across’’ directions is oval along the lateral direction, showing that the material has an anisotropic plastic response. Stresses along

the thickness of the plate are the same and the deformed samples have a basically uniform deformation shape. This result is further verified and shown in Fig. 3. In the present paper, all samples are taken along the thickness direction of the plate. This structural steel has an average grain size of about 7 lm. 2.2. Low and high strain-rate experiments Compression tests at strain rates of 10
3/s and 10 /s are performed using an Instron hydraulic testing machine, over the temperature range from 77 to 800 K, with true strains exceeding about 65%. Elevated temperatures are attained with a high-intensity quartz lamp, in a radiant-heating furnace of an argon environment. The temperature is measured using a thermocouple arrangement, and is maintained constant to within ±2 °C. The deformation of the specimen is measured by an LVDT, mounted in the testing machine, and is calibrated and compared with the results of a standard extensometer before the test. The low temperature of 77 K is obtained by immersing the specimen and the testing fixture (AL2O3 ceramic bars) in a bath of liquid nitrogen. Typical true
1

1200 HSLA-65, T0 = 296K

True Stress (MPa)

1000

800

600

400

200

0 0.00

4,200/s

Top Middle Bottom Across Width Width 2 Across 2

0.10

0.20

3,350/s 3,500/s 3,700/s

0.30

0.40

0.50

0.60

0.70

0.80

True Strain

Fig. 2. Stress–strain curves of samples cut from an HSLA-65 plate at indicated directions and locations.

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1200 HSLA-65, T0 = 296K 3,500/s

True Stress (MPa)

1000

800

0.001/s

600

400

Top Bottom Middle

200

0 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

True Strain Fig. 3. Stress–strain curves of samples cut from an HSLA-65 plate at indicated directions and locations.

stress–true strain curves of HSLA-65 at strain rates of 10
3/s and 10
1/s are displayed in Figs. 4 and 5, respectively. The microstructure at a strain rate of 10
1/s and a temperature of 77 K is displayed in Fig. 6.

Dynamic tests at a strain rate of 3000/s are performed using UCSDÕs recovery Hopkinson technique (Nemat-Nasser et al., 1991; Nemat-Nasser and Isaacs, 1997) at temperatures of 77–1000 K, to strains exceeding 60%. For the high strain-rate

1400 -3

HSLA - 65, 10 /s 77K

1200

True Stress (MPa)

1000 213K 296K 400K 700K

800 600

500 K 600K

800K

400 200 0 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

True Strain

Fig. 4. True stress–true strain curves at indicated initial temperatures and a strain rate of 10
3/s.

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H SL A - 65, 10 /s

1200

77K

True Stress (MPa)

1000 213K 296K

800

400K 700K

600

500K

800K

600K

400 200 0 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

True Strain

Fig. 5. True stress–true strain curves at indicated initial temperatures and a strain rate of 10
1/s.

Fig. 6. Microstructure of a sample strained to c = 60% at 77 K and 10
3/s.

a furnace is employed to pre-heat the specimen, while keeping the transmission and incident bars outside the furnace. These bars are then automatically brought into gentle contact with the specimen, just before the stress pulse reaches the specimen-end of the incident bar. The temperature is measured by a thermocouple that also holds the specimen inside the furnace. The true stress–true strain curves at a strain rate of 3000/s are shown in Fig. 7. UCSDÕs recovery Hopkinson technique makes it possible to obtain an isothermal flow stress at high strain rates and various temperatures. The isothermal flow stress of HSLA-65 at a strain rate of 3000/s and temperatures of 77–500 K, is given in Fig. 8, which also includes the corresponding adiabatic curves (dotted curves). 2.3. Experimental results and discussion

tests at elevated temperatures, it is necessary to heat the sample to the required temperature while keeping the incident and transmission bars of the Hopkinson device at a suitably low temperature. To do this, Nemat-Nasser and Isaacs (1997) have developed a novel enhancement of the compression recovery Hopkinson technique (Nemat-Nasser et al., 1991) for high-temperature tests, where

HSLA-65 is a high-strength low-alloy structural steel, with a carbon content of about 0.08%. Based on the iron–carbon phase diagram (Krauss, 1990), HSLA-65 steel has the characteristics of the bcc structure. The performance of steels mainly depends on the properties associated with their microstructure. It is well known that bcc metals

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385

1600 HSLA - 65, 3,000/s

1400 1200 True Stress (MPa)

T0 = 77K

1000

213K 296K 400K 500K

800 600 600K

400

1,000K

700K 800K

900K

200 0 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

True Strain

Fig. 7. Adiabatic stress–strain curves at indicated initial temperatures and a strain rate of 3000/s.

2000 HSLA - 65, 3,000/s Solid Curves: Isothermal Dashed Curves: Adiabatic

1600 True Stress (MPa)

T0 = 77K

Isothermal flow stress

1200 Adiabatic T0 = 296K

800

T0 = 500K

400

0 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

True Strain

Fig. 8. Comparison between adiabatic and isothermal flow stress at 3000/s.

have high temperature- and strain rate-sensitivity, and their mechanical properties are strongly affected by impurities, which are generally considered to be the rate-controlling mechanism of the thermal component of the flow stress. Figs. 4 and 5 show the strong dependence of the flow stress on the temperature. Fig. 5 shows that when

the true strain reaches about 35% at a strain rate of 10
1/s, the flow stress of HSLA-65 steel at 77 K begins to drop with increasing strain, suggesting some kind of microstructural damage at this strain and temperature. To check this, two samples were examined along different directions by an optical microscope with a 1000 amplification, and no

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sharply with increasing temperature. At a strain rate of 3000/s (see Fig. 11), the flow stress decreases with increasing temperature from 77 K to about 600 K. In the temperature range of 600 K to about 900 K, the flow stress is basically constant, with a slight increase at higher temperatures. In order to compare the flow stress of this steel at various temperatures and strain rates, we examine the response at a true strain of 15%, in Fig. 12, concluding that: (1) The flow stress as well as the strength of HSLA-65 depend on the temperature and strain rate; (2) the flow stress is not temperature-sensitive at elevated temperatures and various strain rates; it even slightly increases with increasing temperature in the range from room temperature to about 700 K for strain rates less than 10
1/s, and from 600 to 900 K for a strain rate of 3000/s. This may suggest that, at higher strain rates, the temperature-insensivity range may shift to even higher values. This effect may be due to the dynamic strain aging of the material.

microcracks or other damage were noticed; see Fig. 6. In Fig. 5, the flow stress curves of two samples at 77 K are seen to be essentially the same, confirming that the drop in the stress is a material response. Also, in Fig. 6, no shearbands are seen at 45° to the loading direction, but, instead, grains are seen to have extended in the direction normal to the loading direction. This unusual phenomenon has not been observed at other strain rates at 77 K. The drop in the flow stress at a strain rate of 10
1/s and a temperature of 77 K could be due to dynamic strain aging. Figs. 7 and 8 show that the flow stress at higher strain rates also depends on the temperature. Figs. 9–11 present the flow stress as a function of the temperature for indicated strain rates and strains. As seen in Figs. 9 and 10, when the temperature is increased from 77 K (liquid nitrogen) to about 300 K, the flow stress decreases quickly, but at temperatures between 300 and 700 K, it is basically constant. This shows that, at lower strain rates, the flow stress of HSLA-65 is independent of the temperature change, from room temperature to about 700 K. In this temperature range, welding heating does not have a significant effect on the properties of this steel. Above 700 K, the flow stress decreases

2.3.1. Dynamic strain aging in steel Based on above test results, it appears that the flow stress of HSLA-65 is not temperature-sensitive in a certain temperature range, where the flow

1400 -3

HSLA-65, 10 /s

1200

Flow Stress (MPa)

1000 800 600 Strain

400 200

0.05

0.10

0.20

0.30

0.40

0.50

0.60 0 0

100

200

300

400

500

600

700

800

900

Temperature, (K)

Fig. 9. Flow stress as a function of temperature for indicated strains and 10
3/s strain rate.

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387

1400 -1

HSLA-65, 10 /s

1200

Stress (MPa)

1000 800 600 Strain

400 200

0.05

0.10

0.20

0.40

0.50

0.60

0.30

0 0

100

200

300

400

500

600

700

800

900

Temperature, (K)

Fig. 10. Flow stress as a function of temperature for indicated strains and 10
1/s strain rate.

1600 HSLA - 65, 3,000/s

Flow Stress (MPa)

1400

Strain Strain Strain Strain Strain Strain

1200

1000

= 0.05 = 0.15 = 0.25 = 0.35 = 0.45 = 0.55

800

600

400 0

200

400

600

800

1000

1200

Temperature, (K)

Fig. 11. Flow stress as a function of temperature for indicated strains and 3000/s strain rate.

stress may even increase with the increasing temperature, possibly due to dynamic strain aging. The term ‘‘static strain aging’’ (SSA) generally refers to the transient stress peaks observed in alloys with dilute solute atoms, when a pre-strained spec-

imen is totally or partially unloaded and aged for a prescribed time and then reloaded at the same prestraining strain rate. It is commonly accepted that this effect is related to the pinning of dislocations by diffusing solute atoms during the aging period.

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8,500/s

1200 Flow Stress (MPa)

3,000/s 0.1/s 1000

0.001/s

800

600

400 0

200

400

600 800 Temperature, (K)

1000

1200

Fig. 12. Effect of strain rates on flow stress at a true strain of 15%.

In general, dynamic strain aging is defined as recurrent pinning (serrated flow) of dislocations while arrested at obstacles during their motion that results in plastic straining (Kubin et al., 1992). Dynamic strain aging is generally attributed to the additional resistance to dislocation motion produced by the mobility of solute atoms that can diffuse to dislocations above a certain temperature (Beukel and Kocks, 1982) while the dislocations are ‘‘waiting’’ at their short-range barriers. During this waiting period, a Cottrell atmosphere and/or a core atmosphere can form at dislocations, depending on the temperature and strain rate (Nakada and Keh, 1970). In steel, the occurrence of static and dynamic strain aging results from the diffusion of C and N in the temperature range of 150–300 °C. Cho et al. (2000) report that the substitutional elements, Cr and Ni in 304 stainless steel, produce static strain aging at temperatures of 900–1100 °C. To explain the strain dependence of the peak value of the stress, two models have been examined, one due to Cottrell and the other due to Kocks et al. (Kubin et al., 1992). The Cottrell model assumes that diffusion of solute atoms is assisted by excess vacancies generated during plastic deformation. In interpreting their results, Kocks et al. abandon the ‘‘vacancy hypothesis’’ and pro-

pose a new model according to which the solute transport and arrest at dislocations is controlled by pipe diffusion via forest dislocations that intersect the slip plane. In this model, the solute hardening process is localized at obstacles associated with forest junctions, and the resulting hardening effect is influenced by the evolution of the forest dislocation density, i.e., it is related to the strain hardening. A key feature of thermally activated dislocation motion is that the dislocations spend most of their time interacting with their local obstacles, such as forests of dislocations, vacancies, and solute atoms (Cheng et al., 2001). The dislocation moves in a jerky way. The segregation of solute atoms to the dislocation core occurs during the time when the dislocations are waiting at the local obstacles. Because a strong interaction force exists between dislocations and solute atoms in the dislocation core area, the diffusion of the core atmosphere becomes significant at lower temperatures than is possible for the solute atoms situated outside the core area, provided that the shear stress is suitably high. In general, the flow stress decreases with the increasing temperature. As is seen in Fig. 4, when, at a strain rate of 10
3/s, the test temperature is increased from 77 to 400 K, the flow stress decreases,

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389

again at the same initial temperature. As is seen, the composite flow stress curve of sample 3 follows basically the flow stress curve of sample 4. In Fig. 14, samples 5 and 6 are loaded to a true strain of 60% at initial temperatures of 900 and 296 K, and at a strain rate of 3000/s. Sample 7 is loaded to a true strain of about 22% at an initial temperature 900 K (thin curve). Then it is reloaded at a strain rate of 3000/s and initial temperature of 296 K, resulting in an adiabatic flow stress (again shown by a thin curve). Sample 8 is loaded to a true strain of about 22% at a strain rate of 3000/s and an initial temperature of 296 K, then it is unloaded, brought back to 296 K, and reloaded at the same strain rate (again the thick solid curve). The thin and thick solid curves show that the prior different deformation histories of these two samples do not substantially affect their subsequent response. Results in Figs. 13 and 14 show that, the microstructure of HSLA-65 does not evolve with its pre-loading in a manner which would affect its further mechanical properties.

but as the temperature is further increased, the flow stress remains almost the same over a temperature range from 400 to 600 K. This suggests that, solute atoms interact with the moving dislocations in this temperature range, retarding their motion. With increasing strain rates, higher temperatures are required to drive the solutes to dislocations at sufficient speed. As the solutes catch up with the moving dislocations and pin them down, the flow stress increases. In Fig. 7, when the true strain is less than about 30%, the flow stress at a temperature of 600 K is higher than that at 700 K. But when the true strain is greater than 30%, the flow stress at 600 K is slightly lower than that at 700 and 800 K. When the temperature is further increased to 900 K, the flow stress attains greater values than that at 600 K. In order to investigate the microstructural evolution with temperature, two interrupted tests are performed; see Figs. 13 and 14. In Fig. 13, samples 1 and 2 are loaded to a true strain of 60% at the initial temperatures of 77 and 900 K respectively, both at a strain rate of 3000/s. Then sample 3 is loaded to a true strain of about 22% at an initial temperature of 900 K and is unloaded. It is then cooled to 77 K, and reloaded at the same strain rate. Sample 4 is loaded to a true strain of about 19% at an initial temperature of 77 K and unloaded, then reloaded

2.3.2. Strain-rate effect on flow stress It is known that the flow stress of most materials increases with the increasing strain rate. In Fig. 12, the stress–temperature curves at various indicated strain rates are compared for a fixed strain

2000 HSLA - 65, 3,000/s

True Stress (MPa)

1600

1200

T0 = 77K

Sample 4

Sample 1

800

T0 = 900K

Sample 2 400 Sample 3 0 0.00

0.10

0.20

0.30

0.40

0.50

0.60

True Strain

Fig. 13. Effect of temperature jump from 900 to 77 K on the flow stress.

0.70

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1000

True Stress (MPa)

T0 = 296K

Sample 6 T0 = 900K

800

600 Sample 5 Sample 7

400

Sample 8

200

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

True Strain

Fig. 14. Effect of temperature jump from 900 to 296 K on the flow stress.

of 15%. As is seen, the flow stress strongly depends on the strain rate, especially when the temperature exceeds 500 K and the dynamic strain-aging range shifts to higher temperatures with an increasing strain rate. In Figs. 15 and 16, the strain-rate effect on the flow stress is further examined by changing the strain rate from 10
3/s to 3000/s. In these two

figures, the stress–strain curves for 3000/s and 10
3/s strain rates, are first obtained at the common initial temperature of 296 K (samples 9 and 10), and then sample 11 is first loaded to a true strain of 21% at a strain rate of 3000/s, unloaded, then reloaded at a strain rate of 10
3/s and temperature of 296 K. In Fig. 16, sample 12 is loaded at a

1200 HSLA - 65, T0 = 296K Sample 9

True Stress (MPa)

1000

3,000/s

800

-3

10 /s

600 Sample 10 400

Sample 11

200

0 0.00

0.10

0.20

0.30

0.40

0.50

0.60

True Strain

Fig. 15. Effect of strain rate jump from 3000/s to 10
3/s on flow stress at the same temperature.

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1200 HSLA - 65

True Stress (MPa)

Sample 9 1000

T0 = 296K, 3,000/s

800

T0 = 500K, 10 /s

-3

600

400

Sample 12 Sample 13

200

0 0.00

0.10

0.20

0.30

0.40

0.50

0.60

True Strain

Fig. 16. Effect of strain rate jump from 3000/s to 10
3/s on flow stress for corresponding temperature jump from 296 to 500 K.

strain rate of 10
3/s and temperature of 500 K. Sample 13 is first loaded to a true strain of 21% at a strain rate of 3000/s and an initial temperature of 296 K, then it is unloaded, and then reloaded again at a temperature of 500 K and a strain rate of 10
3/s. As is seen in Figs. 15 and 16, both samples 11 and 13 closely follow the flow stress of 296 and 500 K at a strain rate of 10
3/s. It is seen that there are no essential differences between the responses of samples 10–13 in reloading. These interrupted test results verify that, pre-straining at different strain rates does not seem to result in microstructural changes of sufficient degree to change the materialÕs subsequent response.

3. Physically-based constitutive model 3.1. Evaluation of plastic work-heat conversion factor Plastic deformation generates heat, which is either dissipated to the surroundings or is used to increase the temperature of the material. When the rate of heat generation is greater than the rate of heat loss, the temperature of the material increases. This generally happens at high strain

rates. For materials whose flow stress is temperature dependent, a continuous rise in temperature during deformation results in simultaneous lowering of the flow stress. The temperature rise of a sample can be calculated from Z c b DT ¼ s dc; ð3:1Þ 0 0 q Cv where q 0 is the mass density (7.8 g/cc), CV is the temperature-dependent heat capacity (taken as 0.5 J/g K at room temperature), c is the plastic strain, s is the flow stress in MPa, and b is the fraction of the plastic work which is converted into heat. The value of b is determined experimentally. Data reported by Kapoor and Nemat-Nasser (1998) for several metals suggest that, for large strains (e.g., c P 20%), b is essentially 1. This has also been verified to be the case for several other polycrystalline metals, see Nemat-Nasser et al. (1999), Nemat-Nasser and Isaacs (1997), and Nemat-Nasser and Guo (1999). In the present case, we have also found that b  1.0. To examine whether or not b  1.0 for HSLA65 steel, an indirect experiment is performed. The area under the true stress–true strain curve gives the plastic work per unit volume in uniaxial deformation. In Fig. 17, three samples (designated as

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1400 HSLA - 65, 3,000/s o

1200

40

60MPa True Stresss (MPa)

45

Temp. Rise = 45.5 C

1000

35 o

T0 = 23.2 C 30

800 Sample 14

600

Sample 15

Sample 16

25 20 15

400

Temperature Rise ( oC)

392

10 200 5 0 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0 0.70

True Strain

Fig. 17. Verification of heat conversion.

14, 15, and 16) are loaded at the same strain rate of 3000/s. Sample 14 is loaded to a true strain of about 65% at an initial temperature of 23.2 °C (room temperature). The corresponding true stress–true strain curve is displayed by a dotted curve in Fig. 17. This is essentially an adiabatic true stress–true strain relation for HSLA-65. The temperature rise in this adiabatic test is calculated using Eq. (3.1), with b  1.0. Samples 15 and 16 are first loaded to a true strain of 21%, starting at room temperature (23.2 °C). Their true stress–true strain relations are shown by thick and thin solid curves in Fig. 17. These curves fall on the curve corresponding to sample 14, showing the reproducibility of the test results. The temperature rise at a true strain of 21% is 45.5 °C, calculated by Eq. (3.1) with b  1.0. Then sample 16 is heated to 68.7 °C (45.5 + 23.2) that corresponds to the initial temperature of 23.2 °C, and is reloaded at the same strain rate, producing the second thick curve, shown in Fig. 17. This curve follows closely the adiabatic curve of sample 14. As a check, sample 15 is reloaded at its initial room temperature (23.2 °C), and the corresponding true stress–true strain curve is displayed by the thin solid curve marked sample 15. The stress difference between the adiabatic

curve and this isothermal curve is measured to be about 60 MPa, for a strain increment of 21%. It is clear that this stress difference (60 MPa) is due to thermal softening of the material. Two important conclusions are drawn from these results: (1) if there was any recovery between unloading and reloading, it did not affect the flow stress noticeably, as the interrupted curve of sample 16 follows the uninterrupted curve of sample 14; and (2) essentially the entire plastic work is converted to heat with a negligibly small amount being stored in the sample as the elastic energy of the dislocations and other defects, or lost through sample boundaries. 3.2. Physically-based constitutive model The experimental results described above reveal the following characteristics for HSLA-65 steel: (1) The plastic flow stress of this HSLA-65 strongly depends on the temperature and the strain rate. Its temperature sensitivity is greater for temperatures below about 400 K, and minimal when the temperature exceeds 400 K; (2) Dynamic strain aging occurs at almost all strain rates over a temperature range from 400 to 1000 K, with the temperature range in dynamic strain aging

S. Nemat-Nasser, W.-G. Guo / Mechanics of Materials 37 (2005) 379–405

393

shifting to higher temperatures with increasing strain rates; and (3) The microstructure of the material does not evolve in a manner to affect its flow stress, as the temperature and strain rate are changed. A suitable constitutive model for this material should therefore include all the above effects. Based on the concept of dislocation kinetics, and guided by experimental results, a physically-based model is developed by Nemat-Nasser and Isaacs (1997), Nemat-Nasser et al. (1999) and NematNasser and Guo (1999), and applied to several polycrystalline metals. A similar model which includes all the characteristics observed in HSLA65 structural steel does not exist. In the present work we seek to incorporate the experimental understanding presented above for HSLA-65 steel, into the constitutive model suggested by Nemat-Nasser and co-workers. However, we will not include the dynamic strain aging effects in the model. Consider the plastic flow in the range of temperatures and strain rates where diffusion and creep are not dominant, and the deformation occurs basically by the motion of dislocations. Here, for HSLA-65, we assume that the flow stress, s, consists of two parts: One part is essentially due to the short-range barriers to the dislocation motion, which may include the Peierls stress, point defects such as vacancies and self-interstitials, other dislocations which intersect the slip plane, alloying elements, and solute atoms (interstitial and substitutional). We denote this by s*. The second part is the athermal component, sa, mainly due to the long-range effects such as the elastic stress field of dislocation forests and grain boundaries. Thus, the flow stress is written as

can evolve differently for different loading conditions, that is, for different histories of c_ and T.

s ¼ sa þ s :

s a  a0 þ a1 c n þ ;

ð3:2Þ

In this formulation, the total flow stress of a material, s, is a function of the strain rate, c_ , temperature, T, and some internal microstructural parameters. The microstructure here refers to the grain sizes, the distribution of second-phase particles or precipitates, and the distribution and density of dislocations. In general, the most commonly used microstructural parameter is the average dislocation density, q. The microstructure

3.3. Effect of long-range barriers, sa The athermal part, sa, of the flow stress, s, is independent of the strain rate, c_ . The temperature effect on sa is only through the temperature dependence of the elastic modulus, especially the shear modulus, l(T) (Conrad, 1970). sa mainly depends on the microstructure of the material, e.g., the dislocation density, grain sizes, point defects, and various solute atoms such as those listed in Table 1. Based on linear elasticity, sa would be proportional to l(T). Hence, we set sa ¼ f ðq; d G ; . . .ÞlðT Þ=l0 ;

ð3:3Þ

where q is the average dislocation density, dG is the average grain size, the dots stand for parameters associated with other impurities, and l0 is a reference value of the shear modulus. In a general loading, the strain c represents the effective plastic strain (see Section 5) which is a monotonically increasing quantity in plastic deformation. In the present case, c defines the loading path and is also a monotonically increasing quantity, since c_ > 0. Therefore, it can be used as a load parameter to define the variation of the dislocation density, the average grain size, and other parameters which affect sa, i.e., we may set sa ¼ f ðqðcÞ; d G ðcÞ; . . .ÞlðT Þ=l0 ¼ f^ ðcÞlðT Þ=l:

ð3:4Þ

Furthermore, as a first approximation, we may use a simple power-law representation of f^ ðcÞ, and choose an average value for l0 so that l(T)/ l0  1. Then, sa may be written as ð3:5Þ

where a0, a1 and n are free parameters which must be fixed experimentally. We emphasize that the effective plastic strain, c, or any plastic-strain components cannot in general represent the microstructure, and here c is used strictly as a load parameter. To identify the constitutive parameters for the athermal stress in Eq. (3.4), we examine the variation of the flow stress with the temperature, as

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shown in Figs. 9–11. See especially Fig. 11, which also includes estimates obtained from isothermal experimental data in Fig. 8. To get the athermal flow stress, we focus attention on the flow stress at a high strain rate, say, the results in Fig. 11. When the temperature is suitably high and the flow stress is rather low, the change in the temperature due to the plastic work would be relatively small and can be estimated directly from the corresponding stress–strain curves, without introducing errors of any significance to the analysis. In view of Figs. 9–11, observe that the flow stress is essentially independent of the temperature, beyond a certain critical temperature, e.g., 350 K at strain rates of 10
3/s and 10
1/s, and 700 K at a strain rate of 3000/s. Because dynamic strain aging occurs at all strain rates and at greater temperatures, we evaluate the parameters of the athermal stress component using the results of Fig. 11, from which the high-temperature limiting values of the flow stress are extracted and plotted in Fig. 18, as stress versus the corresponding plastic strain. The points in Fig. 18 nicely fit a simple power law sa ¼ 760;

n ¼ 0:15:

ð3:6Þ

750 HSLA - 65, T0 = 700K, 3,000/s

700 Athermal Stress (MPa)

sa ¼ s0 cn ;

This is taken as the athermal part of the flow stress for this material. From Eq. (3.4), we observe that the function f depends on at least two length scales, one associated with the average dislocation density, and the other with the average grain size. Within the range of our experimental results, both the dislocation density and the grain sizes change with deformation. Since the elastic interaction forces between two isolated dislocations are inversely proportional to their spacing, it is often assumed, after Taylor (1934, 1938), that the flow stress (here only the athermal part) should also display a similar relation and hence be proportional to the square root of the average dislocation density. Be that as it may, there is no reason to expect that the average dislocation density, q(c), should have any pre-defined dependence on the load parameter, i.e., the effective plastic strain c. Indeed, our experimental results give the relation (3.6), with the exponent n ffi 0.15, for the present material, and for other materials, different exponents have been obtained experimentally; see, e.g. (NematNasser and Isaacs, 1997; Nemat-Nasser et al., 1991, 1999, 2001).

650

600

550 Experimental results Equation

500

450 0.00

0.10

0.20

0.30

0.40

0.50

a

= 760

0.60

True Strain Fig. 18. Limiting values of flow stress as a function of strain.

0. 15

0.70

S. Nemat-Nasser, W.-G. Guo / Mechanics of Materials 37 (2005) 379–405

3.4. Thermally-activated component of the flow stress, s* As mentioned before, the thermally-activated part of the flow stress, s*, represents the resistance to the motion of dislocations by the short-range barriers, such as the Peierls stress, point defects, and other dislocations which intersect the slip planes. The quantity s*, in general, is a function of temperature, T, strain rate, c_ , and internal variables characterizing the microstructure of the material. As is discussed in connection with the interrupted test results given in Figs. 13–16, the microstructure of this material does not seem to be very sensitive to temperature and the strain-rate histories. Therefore, s* in the present case is expected to depend on the current values (rather than the history) of c_ and T, and the structure of the short-range barriers. To obtain a relation between c_ , T, and s*, let DG be the energy that a dislocation must have to overcome its short-range barrier by its thermal activation. Kocks et al. (1975) suggest the following relation between DG and s*, representing a typical barrier encountered by a dislocation:    p q s DG ¼ G0 1
; G0 ¼ ^sbk‘ ¼ ^sV  ; ð3:7Þ ^s where 0 < p 6 1 and 1 6 q 6 2 define the profile of the short-range barrier, ^s is the shear stress above which the barrier is crossed by a dislocation without any assistance from thermal activation, and G0 is the energy required for a dislocation to overcome the barrier solely by its thermal activation; b is the magnitude of the Burgers vector; k and ‘ are the average effective barrier width and spacing, respectively; and V* is thepactivation volume. We ffiffiffiffiffiffi note in passing that ‘ ¼ 3 V  provides a natural length scale in this physics-based model. We define the plastic strain rate by c_ ¼ bqmt ¼ bqm x0 expð
DG=kT Þ, and set   DG ; ð3:8Þ c_ ¼ c_ r exp
kT where c_ r ¼ qm; bt; qm is the average density of the mobile dislocations and t ¼ ‘x0 expð
DG=kT Þ is their average velocity, with x0 being the attempt

395

frequency; and k is the Boltzmann constant. From Eqs. (3.7) and (3.8), obtain "  1=q #1=p _ kT c ln : ð3:9Þ s ¼ ^s 1

G0 c_ r In Eq. (3.9), the parameters p and q define the profile of the short-range energy barrier to the motion of dislocations. Ono (1968) and Kocks et al. (1975) suggest that p = 2/3 and q = 2 are suitable values for these parameters for many metals. Nemat-Nasser and co-workers (1997,1998,1999, 2001) have verified this for several metals. Here, for HSLA-65, we also use the same values for p and q in (3.9). The parameters k/G0 and c_ r characterize the temperature and strain-rate sensitivity of the material. Greater temperature sensitivity is associated with the larger k/G0, whereas larger c_ r corresponds to smaller strain-rate sensitivity. The product ðk=G0 Þ= ln c_ r can be estimated directly from the experimental data of Fig. 19. The steps are as follows. The experimental data in Fig. 19 are obtained by subtracting sa, given by Eq. (3.6), from the data in Fig. 11. The results represent the variation of s* with temperature, for indicated strains and a strain rate of 3000/s. It is seen that dynamic strain aging occurs beyond a temperature of about 800 K at this strain rate. We therefore exclude the experimental data for temperatures exceeding 800 K, as we do not wish to include the dynamic strain aging in the present case; see Cheng and Nemat-Nasser (2000) for modeling that includes DSA effects. Therefore, for temperatures less than 800 K, the results in Fig. 19 can be represented by a single curve (solid curve in Fig. 19), given by s ¼ 1450½1
ð0:00125T Þ1=2 3=2 :

ð3:10Þ

Comparing Eqs. (3.9) and (3.10), we conclude that ^s ¼ 1450 MPa, and that


k c_ ln ¼ 0:00125: G0 c_ r

ð3:11Þ

In Eq. (3.11), G0 is the energy of the Peierls barrier per atom, and will be taken to be about 0.8 eV/ atom for the present application to steel. The reference strain rate, c_ r , on the other hand, must be

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1000 HSLA - 65, 3,000/s

Thermally-activated Stress, (MPa)

900 800

τ = 1450 [1 − (0.00125 T ) ∗

700

1/2 3/2

]

600 500

Strain Strain Strain Strain Strain Strain

= 0.05 = 0.15 = 0.25 = 0.35 = 0.45 = 0.55

400 300 200 100 0 0

200

400

600

800

1000

1200

-100 Temperature, (K) Fig. 19. Thermally activated part of flow stress as a function of temperature for any strain.

estimated by direct comparison with experiments. An order-of-magnitude estimate is obtained by ˚ , x0 ffi 1012/s, qm ffi 1015/ noting that d = b ffi 3.3 A 2 3 m , and ‘0 = O(10 ) lattice spacing, leading to c_ r ¼ Oð108 Þ=s. From Eqs. (3.10) and (3.11), and with G0  0.8 eV/atom, it follows that k=G0 ¼ 10:6  10


5


1

K ;

8

c_ r ¼ 4  10 :

ð3:12Þ

Note that, in Eq. (3.10), 0.00125 is given by T
1 c ; see Fig. 19 and Eq. (3.13) below. Finally, note that, in Eq. (3.9), the second term on the right-hand side, i.e., the thermal component of the flow stress, is non-negative, and should be set equal to zero when the temperature exceeds a corresponding critical value which is strain ratedependent, and which is given by  Tc ¼




k c_ ln G0 c_ r


1 :

ð3:13Þ

For c_ ¼ 0:001, 0.1, 3000, and 8500/s, this gives, Tc  350, 430, 800, and 900 K, respectively. Now, the final constitutive relation (denoted as the PB-model, for physics-based) for this material becomes, for T 6 Tc,

s ¼ 760c0:15 (



c_ þ 1450 1

10:6  10 T ln 4  108

where, T ¼ T 0 þ 0:267 have


5

Rc 0

1=2 )3=2 ; ð3:14Þ

s dc and for T > Tc, we

ð3:15Þ s ¼ 760c0:15 ;


1
5 c_ , and b/q 0 Cv ffi where, T c ¼
10:610 ln 4108 0.267 K/MPa. Figs. 20–24 compare the experimental results with the PB-model predictions at strain rates of 10
3/s to 3000/s, for indicated initial temperatures. To further verify the predictability of this model, independent tests at an 8500/s strain rate and various initial temperatures are performed, and the results are displayed in Fig. 25, together with the corresponding model predictions. As is seen, good correlation between these data and the model predictions is obtained. As pointed out before, the PB-model does not include the dynamic strain aging effects, which occur in the temperature range of 400–800 K, at the low strain rates of 10
3/s and 10
1/s. In Figs. 20 and

S. Nemat-Nasser, W.-G. Guo / Mechanics of Materials 37 (2005) 379–405

397

1400 77K

True Stress (MPa)

1200 1000 213K

800

296K 400K

600 400 -3

HSLA - 65, 10 /s Point Curves: Experiments Solid Curves: PB Model Predictions

200 0 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

True Strain

Fig. 20. Comparison of PB-model predictions with experimental results at a strain rate of 0.001/s and indicated initial temperatures.

1600 1400

77K

True Stress (MPa)

1200 1000

213K 296K 400K 500K

800 600 400

-1

HSLA - 65, 10 /s Point Curves: Experiments Solid Curves: PB Model Predictions

200 0 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

True Strain

Fig. 21. Comparison of PB-model predictions with experimental results at a strain rate of 0.1/s and indicated initial temperatures.

21 we have shown the experimental results for these low strain rates. Aside from the effect of dynamic strain aging, the model predictions are in reasonable agreement with the experimental results. Here we note that Eq. (3.10) and the experimental data of Fig. 19 suggest that the activation vol-

ume V * and hence the corresponding length scale ‘* are essentially constant in the present case. This is not, however, generally the case for other materials, as has been shown for titanium and copper in (Nemat-Nasser and Isaacs, 1997; Nemat-Nasser and Li, 1998).

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S. Nemat-Nasser, W.-G. Guo / Mechanics of Materials 37 (2005) 379–405 1600 1400 1200 True Stress (MPa)

T0 = 77K

1000 296K

800

500K 700K

600 400

HSLA - 65, 3,000/s Point Curves: Experiments Solid Curves: PB Model Predictions

200 0 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

True Strain

Fig. 22. Comparison of PB-model predictions with experimental results at a strain rate of 3000/s and indicated initial temperatures.

1200

True Stress (MPa)

1000

213K 400K

800

800K 600K

600

400 HSLA - 65, 3,000/s Point Curves: Experiments Solid Curves: PB Model Predictions

200

0 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

True Strain

Fig. 23. Comparison of PB-model predictions with experimental results at a strain rate of 3000/s and indicated initial temperatures.

4. Assessment by Johnson–Cook model In order to assess the PB-model, we now consider the Johnson–Cook (JC) (Johnson and Cook, 1983) model that has enjoyed much success in various applications (Liang and Khan, 1999), and

apply it to model our experimental results for HSLA-65. As suggested by Johnson and Holmquist (1988), care must be exercised when using extrapolation to estimate the parameters of this model. In the JC-model, the von Mises effective stress, s, is expressed as,

S. Nemat-Nasser, W.-G. Guo / Mechanics of Materials 37 (2005) 379–405

399

2000 HSLA - 65, 3,000/s Light Curves: Experiments Solid Curves: PB Model Predictions

1600 True Stress (MPa)

T0 = 77K

1200 T0 = 296K

800

T0 = 500K

400

0 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

True Strain

Fig. 24. Comparison of PB-model predictions with experimental results at a strain rate of 3000/s and indicated (isothermal) temperatures.

1600 1400

True Stress (MPa)

1200

T0 = 77K

1000

296K 500K

800

700K

600 400

HSLA - 65, 8,500/s Point Curves: Experiments Solid Curves: PB Model Predictions

200 0 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

True Strain

Fig. 25. Comparison of PB-model predictions with experimental results at a strain rate of 8500/s and indicated initial temperatures.

s ¼ ðA þ Bcn Þð1 þ C ln e_  Þð1
T m Þ;

ð4:1Þ

where c as before is the effective plastic strain, e_  ¼ c_ =_c0 is the dimensionless strain rate, (_c0 is normally taken to be 1.0/s), and

T ¼

T
Tr ; Tm
Tr

ð4:2Þ

where Tr is a reference temperature and Tm (=1773 K) is the melting temperature of the material.

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S. Nemat-Nasser, W.-G. Guo / Mechanics of Materials 37 (2005) 379–405

Tr must be chosen as the lowest temperature of interest or the lowest temperature of the experiments because the parameter m is normally less than one, and T should be greater than or equal to Tr for Eq. (4.1) to be valid; in the present case, we take Tr = 50 K. The five material constants, A, B, n, C and m, are obtained such as to achieve good correlation with the experimental results. Table 2 gives the final values of these parameters. The final expression for the Johnson–Cook model for HSLA-65 now is, s ¼ ð790 þ 1320c0:25 Þð1 þ 0:022 ln e_  Þð1
T 0:35 Þ T
50 : T ¼ 1723 ð4:3Þ In this equation, s, c, and e_ are axial true stress, true strain, and the numerical value of the strain

rate (since it is normalized by the 1/s strain rate), respectively. Figs. 26–31 compare the experimental results with the model predictions at strain rates of 10
3/s to 8500/s, for indicated initial temperatures.

5. Application of model to three-dimensional deformation To apply the model for the three-dimensional calculations, we view s and c_ as the effective von Mises stress and strain rate, defined by  1=2  1=2 3 0 0 2 p p s¼ r r D D ; c_ ¼ ; ð5:1Þ 2 ij ij 3 ij ij where r0ij , i, j = 1, 2, 3, are the rectangular Cartesian components of the deviatoric part of the true stress tensor, and Dpij are the components of the deviatoric part of the plastic deformation rate tensor.

Table 2 Values of the parameters in the JC-model for HSLA-65 A

B

n

C

m

Tr (K)

Tm (K)

790

1320

0.25

0.022

0.35

50

1773

1400 77K

1200

True Stress (MPa)

1000 213K

800

296K 400K

600 400 -3

HSLA - 65, 10 /s Point Curves: Experiments Solid Curves: JC Model Predictions

200 0 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

True Strain

Fig. 26. Comparison of JC-model predictions with experimental results at a strain rate of 0.001/s and indicated initial temperatures.

S. Nemat-Nasser, W.-G. Guo / Mechanics of Materials 37 (2005) 379–405

401

1600 1400

77K

True Stress (MPa)

1200 1000

213K 296K 400K 500K

800 600 400

-1

HSLA - 65, 10 /s Point Curves: Experiments Solid Curves: JC Model Predictions

200 0 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

True Strain

Fig. 27. Comparison of JC-model predictions with experimental results at a strain rate of 0.1/s and indicated initial temperatures.

1600 1400 1200 True Stress (MPa)

T0 = 77K

1000 296K

800

500K 700K

600 400

HSLA - 65, 3,000/s Point Curves: Experiments Solid Curves: JC Model Predictions

200 0 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

True Strain

Fig. 28. Comparison of JC-model predictions with experimental results at a strain rate of 3000/s and indicated initial temperatures.

In (5.1), repeated indices are summed over 1, 2, and 3. For uniaxial tests, s is the axial stress and c_ is the axial strain rate. To illustrate the use of model results in threedimensional settings, consider a plasticity model

D

in which the Jaumann rate of the true stress, rij , is related to the elastic deformation rate tensor by D

rij ¼ C ijkl ðDkl
Dpkl Þ: D

rij ¼ r_ ij
W ik rkj þ rik W kj ;

ð5:2Þ

402

S. Nemat-Nasser, W.-G. Guo / Mechanics of Materials 37 (2005) 379–405 1200

True Stress (MPa)

1000

213K 400K

800

600K

600

400 HSLA - 65, 3,000/s Point Curves: Experiments Solid Curves: JC Model Predictions

200

0 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

True Strain

Fig. 29. Comparison of JC-model predictions with experimental results at a strain rate of 3000/s and indicated initial temperatures.

2000 HSLA - 65, 3,000/s Light Curves: Experiments Solid Curves: JC Model Predictions

1600

True Stress (MPa)

T0 = 77K

1200 T0 = 296K

800

T0 = 500K

400

0 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

True Strain

Fig. 30. Comparison of JC-model predictions with experimental results at a strain rate of 3000/s and indicated (isothermal) temperatures.

where Cijkl is the instantaneous elasticity tensor, Dij and Wij are the deformation rate and spin tensors, given in terms of the velocity gradient oti Lij ¼ ox , by Dij ¼ 12 ½Lij þ Lji , and W ij ¼ 12 ½Lij
j Lji , respectively. We now consider the simplest

model for the deviatoric plastic deformation rate, Dpij , as follows: lij Dpij ¼ c_ pffiffiffiffiffiffiffiffi ; 2=3

r0ij lij ¼ pffiffiffiffiffiffiffiffi ; 2=3s

ð5:3Þ

S. Nemat-Nasser, W.-G. Guo / Mechanics of Materials 37 (2005) 379–405

403

1600 1400

True Stress (MPa)

1200

T0 = 77K

1000

296K 500K

800

700K

600 400

HSLA - 65, 8,500/s Point Curves: Experiments Solid Curves: JC Model Predictions

200 0 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

True Strain

Fig. 31. Comparison of JC-model predictions with experimental results at a strain rate of 8500/s and indicated initial temperatures.

with c_ given by (3.8). For DH-36 and using the PBmodel parameters, we have 8 9 "   #2 = < 105 0:15 2=3 s
760c 1
c_ ¼ 4  108 exp
: 10:6 ; 1450 c¼

Z t 0

2 p p D D 3 ij ij

1=2 dt; ð5:4Þ

where s is measured in MPa, and t measures the actual time. 5.1. Comparison of model predictions An assessment of the predictions of the two models has been made by comparing the model results to the experimental data of HSLA-65 steel under different temperatures and different strain rates. Both models show good agreement with the Hopkinson bar data, when dynamic strain aging is excluded. Compared with the PB-model, it is seen that, at the low strain rates of 10
3/s and 10
1/s, the JCmodel has less accurate predictions for temperatures between 77 and 400 K. The reason for this is that the strain-rate constant C has been chosen such that good correlation with the high strain-rate

data is achieved, as the primary application of the model is at higher strain rates. This then leads to a poor prediction of the flow stress at low strain rates. Since the PB-model is based on various aspects of the kinetics and kinematics of dislocation motion, it seems to have a better predictive capability over a broader range of strain rates.

6. Conclusions To understand and model the thermomechanical response of HSLA-65 steel, uniaxial compression tests are performed on cylindrical samples. True strains exceeding 60% are achieved in these tests, over the range of strain rates from 0.001/s to about 8500/s, and at initial temperatures from 77 to 1000 K. In an effort to understand the underlying deformation mechanisms, some interrupted tests with temperature and strain rate jumps are also performed. The microstructure of undeformed and deformed samples is examined. Several noteworthy conclusions are as follows: 1. The experimental results show that the mechanical properties of the HSLA-65 baseplate depend on the material orientation, that is, the steel plate shows an anisotropic behavior.

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2. This steel displays good ductility and plasticity (strain > 60%) at low temperatures (even at 77 K) and high strain rates, without displaying any noticeable damage or microcracks. 3. When the temperature exceeds the room temperature, the flow stress of this material decreases at a rather low rate (especially at low strain rates, i.e., below about 0.1/s) with increasing temperature. Thus, the strength of this material is not temperature-sensitive at high temperatures, showing that the material has good weldability. 4. Dynamic strain aging occurs at temperatures between 500 and 1000 K and the range of strain-rates from 0.001/s to 3000/s, with the peak value of the stress shifting to higher temperatures with increasing strain rates. 5. The microstructure of this material does not evolve with the changes of the strain rate and the temperature, based on our experiments. 6. Based on the experimental results, a physicallybased model is developed. In the absence of dynamic strain aging, the model predictions are in good agreement with the experimental results over a wide range of temperatures and strain rates. 7. As an alternative to this model, the Johnson– Cook model is considered and its free parameters are estimated using our data. Both models show good agreement with the Hopkinson bar data, with the physics-based model having better correlation with the experimental results over a broader range of strain rates. Acknowledgments The authors would like to thank Mr. Jon Isaacs for his assistance in sample preparation. This work was supported by the Naval Surface Warfare Center, Carderock Division (NSWCD) under a program directed by Mr. Scott Natkow, through contract N00167-03-M-0120 with the University of California, San Diego. References Beukel, A.V.D., Kocks, U.F., 1982. The strain dependence of static and dynamic strain-aging. Acta Metall. 30, 1027– 1034.

Cheng, J.Y., Nemat-Nasser, S., 2000. A model for experimentally-observed high-strain-rate dynamic strain aging in titanium. Acta Mater. 48, 3131–3144. Cheng, J.Y., Nemat-Nasser, S., Guo, W.G., 2001. A unified constitutive model for strain-rate and temperature dependent behavior of molybdenum. Mech. Mater. 33, 603–616. Cho, S.-H., Yoo, Y.-C., Jonas, J.J., 2000. Static and dynamic strain aging in 304 austenitic stainless steel at elevated temperature. J. Mater. Sci. Lett. 19, 2019–2022. Conrad, H., 1970. The athermal component of the flow stress in crystalline solids. Mater. Sci. Eng. 6, 260–264. Dhua, S.K., Mukerjee, D., Sarma, D.S., 2001. Influence of tempering on the microstructure and mechanical properties of HSLA-100 steel plate. Metall. Trans. A 32A, 2259–2270. Johnson, G.R., Cook, W.H., 1983. A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures. In: Proceedings of the Seventh International Symposium on Ballistic, The Hague, The Netherlands, pp. 541–547. Johnson, G.R., Holmquist, T.J., 1988. Evaluation of cylinderimpact test data for constitutive model constants. J. Appl. Phys. 64 (8), 3901–3910. Kapoor, R., Nemat-Nasser, S., 1998. Determination of temperature rise during high strain rate deformation. Mech. Mater. 27, 1–12. Kocks, U.F., Argon, A.S., Ashby, M.F., 1975. Thermodynamics and kinetics of slip. Progr. Mater. Sci. 19, 1–271. Krauss, G., 1990. Microstructures, processing, and properties of steel. ASM Handbook 1, 126–139. Kubin, L.P., Estrin, Y., Perrier, C., 1992. On static strain aging. Acta Metall. 40, 1037–1044. Liang, R.Q., Khan, A.S., 1999. A critical review of experimental results and constitutive models for BCC and FCC metals over a wide range of strain rates and temperatures. Int. J. Plast. 15, 963–980. Militzer, M., Hawbolt, E.B., Meadowcroft, T.R., 2000. Microstructural model for hot strip rolling of high-strength lowalloy steels. Metall. Trans. A 31A, 1247–1259. Nakada, Y., Keh, A.S., 1970. Serrated flow in Ni–C alloys. Acta Metall. 18, 437–443. Nemat-Nasser, S., Guo, W.G., 1999. Flow stress of commercially pure niobium over a broad range of temperatures and strain rates. Mater. Sci. Eng. A 284, 202–210. Nemat-Nasser, S., Isaacs, J.B., 1997. Direct measurement of isothermal flow stress of metals at elevated temperatures and high strain rates with application to Ta and Ta–W alloys. Acta Mater. 45, 907–919. Nemat-Nasser, S., Li, Y.L., 1998. Flow stress of fcc polycrystals with application to OFHC Cu. Acta Mater. 46, 565–577. Nemat-Nasser, S., Isaacs, J.B., Starrett, J.E., 1991. Hopkinson techniques for dynamic recovery experiments. Proc. R. Soc. Lond. 435(A), 371–391. Nemat-Nasser, S., Guo, W.G., Liu, M.Q., 1999. Experimentally-based micromechanical modeling of dynamic response of molybdenum. Scripta Mater. 40, 859–872. Nemat-Nasser, S., Guo, W.G., Kihl, D.P., 2001. Thermomechanical response of AL-6XN stainless steel over a wide

S. Nemat-Nasser, W.-G. Guo / Mechanics of Materials 37 (2005) 379–405 range of strain rates and temperatures. J. Mech. Phys. Solids. 49, 1823–1846. Ono, K., 1968. Temperature dependence of dispersed barrier hardening. J. Appl. Phys. 39, 1803–1806.

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Taylor, G.I., 1934. The mechanism of plastic deformation of crystals-I, II. Proc. R. Soc. Lond. A 145, 362–404. Taylor, G.I., 1938. Plastic strain in metals. J. Inst. Metals 62, 307–324.