DYNAMIC CONSTITUTIVE EQUATION OF THE AUTO-BODY STEEL SHEET WITH THE VARIATION OF TEMPERATURE
Hoon Huh*, Jung-Han Song*, Hee-Jong Lee* and Sung-Ho Park**
*Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Daeduck Science Town, Daejeon, 305-701, Korea, *[email protected],
**[email protected], ***[email protected]
**POSCO Automotive Steel Research Laboratory, 699, Gumho-dong, Gwangyang-si, Jeonnam, 545-090, Korea, [email protected]
ABSTRACT-This paper is concerned with the empirical constitutive equation for the flow stress of steel sheets used in an auto-body with the variation of temperature and strain-rate. Dynamic tensile tests are carried out with steel sheets at various temperatures.
Both the strain-rate sensitivity and the temperature sensitivity of the flow stress are investigated at the intermediate strain-rate. In order to describe the thermo-mechanical behavior of the steel sheet at the intermediate strain-rate accurately, a hardening equation is newly suggested by modifying the well-known Khan–Huang model. The new model suggested gives good correlation with experimental results at various intermediate strain- rates and temperatures.
INTRODUCTION: An accurate stress–strain curve at the intermediate strain-rate has to be utilized in the crash analysis in order to achieve the light-weight and safe design of an auto-body. Some empirical constitutive models such as the Cowper–Symonds model [Jones, 1989], the Johnson–Cook model [Johnson and Cook, 1985] and the Khan–Huang model [Liang and Khan, 1999] have been suggested in order to apply the dynamic thermo-mechanical behavior of the flow stress to the numerical simulation of structures.
Since these models are based on the experimental result obtained from the Hopkinson bar test at the high strain-rate, the flow stress at the intermediate strain-rate of the car crash should be estimated by interpolating the flow stress obtained from the quasi-static test and the high strain-rate test. This interpolation scheme is not able to accurately describe the variation of the flow stress at the range of intermediate strain-rates.
In this paper, an empirical constitutive equation is newly suggested in order to accurately represent the dynamic thermo-mechanical response of the flow stress at the intermediate strain-rate. Dynamic tensile tests were carried out with steel sheets for an auto-body such as SPRC35R, SPRC45E and TRIP60 at various temperatures. The strain-rate sensitivity and the temperature sensitivity of the flow stress are investigated at the intermediate strain-rate and those characteristics are considered to suggest a hardening equation coupled with the strain, the strain-rate and the temperature. The new model suggested gives good correlation with experimental results at the intermediate strain-rate of the car crash and the working temperatures.
EXPERIMENTS AND RESULTS: Dynamic tensile tests were carried out with the variation of the environmental temperate from –40℃ to 200℃ while the strain-rate is
139
0.0 0.1 0.2 0.3 0.4 0.5 0
100 200 300 400 500 600 SPRC45E
Engineering Stress (MPa)
Engineering Strain 200 /sec 100 /sec 10 /sec 1 /sec 0.1 /sec 0.001 /sec
- 40 °C
0.0 0.1 0.2 0.3 0.4 0
.5 0
100 200 300 400 500
600 SPRC45E 25 °C
Engineering Stress (MPa)
Engineering Strain 200 /sec 100 /sec 10 /sec 1 /sec 0.1 /sec 0.001 /sec
0.0 0.1 0.2 0.3 0.4 0.5
0 100 200 300 400 500 600
SPRC45E 200 °C
200 /sec 100 /sec 10 /sec 1 /sec 0.1 /sec 0.001 /sec
Engineering Stress (MPa)
Engineering Strain
Fig. 1 Stress–strain curves of SPRC45E with the variation of strain rate and temperature
-50 0 50 100 150 200
0 200 300 400 500 600
Experimental results Fitted curve
200/sec 100/sec 10/sec 1/sec 0.1/sec 0.001/sec ε =0 (Yield)
Flow Stress (MPa)
Temperature (°C) SPRC35R
-50 0 50 100 150 200
0 200 300 400 500 600
200/sec 100/sec 10/sec 1/sec 0.1/sec 0.001/sec
Experimental results Fitted curve
ε =0 (Yield) SPRC45E
Flow Stress (MPa)
Temperature (°C)
-50 0 50 100 150 200
0 300 400 500 600 700
Experimental results Fitted curve
Flow Stress (MPa)
Temperature (°C) ε =0 (Yield) TRIP60
200/sec 100/sec 10/sec 1/sec 0.1/sec 0.001/sec
Fig. 2 Strain rate and temperature sensitivity of yield stress for steel sheets
assigned to be ranged from 0.001/s to 200/s. A servo-hydraulic type material testing machine is utilized for the dynamic material test [Huh et al., 2004].
Stress–strain curves of SPRC45E are shown with the variation of the strain-rate and temperature in Fig. 1. The variation of the yield stress with respect to the strain-rate and temperature is depicted for the temperature sensitivity in Fig. 2. The figures represent that strain-rate sensitivity is decreased as the temperature increase while the thermal sensitivity is increased as the strain-rate increases. The figures also indicate that the low strength steel is more sensitive to the strain-rate and the temperature than the high strength steel.
CONSTITUTIVE EQUATION: In order to describe the strain-rate and temperature dependent behavior of the flow stress at the intermediate strain-rate, a constitutive equation is newly suggested by modifying the well-known Khan–Huang model. The suggested model is represented by Eqn. (1).
(
C( ) ) (
T m)
B D
A n p
n
p
*
* 0
*
1 ln
ln 1
1 ln 0 ⎟⎟ + −
⎠
⎞
⎜⎜
⎝
⎛
⎟⎟⎠
⎜⎜ ⎞
⎝
⎛ − +
= ε ε ε
σ & & (1)
( )
εε ε
ε& & & , 1 , ln&
, s / 10 ,
where D0p = 6 * = ref n=q1⋅ −T*q2 m=m1 +m2⋅ (2) In the above model, the strain-rate hardening is interpolated with exponent function of the normalized strain-rate while the Khan−Huang model interpolates the flow stress as a linear function with the log scale of the strain-rate. The strain-rate and temperature sensitivity are also considered by coupling the strain, the strain-rate and the temperature as represented by Eqn. (2). Flow stress curves obtained from the experiment are compared with those interpolations by the suggested model and the Khan−Huang model
140
0.00 0.05 0.10 0.15 0.20
0 100 200 300 400 500 600 700
True Strain
True Stress (MPa)
200 °C SPRC45E
200/sec 100/sec 10/sec 1/sec 0.1/sec 0.001/sec Stress Curves from Experiments
Khan-Huang Model
0.00 0.05 0.10 0.15 0.20
0 100 200 300 400 500 600 700
True Strain
True Stress (MPa)
200/sec 100/sec 10/sec 1/sec 0.1/sec 0.001/sec Stress Curves from Experiments
Khan-Huang Model
- 40 °C SPRC45E
0.00 0.05 0.10 0.15 0.20
0 100 200 300 400 500 600 700
SPRC45E 25 °C
True Stress (MPa) 200/sec
100/sec 10/sec 1/sec Stress Curves from Experiments 0.1/sec
Khan-Huang Model 0.001/sec
True Strain
(a) Khan–Huang model
0.00 0.05 0.10 0.15 0.20
0 100 200 300 400 500 600 700
Stress Curves from Experiments Modified Khan-Huang Model
200/sec 100/sec 10/sec 1/sec 0.1/sec 0.001/sec
0.00 0.05 0.10 0.15 0.20
0 100 200 300 400 500 600 700
200/sec 100/sec 10/sec 1/sec 0.1/sec 0.001/sec Stress Curves from Experiments
Modified Khan-Huang Model
True Stress (MPa)
True Strain - 40 °C SPRC45E
0.00 0.05 0.10 0.15 0.20
0 100 200 300 400 500 600 700
200/sec 100/sec 10/sec 1/sec 0.1/sec 0.001/sec Stress Curves from Experiments
Modified Khan-Huang Model
True Strain
True Stress (MPa)
200 °C SPRC45E SPRC45E 25 °C
True Stress (MPa)
True Strain
(b) modified Khan–Huang model
Fig. 3 Comparison of the interpolated flow stress with respect to the constitutive model
as shown in Fig. 3. The comparison demonstrates that the suggested model gives relatively accurate description of experimental results at various strain-rates and temperatures.
CONCLUSION: This paper newly proposes an empirical constitutive equation by modifying the Khan–Huang model. Dynamic tensile tests were carried out with several steel sheets at various strain-rates and temperatures. The strain-rate and the temperature sensitivity of the flow stress are evaluated at the intermediate strain-rate in order to formulate the suggested hardening equation by coupling the strain, the strain-rate and the temperature. The model suggested nicely describes the dynamic thermo-mechanical behavior of steel sheets at the intermediate strain-rate and working temperature. The model provides indispensable information for accurate crash simulation of the auto-body structure.
REFERENCES:
Jones, N., 1989, Structural Impact, Cambridge University Press, Cambridge, U.K.
Johnson, G. R. and Cook, W. H., 1985, “Fracture Characteristics of Three Metals Subjected to Various Strains, Strain rates, Temperatures and Pressures”, Eng. Frac.
Mech., 21, 31.
Liang, R. and 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. Plasticity, 15, 963.
Huh, H., Lim, J. H., Kim, S. B., Han , S. S. and Park, S. H., 2004, “Formability of the Steel Sheet at the Intermediate Strain Rate”, Key Eng. Mater., 274-276, 403.
141
