Description of Hot Compressive Stress-Strain Curves Using Transfer Functions
Abstract
:1. Introduction
2. Materials and Methods
2.1. Steel Fabrication and Sample Preparation
2.2. Modeling Using Transfer Functions and Model Accuracy Estimation Method
3. Results
3.1. Comparison of Measured and Model-Calculated Stress-Strain Curves
3.2. Polynomial Transfer Function Coefficients
3.3. Implemantation of Transfer Functions in other Simulation and Industrial Environents
3.4. Some Aspects of Strain Rate Input Signal
3.5. Stress Prediction for Non-Constant Strain Rates
3.6. Zero Pole Analyses of Obtained Transfer Functions
4. Discussion
5. Conclusions
- Continuous type TF can accurately describe hot-deformation stress-strain relationship, but the individual model has to be used for specific temperature and strain-rate conditions.
- A disadvantage of the proposed TF model is a higher number of model parameters.
- Choices of strain or strain rate for input signal leads to comparable accuracy of obtained models. Selection of strain rate however leads to more dense roots of TF polynomials.
- Similar effect on roots of TF polynomials is obtained by substitution of time with strain in Laplacian transformation.
- Second or third order TF polynomials seem to offer appropriate dynamics for stress strain relationship description, but later offer better accuracy.
- Once TF polynomials are determined, one can easily predict strain response on any ‘reasonable’ strain-rate signal or measurement.
- Transfer Functions models are nowadays supported by most PLCs, PACs and other numerical computational environments and therefore description of stress-strain relationship by TF is supported down to the industrial level. If not by TF, one can transform TF to the preferable model form.
Author Contributions
Funding
Conflicts of Interest
References
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C | Si | Mn | Cr | V |
---|---|---|---|---|
0.50 | 0.30 | 0.95 | 1.00 | 0.15 |
AARE/% | T = 1000 °C | T = 1050 °C | T = 1100 °C | T = 1150 °C | T = 1200 °C |
---|---|---|---|---|---|
= 0.01 s−1 | 0.82 | 0.93 | 1.05 | 0.64 | 0.69 |
= 0.1 s−1 | 1.28 | 1.00 | 1.04 | 0.70 | 0.83 |
= 1 s−1 | 1.02 | 0.59 | 1.10 | 1.55 | 1.37 |
= 10 s−1 | 1.58 | 1.58 | 0.22 | 0.41 | 0.25 |
a3 | a2 | a1 | a0 | b2 | b1 | b0 | |
---|---|---|---|---|---|---|---|
1000//0.01 | −7.376 × 102 | 1.338 × 106 | 3.081 × 107 | 5.711 × 107 | 2.403 × 102 | 3.439 × 103 | 8.941 × 103 |
1000//0.1 | −1.066 × 102 | 1.211 × 105 | 5.752 × 106 | 3.390 × 107 | 2.637 × 102 | 5.241 × 103 | 4.145 × 104 |
1000//1 | −5.758 × 10−1 | 1.461 × 104 | 1.684 × 105 | 1.111 × 105 | 1.611 × 102 | 8.182 × 102 | 1.136 × 103 |
1000//10 | 1.589 × 10−1 | 4.730 × 102 | 4.427 × 103 | 2.160 × 104 | 3.500 × 101 | 2.295 × 102 | 1.048 × 103 |
1050//0.01 | −4.028 × 102 | 1.394 × 106 | 4.114 × 107 | 3.211 × 107 | 3.436 × 102 | 5.428 × 103 | 8.683 × 103 |
1050//0.1 | −6.808 × 101 | 8.533 × 104 | 1.570 × 106 | 1.409 × 107 | 1.758 × 102 | 1.603 × 103 | 1.915 × 104 |
1050//1 | 2.854 | 1.672 × 104 | 2.472 × 105 | 1.950 × 104 | 2.193 × 102 | 1.520 × 103 | 7.147 × 102 |
1050//10 | 8.608 × 10− | 4.429 × 102 | 3.892 × 103 | 1.921 × 104 | 3.657 × 101 | 2.264 × 102 | 1.091 × 103 |
1100//0.01 | −2.596 × 10−2 | 1.429 × 106 | 5.329 × 107 | 6.296 × 107 | 5.179 × 102 | 9.334 × 103 | 1.830 × 104 |
1100//0.1 | −4.620 × 101 | 1.585 × 105 | 3.688 × 106 | 9.175 × 106 | 3.933 × 102 | 4.126 × 103 | 1.574 × 104 |
1100//1 | 9.209 | 1.091 × 104 | 1.447 × 105 | −1.203 × 105 | 1.709 × 102 | 8.852 × 102 | 3.621 × 10−14 |
1100//10 | 1.403 | 3.963 × 102 | 4.189 × 103 | 3.550 × 103 | 4.268 × 101 | 2.556 × 102 | 3.353 × 102 |
1150//0.01 | 9.378 × 101 | 8.453 × 105 | 4.222 × 107 | 5.073 × 107 | 4.217 × 102 | 9.412 × 103 | 1.923 × 104 |
1150//0.1 | −4.426 × 101 | 1.606 × 105 | 5.077 × 106 | 1.299 × 107 | 5.316 × 102 | 7.034 × 103 | 2.646 × 104 |
1150//1 | −2.684 | 1.081 × 104 | 1.265 × 105 | 3.472 × 105 | 1.861 × 102 | 9.997 × 102 | 4.460 × 103 |
1150//10 | 4.321 × 10−1 | 3.464 × 102 | 3.132 × 103 | 4.888 × 103 | 4.140 × 101 | 2.256 × 102 | 4.773 × 102 |
1200//0.01 | 8.630 × 101 | 1.008 × 106 | 4.872 × 107 | 5.894 × 107 | 5.719 × 102 | 1.432 × 104 | 2.712 × 104 |
1200//0.1 | −1.980 × 101 | 1.036 × 105 | 5.033 × 106 | 7.441 × 106 | 4.679 × 102 | 8.789 × 103 | 1.913 × 104 |
1200//1 | −1.317 | 8.841 × 103 | 1.317 × 105 | −1.375 × 105 | 2.231 × 102 | 1.085 × 103 | 1.745 × 10−14 |
1200//10 | 8.736 × 10−1 | 3.175 × 102 | 2.830 × 103 | 1.370 × 104 | 4.218 × 101 | 2.669 × 102 | 1.293 × 103 |
AARE/% | T = 1000 °C | T = 1050 °C | T = 1100 °C | T = 1150 °C | T = 1200 °C |
---|---|---|---|---|---|
= 0.01 s−1 | 3.25 | 4.10 | 2.54 | 1.61 | 1.55 |
= 0.1 s−1 | 5.16 | 2.32 | 3.95 | 2.39 | 2.11 |
= 1 s−1 | 2.82 | 2.19 | 5.96 | 7.13 | 7.55 |
= 10 s−1 | 2.78 | 3.10 | 1.96 | 3.30 | 7.17 |
a2 | a1 | a0 | b1 | b0 | |
---|---|---|---|---|---|
1000/0.01 | 1.914 × 103 | 4.468 × 105 | −1.490 × 105 | 5.107 × 101 | 6.652 × 10−10 |
1000/0.1 | 2.621 × 102 | 3.010 × 104 | 1.524 × 105 | 2.740 × 101 | 1.887 × 102 |
1000/1 | −4.969 | 1.819 × 104 | 4.483 × 105 | 2.431 × 102 | 2.577 × 103 |
1000/10 | 2.504 × 10−1 | 4.836 × 102 | 2.183 × 104 | 7.601 × 101 | 9.907 × 102 |
1050/0.01 | 2.791 × 103 | 1.998 × 105 | −1.068 × 105 | 2.685 × 101 | 8.790 × 10−15 |
1050/0.1 | 3.202 × 102 | 1.369 × 104 | 8.552 × 104 | 1.382 × 101 | 1.193 × 102 |
1050/1 | 1.270 × 10−1 | 2.029 × 104 | 5.492 × 105 | 3.077 × 102 | 3.834 × 103 |
1050/10 | 1.422 × 10−1 | 4.562 × 102 | 2.304 × 104 | 8.508 × 101 | 1.213 × 103 |
1100/0.01 | 2.187 × 103 | 1.353 × 105 | 5.250 × 104 | 2.381 × 101 | 2.367 × 101 |
1100/0.1 | 3.216 × 102 | 1.253 × 104 | 1.033 × 104 | 1.400 × 101 | 2.412 × 101 |
1100/1 | 1.520 | 1.741 × 104 | 8.068 × 105 | 3.791 × 102 | 7.059 × 103 |
1100/10 | 1.382 | 5.424 × 102 | 2.676 × 104 | 1.171 × 102 | 1.781 × 103 |
1150/0.01 | 1.459 × 103 | 1.323 × 105 | 8.470 × 104 | 2.963 × 101 | 3.988 × 101 |
1150/0.1 | 2.515 × 102 | 1.150 × 104 | 1.901 × 104 | 1.593 × 101 | 4.216 × 101 |
1150/1 | −4.192 | 1.393 × 104 | 5.976 × 105 | 3.307 × 102 | 5.954 × 103 |
1150/10 | 2.924 × 10−1 | 2.871 × 102 | 3.799 × 103 | 3.541 × 101 | 2.945 × 102 |
1200/0.01 | 1.377 × 103 | 1.151 × 105 | 7.191 × 104 | 3.400 × 101 | 4.051 × 101 |
1200/0.1 | 1.674 × 102 | 1.320 × 104 | 9.008 × 103 | 2.299 × 101 | 2.873 × 101 |
1200/1 | 3.462 × 10−1 | 1.049 × 104 | 5.737 × 105 | 3.761 × 102 | 7.447 × 103 |
1200/10 | −1.229 | 8.061 × 102 | 1.636 × 105 | 7.588 × 102 | 1.437 × 104 |
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Vode, F.; Malej, S.; Arh, B.; Tehovnik, F.; Podgornik, B. Description of Hot Compressive Stress-Strain Curves Using Transfer Functions. Metals 2019, 9, 290. https://doi.org/10.3390/met9030290
Vode F, Malej S, Arh B, Tehovnik F, Podgornik B. Description of Hot Compressive Stress-Strain Curves Using Transfer Functions. Metals. 2019; 9(3):290. https://doi.org/10.3390/met9030290
Chicago/Turabian StyleVode, Franci, Simon Malej, Boštjan Arh, Franc Tehovnik, and Bojan Podgornik. 2019. "Description of Hot Compressive Stress-Strain Curves Using Transfer Functions" Metals 9, no. 3: 290. https://doi.org/10.3390/met9030290