Figure 1.
Schematic diagram of uniaxial tensile sample.
Figure 1.
Schematic diagram of uniaxial tensile sample.
Figure 2.
MTS-Landmark 810 universal testing machine.
Figure 2.
MTS-Landmark 810 universal testing machine.
Figure 3.
Specimen installation diagram.
Figure 3.
Specimen installation diagram.
Figure 4.
The schematic diagram of test loading-unloading process.
Figure 4.
The schematic diagram of test loading-unloading process.
Figure 5.
Hysteresis curves with different strain amplitudes at strain rate of 0.0025%/s ( is stress, is strain, is strain amplitude).
Figure 5.
Hysteresis curves with different strain amplitudes at strain rate of 0.0025%/s ( is stress, is strain, is strain amplitude).
Figure 6.
Hysteresis curves with different strain amplitudes at strain rate of 0.005%/s ( is stress, is strain, is strain amplitude).
Figure 6.
Hysteresis curves with different strain amplitudes at strain rate of 0.005%/s ( is stress, is strain, is strain amplitude).
Figure 7.
Hysteresis curves with different strain amplitudes at strain rate of 0.01%/s ( is stress, is strain, is strain amplitude).
Figure 7.
Hysteresis curves with different strain amplitudes at strain rate of 0.01%/s ( is stress, is strain, is strain amplitude).
Figure 8.
Schematic diagram of spring-pot element.
Figure 8.
Schematic diagram of spring-pot element.
Figure 9.
Fractional Maxwell model of M2052 damping alloy.
Figure 9.
Fractional Maxwell model of M2052 damping alloy.
Figure 10.
Microstructure of twins. (a) Schematic diagram of twin microstructure; (b) Electron micrograph of twin structure.
Figure 10.
Microstructure of twins. (a) Schematic diagram of twin microstructure; (b) Electron micrograph of twin structure.
Figure 11.
Twins movement under stress [
30]. (
a) The original lattice structure. (
b) The deformed lattice structure. (
c) The twin crystal lattice structure. (
d) The larger twin crystal lattice structure. (
e) The divided of twin crystal lattice structure. (
f) An enlarged local view of the twin boundary.
Figure 11.
Twins movement under stress [
30]. (
a) The original lattice structure. (
b) The deformed lattice structure. (
c) The twin crystal lattice structure. (
d) The larger twin crystal lattice structure. (
e) The divided of twin crystal lattice structure. (
f) An enlarged local view of the twin boundary.
Figure 12.
The relation of stress-induced martensite and strain-induced martensite. ( is stress. is the critical stress of stress-induced martensite. is the nominal stress of stress-induced martensite at T2. is the stress of strain-induced martensite at T2. is the stress required for the transformation of austenite. T is temperature. Ms is the start temperature of the martensitic transformation. Md is the end temperature of the martensitic transformation. T1 is the room temperature.).
Figure 12.
The relation of stress-induced martensite and strain-induced martensite. ( is stress. is the critical stress of stress-induced martensite. is the nominal stress of stress-induced martensite at T2. is the stress of strain-induced martensite at T2. is the stress required for the transformation of austenite. T is temperature. Ms is the start temperature of the martensitic transformation. Md is the end temperature of the martensitic transformation. T1 is the room temperature.).
Figure 13.
Schematic diagram of energy conversion.
Figure 13.
Schematic diagram of energy conversion.
Figure 14.
The fitting curve of stress in loading stage for fixed C = 100.
Figure 14.
The fitting curve of stress in loading stage for fixed C = 100.
Figure 15.
The fitting curve of stress in loading stage for fixed .
Figure 15.
The fitting curve of stress in loading stage for fixed .
Figure 16.
Flow chart of genetic optimization.
Figure 16.
Flow chart of genetic optimization.
Figure 17.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 17.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 18.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 18.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 19.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 19.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 20.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 20.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 21.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 21.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 22.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 22.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 23.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 23.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 24.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 24.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 25.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 25.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 26.
The deviation diagram of uniaxial cyclic tensile test data with the fitting curve of fractional Maxwell model in the loading stage (x-axial is strain (%), y-axial is stress (MPa), (%/s) is strain rate).
Figure 26.
The deviation diagram of uniaxial cyclic tensile test data with the fitting curve of fractional Maxwell model in the loading stage (x-axial is strain (%), y-axial is stress (MPa), (%/s) is strain rate).
Figure 27.
Fitting diagram of correction terms (x-axial is strain (%), y-axial is stress (MPa), (%/s) is strain rate, red is the fitting curve and blue is the original deviation value).
Figure 27.
Fitting diagram of correction terms (x-axial is strain (%), y-axial is stress (MPa), (%/s) is strain rate, red is the fitting curve and blue is the original deviation value).
Figure 28.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 28.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 29.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 29.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 30.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 30.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 31.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 31.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 32.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 32.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 33.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 33.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 34.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 34.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 35.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 35.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 36.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 36.
Comparison of experimental stress–strain curves and model predictions at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 37.
The unloading deviation curve and the unloading correction term fitting curve. (a) the unloading deviation curve; (b) unloading correction term fitting curve.
Figure 37.
The unloading deviation curve and the unloading correction term fitting curve. (a) the unloading deviation curve; (b) unloading correction term fitting curve.
Figure 38.
The modified fractional Maxwell model curve fitted according to the unloading section compared with the experiment data at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 38.
The modified fractional Maxwell model curve fitted according to the unloading section compared with the experiment data at and . ( is strain, (MPa) is stress, is strain rate, is strain amplitude, the fitting curve is red and the experiment data is black.).
Figure 39.
The schematic of classic Maxwell model.
Figure 39.
The schematic of classic Maxwell model.
Figure 40.
The schematic of Maxwell 3 parameters model.
Figure 40.
The schematic of Maxwell 3 parameters model.
Figure 41.
The schematic of Maxwell four-parameter model.
Figure 41.
The schematic of Maxwell four-parameter model.
Figure 42.
The optimal Maxwell models fit to the loading experimental data at and . ( is strain rate, is strain amplitude.).
Figure 42.
The optimal Maxwell models fit to the loading experimental data at and . ( is strain rate, is strain amplitude.).
Figure 43.
The value of at different conditions and the fitting of the mean value of .
Figure 43.
The value of at different conditions and the fitting of the mean value of .
Figure 44.
The value of C at different conditions and the fitting of the mean value of C.
Figure 44.
The value of C at different conditions and the fitting of the mean value of C.
Figure 45.
The value of a at different conditions and the fitting of the mean value of a.
Figure 45.
The value of a at different conditions and the fitting of the mean value of a.
Figure 46.
The value of b at different conditions and the fitting of the mean value of b.
Figure 46.
The value of b at different conditions and the fitting of the mean value of b.
Figure 47.
The value of c at different conditions and the fitting of the mean value of c.
Figure 47.
The value of c at different conditions and the fitting of the mean value of c.
Table 1.
Natural properties of M2052 damping alloy.
Table 1.
Natural properties of M2052 damping alloy.
Name | Tensile Strength | Young’s Modulus | Poisson’s Ratio | Yield Strength | Density |
---|
Value | 540 MPa | 68.5 GPa | 0.338 | 205 MPa | 7.25 g/cm3 |
Table 2.
Hysteresis area under different strain rates and strain amplitudes (unit: 10 kJ/m3).
Table 2.
Hysteresis area under different strain rates and strain amplitudes (unit: 10 kJ/m3).
Strain Amplitudes | 0.05% | 0.1% | 0.15% |
---|
Strain Rates | 0.0025%/s | 0.041872 | 0.147943 | 0.503593 |
0.005%/s | 0.036426 | 0.180372 | 0.3791682 |
0.01%/s | 0.071483 | 0.319285 | 0.653956 |
Table 3.
The slope of curve under different strain rates and strain amplitudes.
Table 3.
The slope of curve under different strain rates and strain amplitudes.
Strain Amplitudes | 0.05% | 0.1% | 0.15% |
---|
Strain Rates | 0.0025%/s | 730.61979 | 687.76791 | 669.23966 |
0.005%/s | 724.0259 | 694.45375 | 680.06215 |
0.01%/s | 703.77106 | 689.28956 | 681.57759 |
Table 4.
The strain amplitude error between the set value and the measured value.
Table 4.
The strain amplitude error between the set value and the measured value.
Number | | | | Error (%) | The Relative Error (%) |
---|
1 | 0.0025 | 0.05% | 0.04748 | 0.00437 | 0.0504 |
2 | 0.1% | 0.09132 | 0.00797 | 0.0868 |
3 | 0.15% | 0.14239 | 0.00681 | 0.05073 |
4 | 0.005 | 0.05% | 0.04332 | 0.00372 | 0.1336 |
5 | 0.1% | 0.09123 | 0.00696 | 0.0877 |
6 | 0.15% | 0.14132 | 0.00779 | 0.05787 |
7 | 0.01 | 0.05% | 0.05668 | −0.01243 | −0.1336 |
8 | 0.1% | 0.11754 | −0.01848 | −0.1754 |
9 | 0.15% | 0.1558 | −0.01686 | −0.03867 |
Table 5.
Fractional Maxwell model fitting coefficient.
Table 5.
Fractional Maxwell model fitting coefficient.
Number | | | Generations | Fval | | C |
---|
1 | 0.0025 | 0.05 | 18,191 | 1.9510 | 0.0102 | 12,380.56 |
2 | 0.1 | 11,591 | 1.123 | 0.0044 | 8929 |
3 | 0.15 | 19,989 | 3.8929 | 0.0046 | 12,374 |
4 | 0.005 | 0.05 | 28,735 | 2.0648 | 0.0125 | 21,216.625 |
5 | 0.1 | 20,263 | 3.98 | 0.0016 | 15,824.938 |
6 | 0.15 | 10,449 | 3.92 | 0.0011 | 11,160.78 |
7 | 0.01 | 0.05 | 34,455 | 0.7885 | 7.2797 × 10−4 | 23,822 |
8 | 0.1 | 44,922 | 2.6657 | 5.1945× 10−4 | 29,563 |
9 | 0.15 | 27,016 | 0.7995 | 0.0079 | 20,059.3 |
Table 6.
Fitting coefficient and evaluation index of the modified item.
Table 6.
Fitting coefficient and evaluation index of the modified item.
Number | | | a | b | c | SSE (Variance) | R2 (Determinate Coefficient) | RMSE (Mean Square Root) |
---|
1 | 0.0025 | 0.05% | 1.777 | 50.99 | −0.3816 | 0.0037 | 0.9999 | 0.0046 |
2 | 0.1% | 1.99 | 28.66 | −0.3183 | 1.47 | 0.9917 | 0.0609 |
3 | 0.15% | 2.45 | 22.61 | 0.0504 | 258.2 | 0.5804 | 0.579 |
4 | 0.005 | 0.05% | 1.937 | 34.69 | −0.164 | 30.99 | 0.4756 | 0.5836 |
5 | 0.1% | 2.408 | 31.6 | −0.1376 | 60.24 | 0.6412 | 0.5488 |
6 | 0.15% | 2.255 | 16.97 | 0.1778 | 162.3 | 0.3146 | 0.7405 |
7 | 0.01 | 0.05% | 0.8868 | 52.32 | −0.5697 | 17.88 | 0.2591 | 0.5504 |
8 | 0.1% | 1.935 | 21.92 | −0.0335 | 34.78 | 0.4599 | 0.5407 |
9 | 0.15% | 0.9247 | 18.7 | −0.1211 | 56.27 | 0.2077 | 0.5639 |
Table 7.
The evaluation index of modified fractional Maxwell model.
Table 7.
The evaluation index of modified fractional Maxwell model.
Number | | | SSE (Variance) | RMSE (Mean Square Root) | R2 (Determinate Coefficient) |
---|
1 | 0.0025 | 0.05 | 181.0160 | 0.4897 | 0.9933 |
2 | 0.1 | 324.4874 | 0.3986 | 0.9989 |
3 | 0.15 | 986.723 | 0.8313 | 0.9991 |
4 | 0.005 | 0.05 | 101.1080 | 0.5266 | 0.9938 |
5 | 0.1 | 286.8182 | 0.6895 | 0.9983 |
6 | 0.15% | 643.7925 | 1.0537 | 0.999 |
7 | 0.01 | 0.05 | 57.6222 | 0.4237 | 0.9972 |
8 | 0.1 | 277.4389 | 1.0837 | 0.9986 |
9 | 0.15 | 1843.8 | 5.1216 | 0.9975 |
Table 8.
Parameters and evaluation index of Maxwell models.
Table 8.
Parameters and evaluation index of Maxwell models.
Number | Name | E1 (GPa) | E2 (GPa) | | | SSE (Variance) | R2 (Determinate Coefficient) | RMSE (Mean Square Root) |
---|
1 | Two-parameter model | 68.5 | - | 3.993 × 105 | - | 1.926 × 108 | 0.9969 | 976.5 |
2 | Three-parameter model | 2.22 × 10−14 | 68.5 | 75.06 | - | 1.925 × 108 | 0.9969 | 976.1 |
3 | Four-parameter model | 1.243 | 67.257 | 1205 | 1503 | 2.235 × 108 | 0.9964 | 1057 |
Table 9.
The average values of model parameters under the same strain amplitudes but different strain rates.
Table 9.
The average values of model parameters under the same strain amplitudes but different strain rates.
Number | Strain Amplitude (%) | Strain Rate (%/s) | α | C | a | b | c |
---|
1 | 0.05% | 0.0025 | 0.0102 | 12,380.56 | 1.777 | 50.99 | −0.3816 |
2 | 0.005 | 0.0125 | 21,216.625 | 1.937 | 34.69 | −0.164 |
3 | 0.01 | 7.28 × 10−4 | 23,822 | 0.8868 | 52.32 | −0.5697 |
4 | Average | 0.007809 | 19,139.72833 | 1.5336 | 46 | −0.371767 |
5 | 0.10% | 0.0025 | 0.0044 | 8929 | 1.99 | 28.66 | −0.3183 |
6 | 0.005 | 0.0016 | 15,824.938 | 2.408 | 31.6 | −0.1376 |
7 | 0.01 | 5.19 × 10−4 | 29,563 | 1.935 | 21.92 | −0.0335 |
8 | Average | 0.002173 | 18,105.65 | 2.111 | 27.39333 | −0.16313 |
9 | 0.15% | 0.0025 | 0.0046 | 12,374 | 2.45 | 22.61 | 0.0504 |
10 | 0.005 | 0.0011 | 11,160.78 | 2.255 | 16.97 | 0.1778 |
11 | 0.01 | 0.0079 | 20,059.3 | 4.886 | 17.34 | 0.07346 |
12 | Average | 0.004533 | 14,531.36 | 3.197 | 18.97333 | 0.100553 |
Table 10.
Model parameters under different loading conditions after averaging.
Table 10.
Model parameters under different loading conditions after averaging.
Number | Strain Rate (%/s) | Strain Amplitude (%) | α | C | a | b | c |
---|
1 | 0.0025 | 0.05 | 0.007809 | 19,139.72833 | 1.5336 | 46 | −0.371767 |
2 | 0.1 | 0.002173 | 18,105.65 | 2.111 | 27.39333 | −0.16313 |
3 | 0.15 | 0.004533 | 14,531.36 | 3.197 | 18.97333 | 0.100553 |
4 | 0.005 | 0.05 | 0.007809 | 19,139.72833 | 1.5336 | 46 | −0.371767 |
5 | 0.1 | 0.002173 | 18,105.65 | 2.111 | 27.39333 | −0.16313 |
6 | 0.15 | 0.004533 | 14,531.36 | 3.197 | 18.97333 | 0.100553 |
7 | 0.01 | 0.05 | 0.007809 | 19,139.72833 | 1.5336 | 46 | −0.371767 |
8 | 0.1 | 0.002173 | 18,105.65 | 2.111 | 27.39333 | −0.16313 |
9 | 0.15 | 0.004533 | 14,531.36 | 3.197 | 18.97333 | 0.100553 |
Table 11.
The error of mean parameters fitting curve with experimental curve and original model parameter fitting curve.
Table 11.
The error of mean parameters fitting curve with experimental curve and original model parameter fitting curve.
Number | | | MSE | SSE | R2 | MSE | SSE | R2 |
---|
Compared with Experimental Curve | Compared with Original Model Parameter Fitting Curve |
---|
1 | 0.0025 | 0.05% | 0.3401 | 59.5182 | 0.9955 | 0.0493 | 8.6320 | 0.9999 |
2 | 0.1% | 0.3633 | 145.3041 | 0.9991 | 0.1087 | 43.4667 | 0.9999 |
3 | 0.15% | 1.2377 | 736.4596 | 0.9994 | 0.7022 | 417.7823 | 0.9999 |
4 | 0.005 | 0.05% | 0.3900 | 36.6571 | 0.9954 | 0.0600 | 5.6439 | 0.9999 |
5 | 0.1% | 0.4237 | 86.0018 | 0.9988 | 0.1286 | 26.1143 | 0.9998 |
6 | 0.15% | 1.0439 | 312.1178 | 0.9992 | 0.5027 | 150.2989 | 0.9999 |
7 | 0.01 | 0.05% | 0.5603 | 34.7357 | 0.9976 | 0.2715 | 16.8323 | 0.9999 |
8 | 0.1% | 0.4168 | 50.8456 | 0.9993 | 0.1352 | 16.4890 | 0.9999 |
9 | 0.15% | 2.7624 | 497.2249 | 0.9993 | 2.4484 | 440.7147 | 0.9996 |
Table 12.
The fitting function of each parameter.
Table 12.
The fitting function of each parameter.
Number | Name | Function |
---|
1 | | |
2 | | |
3 | | |
4 | | |
5 | | |