1 Introduction
2 General considerations
2.1 Basic kinematics
2.2 Thermodynamics
2.3 Parameter fitting
2.4 Root mean squared error and correlation matrix
3 Damage models
3.1 Choice of basic hyperelastic models
3.2 Damage models with virgin state variable
3.3 Damage models with damage variable
3.4 Damage models with amplification variable
4 Experiments
No. | Uniaxial | Planar | Equibiaxial |
---|---|---|---|
1 | 21.48 | 18.36 | 10.22 |
2 | 47.58 | 40.13 | 21.48 |
3 | 79.28 | 65.99 | 33.90 |
4 | 117.79 | 96.76 | 47.58 |
5 | 164.58 | 133.47 | 62.66 |
6 | 221.41 | 177.35 | 79.28 |
7 | 290.45 | 229.89 | 97.60 |
8 | 374.33 | 292.94 | 117.79 |
9 | 476.22 | 368.71 | 140.05 |
10 | 600.00 | 459.93 | 164.58 |
5 Results and discussion
5.1 Stiff compound
Mod. No. | Reference | Basic Mod. | Parameters | Cost function | RMSE | RMSPE | Mean correlation | Mod. Calls |
---|---|---|---|---|---|---|---|---|
1.6* | [15] modified | (b) | 2+3 | 0.1204 | 0.4180 | 0.5911 | 0.3963 | 18 |
3.2 | [34] | (c) | 2+3 | 0.1205 | 0.4185 | 0.4449 | 0.8034 | 67 |
1.4 | [42] | (b) | 3+3 | 0.1205 | 0.4186 | 0.5757 | 0.3435 | 18 |
1.6 | [15] | (b) | 1+3 | 0.1206 | 0.4188 | 0.6337 | 0.4586 | 16 |
1.4* | [42] modified | (b) | 2+3 | 0.1222 | 0.4245 | 0.5684 | 0.3968 | 20 |
3.2* | [34] modified | (c) | 2+3 | 0.1235 | 0.4291 | 0.6571 | 0.6232 | 60 |
1.1 | [11] | (b) | 3+3 | 0.1242 | 0.4316 | 0.4930 | 0.3716 | 20 |
1.5 | [5] | (b) | 3+3 | 0.1268 | 0.4407 | 0.4983 | 0.3710 | 23 |
1.2 | [13] | (c) | 2+3 | 0.1316 | 0.4570 | 0.5881 | 0.3801 | 20 |
1.3 | [33] | (c) | 2+3 | 0.1332 | 0.4627 | 0.5891 | 0.3752 | 22 |
2.3 | [8] | (a) | 2+5 | 0.1427 | 0.4949 | 2.0293 | 0.3810 | 25 |
2.1 | [38] | (a) | 2+5 | 0.1460 | 0.5064 | 0.6444 | 0.4112 | 32 |
2.5 | [2] | (a) | 2+5 | 0.1497 | 0.5191 | 2.6650 | 0.4099 | 22 |
2.2 | [29] | (a) | 2+5 | 0.1519 | 0.5268 | 2.9646 | 0.4323 | 40 |
2.4 | [4] | (a) | 2+5 | 0.1696 | 0.5889 | 0.6908 | 0.3626 | 43 |
2.4* | [4] modified | (a) | 2+5 | 0.1696 | 0.5889 | 0.6908 | 0.3627 | 72 |
3.1b | [23] | (a) | 3+5 | 0.1911 | 0.6620 | 1.3833 | 0.5436 | 25 |
2.6 | [24] | (a) | 2+5 | 0.1914 | 0.6646 | 1.5092 | 0.3957 | 38 |
3.1a | [23] | (b) | 3+3 | 0.1927 | 0.6691 | 1.3027 | 0.5624 | 44 |
5.2 Soft compound
Mod. No. | Reference | Basic Mod. | Parameters | Cost Function | RMSE | RMSPE | Mean Correlation | Mod. Calls |
---|---|---|---|---|---|---|---|---|
1.4 | [42] | (b) | 3+3 | 0.0379 | 0.0964 | 0.4382 | 0.3989 | 108 |
1.4* | [42] | (b) | 2+3 | 0.0380 | 0.0965 | 0.4413 | 0.4506 | 27 |
1.6* | [15] modified | (b) | 2+3 | 0.0381 | 0.0974 | 0.4260 | 0.4229 | 33 |
1.6 | [15] | (b) | 1+3 | 0.0388 | 0.1010 | 0.3950 | 0.5623 | 20 |
3.2 | [34] | (c) | 2+3 | 0.0389 | 0.0988 | 0.3081 | 0.6043 | 38 |
3.2* | [34] modified | (c) | 2+3 | 0.0394 | 0.1026 | 0.3361 | 0.7258 | 38 |
1.2 | [13] | (b) | 2+3 | 0.0396 | 0.1010 | 0.4741 | 0.4448 | 28 |
1.3 | [33] | (b) | 2+3 | 0.0398 | 0.1016 | 0.4766 | 0.4462 | 31 |
1.5 | [5] | (b) | 3+3 | 0.0398 | 0.1016 | 0.4766 | 0.3987 | 34 |
1.1 | [11] | (b) | 3+3 | 0.0460 | 0.1233 | 0.5032 | 0.3609 | 43 |
2.3 | [8] | (a) | 2+5 | 0.0468 | 0.1287 | 0.5013 | 0.4818 | 104 |
2.2 | [29] | (c) | 2+3 | 0.0474 | 0.1317 | 0.3954 | 0.3501 | 67 |
3.1a | [23] | (a) | 3+5 | 0.0486 | 0.1316 | 0.4578 | 0.5520 | 382 |
2.1 | [38] | (c) | 2+3 | 0.0487 | 0.1360 | 0.4740 | 0.4163 | 115 |
2.5 | [2] | (c) | 2+3 | 0.0501 | 0.1371 | 0.5648 | 0.3664 | 17 |
2.6 | [24] | (c) | 2+3 | 0.0508 | 0.1412 | 0.4876 | 0.4030 | 90 |
2.4 | [4] | (c) | 2+3 | 0.0510 | 0.1385 | 0.4866 | 0.3800 | 116 |
2.4* | [4] modified | (c) | 2+3 | 0.0510 | 0.1385 | 0.4866 | 0.3800 | 129 |
3.1b | [23] | (a) | 3+5 | 0.0517 | 0.1454 | 0.5665 | 0.5826 | 77 |