Locking nuts are widely used in industry and any defects from their manufacturing may cause loosening of the connection during their service life. In this study, simulations of the folding process of a nut’s flange made from AISI 1040 steel are performed. Besides the bilinear isotropic hardening rule, Chaboche’s nonlinear kinematic hardening rule is employed with associated flow rule and Hill48 yield criterion to set a plasticity model. The bilinear isotropic hardening rule’s parameters are determined by means of a monotonic tensile test. The Chaboche’s parameters are determined by using a low cycle tension/compression test by applying curve fitting methods on the low cycle fatigue loop. Furthermore, the parameter calibrations are performed in the finite element simulations by using an optimization approach based on the inverse analysis. Dimensional accuracy for the nut is of primary concern due to the tolerance constraints of the nut manufacturers. Experimental diameter and height measurements of the folded locking nut are compared with those obtained from the optimized model. The results reveal that the folding dimensions can be predicted more accurately when the model parameters are determined by using the combined hardening rule. The calibrated parameters are presented for the folding and cycling deformation processes.
Nylock nuts have very intensive usage areas among other lock nuts especially in the automotive industry [
Although finite element (FE) simulation with suitable model and parameters is a useful tool for the plastic deformation prediction, their prediction performances are still dependant on the elastic and inelastic models to be used. A yield function, a hardening rule, and a flow rule must be combined to set a plasticity model. Lots of functions and rules for plasticity were presented in the literature. Thus, composing a suitable model for the case is another handicap. No unique method has been developed yet for selection of a model and determination of its parameters. It depends on the material type and deformation process strictly.
One of the solutions to the problems above is to calibrate the material parameters or change the model with a more advanced one. The complex nature of the advanced models may cause much more time-consumption during their implementation. Inverse analysis is a widely used method for parameter calibration [
During any plastic deformation process, hardening or softening occurs due to locking or releasing of dislocation movements when yield starts. While the isotropic hardening rule governs the evolution of the expansion or contraction of the yield surface, the kinematic hardening rule controls the evolution of the back stress
The aim of this study is to investigate a suitable model and its parameters for the nut flange deformation process leading to an improved dimensional prediction and accurate simulating of the hysteresis loop. The Chaboche kinematic hardening rule (CHAB) and bilinear isotropic (BISO) hardening rule commonly used in the literature are implemented. The novelty of the work is that the models are combined for the nut flange folding simulations which have importance for the manufacturing industry. Then their calibrations are performed in the FE simulations by using an optimization approach based on the inverse analysis. Dimensional accuracy for the nut is of primary concern due to the tolerance constraints of the nut manufacturers. Therefore, the parameters are optimized based on nut’s diameter and length measurements. Finally, the nut flange folding behaviour of AISI 1040 has been simulated by using the optimized material parameters. The investigation shows that the rule’s parameters determined experimentally from a series of strain controlled low cycle uniaxial tension-compression tests can be used instead of more complex deformation processes. Validations are done by checking the dimension and hysteresis loop shapes from experiments and predictions.
The tensile tests were carried out using the specimens given in
The nut (AISI 1040) | The ring (PA6) | |
---|---|---|
Density [ |
7.85 gr/cm3 | 0.952 gr/cm3 |
Young’s Modulus [ |
207 GPa | 1100 MPa |
Poisson ratio [ |
0.28 | 0.42 |
Yield strength, |
415 MPa | 26.2 MPa |
Maximum strength | 651.84 MPa | 72.81 MPa |
Tangent modulus, |
2014.21 MPa | 1070 MPa |
1.11 | -- | |
1.01 | -- | |
1.27 | -- |
True stress-strain data was used for calculations. Kacar and Kılıç explained how to remove the elastic strain in detail [
BISO is good at modelling the material behaviour subjected to any plastic deformation in which just a monotonic loading and elastic unloading case are seen. However, it may be not enough by itself when reversal loads arise. Therefore, it is combined with CHAB. CHAB’s parameters are determined by using a hysteresis loop obtained from the low cycle fatigue testing [
A low cycle fatigue test with tension-compression loads in which the strain is symmetric gives the hysteresis loops as seen in
The maximum stress value in the tensile course is different in the compression course in one loop. Also it is seen that the stress level increases when the cycle is getting closer to the end. The Bauschinger effect and the strain hardening lead to these behaviours. In the compression course, it is very hard to keep the deformation in-plane due to buckling [
In the simulation, the material’s nonlinear mechanical behaviour is set up by using a constitutive model. The Hill48 yield criterion is used in the constitutive model [
A stress state can be transformed to an equivalent stress value by means of a yield criterion’s equation. Thus, it is a convenient tool to compare any stress state to the material’s yield strength to determine whether plastic deformation has started or not. A general comparison formula is given in
where
where
where
Therefore,
The isotropic term and the back stress term representing the kinematic rule are added to the comparison equation of the yield criterion as in
where
where
where
The back stress in
where
Now,
When substituting
Similarly, it is rewritten for three back stress terms,
Actually,
where
While the relationships between the strain and stress can be described by Hooke’s law for elastic behaviour, it is determined by a flow rule for plastic behaviour. A flow rule gives the relationship between the stress and the plastic deformation
FE simulations were performed for the folding and cyclic loading processes. Both models were used in the optimization. The final diameter and height of the locking nut were probed in addition to the stress and deformation results. The proposed optimum parameters were re-simulated in the folding process to obtain results for the diameter, height, and stress state. The simulation results were compared to experimental measurements for validation.
One of the techniques for calibration of just initialized material constants to obtain improvement on the general fit of the model prediction to experimental data is to use the optimization method [
An objective function was set as in
where
Parameters | Value |
---|---|
Estimated number of evaluation | 2000 |
Number of initial samples | 100 |
Number of samples per iteration | 100 |
Maximum allowable pareto percentage | 70% |
Convergence stability percentage | 2% |
Maximum number of iterations | 20 |
The best parameters will be the values which lead the simulation results to (almost) match the experimental results. Our goal is to minimize the difference between the measured and predicted dimensions. The goal function is set 0.5% as the convergence stability criterion. Although maximum iterations are limited to 100 as a stopping criterion for the optimization process, the most probable and physically possible points are found within 20 iterations. The convergence status during the optimization process is given in
Once the simulations are completed on all design of experiment (DOE) points, now a function which will give the relations between input and output variables is fitted by means of response surfaces in the optimization module. These functions will be used to catch the optimum values along any extra points besides DOEs.
A convergence and mesh independence study was conducted in order to improve the computational efficiency as seen in
{ Number of elements} | Change (%) | { Flange diameter (mm)} | { Computation time} | { Result file size (MB)} | { Memory (MB)} |
---|---|---|---|---|---|
1890 | -- | 30.0273 | 2min 29s | 115.83 | 512 |
2514 | 0.64369 | 30.0411 | 8min 21s | 129.63 | 524 |
3756 | −0.040609 | 30.0402 | 9min 40s | 162.63 | 528 |
6240 | −0.0099883 | 30.0400 | 10min 53s | 198.88 | 532 |
11208 | −0.0011667 | 30.0400 | 17min 50s | 226.13 | 540 |
21144 | 0.0005333 | 30.0400 | 43min 28s | 668.38 | 556 |
41016 | 0.0009889 | 30.0400 | 2h 17min | 894.17 | 578 |
The Chaboche’s
Models | Parameters | |||||||
---|---|---|---|---|---|---|---|---|
BISO (initial) | CHAB (initial) | |||||||
Bilinear | 2014.21 | 522.12 | – | – | – | – | – | – |
Chaboche | – | 394.82 | 2632.97 | 18.51 | 2632.96 | 18.87 | 2632.97 | 18.54 |
Combined | 489.17 | 431.87 | 1869.90 | 17.63 | 1869.90 | 17.63 | 1869.86 | 17.33 |
The optimization process modifies the model parameters to get the more accurate folding predictions. For this purpose, a finite element model is prepared. Instead of a 3D model, an axial symmetric 2D model is used to avoid time consuming computations. A cylindrical coordinate system (
Models | Parameters | |||||||
---|---|---|---|---|---|---|---|---|
BISO (initial) | CHAB (initial) | |||||||
Bilinear | −1245.8 | 667.64 | – | – | – | – | – | – |
Chaboche | – | 669.66 | 12274.32 | 63.20 | 12274.43 | 62.95 | 12274.64 | 63.20 |
Combined | −1354.5 | 697.12 | 12112.15 | 63.05 | 12112.18 | 67.02 | 12111.76 | 62.99 |
The Coulomb friction coefficient at the tool and sample interface is assumed to be constant and taken as 0.125 for the AISI 1040 steel [
Permanent deformation of the nut is seen after the punch goes away. When the punch starts to turn back after 20nd solution step, the height also returns from 18.2996 to 18.3918 mm because of the springback (0.5%) due to the recovery of the elastic deformation. It is seen that the springback is one of the important phenomenon on the AISI 1040 steels.
In the optimization, the folding simulations for the nut M20 × 1.5 was used. Finally, the calibrated model parameters were used in another folding simulation for the nut M24 × 1.5 for validation of the calibrated model. Validations on the real components are more reliable since they reflect the deformation conditions the best. The simulated and experimentally measured diameter and height of the nut are compared. Also the stress and strain response of the material at the scoped point are compared with the experimental hardening curve’s shape to investigate the similarity between the material response from the folding and uniaxial test.
The folded height and diameter of the flange are predicted as the output. The difference between the measured and predicted dimensions will be minimized as the goal function. The constrains are applied as:
no constraint for the target for the diameter has been specified between 23.5 to 24 mm, the target for the nylock nut’s height has been specified between 18.5 to 19 mm after folding, the maximum absolute stress has been specified 1016.17 MPa for the cyclic case.
Nut size | Before folding | After folding | ||
---|---|---|---|---|
Inner diameter (mm) | Height (mm) | Inner diameter (mm) | Height (mm) | |
M20 × 1.5 | 25.51 | 22.51 | 23.96 | 18.86 |
M20 × 1.5 | 25.53 | 22.55 | 23.96 | 18.85 |
M20 × 1.5 | 25.51 | 22.50 | 23.96 | 18.79 |
M24 × 1.5 | 30.05 | 25.03 | 28.46 | 21.18 |
M24 × 1.5 | 30.02 | 25.00 | 28.41 | 21.18 |
M24 × 1.5 | 30.05 | 25.11 | 28.44 | 21.30 |
The variables’ lower and upper limits are listed in
Lower | 0 (0) | 300 (0) | 1000 (0) | 0 (0) |
Upper | 4000 (20000) | 600 (1200) | 200000 (300000) | 30000 (30000) |
The optimization module suggests the optimum values as in
Models | Parameters | Outputs | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
{ |
{ |
{ |
{ |
{ |
{ Diameter (mm)} | { Height (mm)} | ||||
Bilinear | 4000 | 300 | – | – | – | – | – | – | 23.388 | 18.289 |
Chaboche | 302.2 | 89331.8 | 24389.5 | 153625.1 | 23200.7 | 59112.1 | 19874.1 | 23.171 | 18.311 | |
Combined | 4000 | 300 | 40931.4 | 18207.5 | 18734.8 | 18072.1 | 1000 | 30000 | 23.337 | 18.289 |
In addition to the folding simulations, the tension-compression test is also simulated to determine the model parameters for cyclic plasticity. A unit cylindrical model is used [
The optimization module suggests the optimum values as in
Models | Parameters | Output | |||||||
---|---|---|---|---|---|---|---|---|---|
Hill48 (MPa) | |||||||||
Bilinear | 15082.8 | 650.26 | -- | -- | -- | -- | -- | -- | 987.98 |
Chaboche | -- | -- | 297120.2 | 77038.95 | 127692.12 | 109.1 | 38490.2 | 11 | 1014.97 |
Combined | 10.4 | 21.1 | 276848.4 | 75568.15 | 118846.27 | 101.8 | 34691.4 | 11 | 998.40 |
The proposed optimum values are re-simulated for verification. The percent true relative errors of the verified outputs are calculated using
Output | Models | Proposed optimum | Verified | Experimental | Error (%) |
---|---|---|---|---|---|
Diameter (mm) | BISO | 23.381 | 23.387 | 23.96 | 2.39 |
CHAB | 23.271 | 23.274 | 23.96 | 2.86 | |
Combined | 23.337 | 23.348 | 23.96 | 2.55 | |
Height (mm) | BISO | 18.489 | 18.478 | 18.86 | 2.03 |
CHAB | 18.311 | 18.305 | 18.86 | 2.94 | |
Combined | 18.383 | 18.483 | 18.86 | 2.00 | |
Stress (MPa) | BISO | 987.98 | 1004.24 | 1016.17 | 1.17 |
CHAB | 1014.97 | 1015.47 | 1016.17 | 0.07 | |
Combined | 998.40 | 1016.69 | 1016.17 | 0.05 |
For validation, a new folding simulation is performed for the nut M24 × 1.5 whose nominal diameter is
Output | Models | Model predictions | Experimental | Error (%) |
---|---|---|---|---|
Diameter (mm) | BISO | 30.89 | 30.04 | 2.83 |
CHAB | 29.17 | 30.04 | 2.90 | |
Combined | 30.02 | 30.04 | 0.07 | |
Height (mm) | BISO | 25.74 | 25.05 | 2.75 |
CHAB | 25.79 | 25.05 | 2.95 | |
Combined | 25.08 | 25.05 | 0.12 |
The relationships between the parameters and the goals are obtained by means of response surface graphics created based on the Kriging method [
When the calibrated parameters for the combined model are used in the folding simulation, the stress and deformation results are obtained as seen in
The Hill48 stress distribution is compared with that of the Von Mises equivalent stress. The Hill48 stress is slightly less than the Von Mises stress because
Comparison of the hardening curves is the best way for validation of the parameters. The curves are compared with experimental points in reference with shape and the peak stress as seen in
The BISO model provides a linear line for hardening both before and after calibration, as expected. This linearity starts from the yield point and continues to increase with the constant slope at increasing strain values. However, the experimental behaviour of the material shows that it has a significant curvature after the yield point. For this reason, the representation ability of BISO is not sufficient. The calibration process could not improve this model. Before and after calibration, the CHAB model has a good representation for hardening behaviour. The fitted model has a bigger deviation to the prediction of the peak stress than the simulation having the calibrated models. The CHAB model starts with over-prediction with increasing strain. The combined model overcomes the over-prediction. The peak tensile stress is seen at 0.5% strain. Fitted CHAB predicts the peak stress as 959.30 MPa, while the experimental peak is 849.45 MPa leading up to 11.45% difference. Calibrated CHAB predicts 855.83 MPa leading to 0.75%. Previously a 1.6% difference was reported by Kang et al. [
When the parameters before calibration are used in the folding simulation, it is seen that there are significant differences in the loop shapes in
The BISO model alone has definitely not been sufficient in modelling the cyclic behaviour as seen in
Summarizing above discussions, it is concluded that one hysteresis loop from the uniaxial strain controlled test is enough to calibrate the parameters as reported by Paul et al. [
This study presents parameter determination and calibration for the nut flange bending process. A plasticity model is set by using the Hill48 yield criterion, combined hardening rules, and associated flow rule for FE simulation of the flange folding process. While the BISO model’s parameters are determined from the monotonic tensile test curve, the CHAB’s parameters are determined by nonlinear regression on the experimental uniaxial hysteresis loops. The optimization process is performed to calibrate the parameters. Experiments are conducted to validate the models. Based on the analysed data, the results reveal the following;
Although a model obtained from the tensile/compression test should not be used directly for the simulation of any multi-axial deformation process, it will be suitable as long as it is calibrated with experimental data. The calibrated model parameters leading to accurate folding or cyclic deformation simulation are presented for the folding process. The calibrated parameters are different for both cases. Therefore, they cannot be used interchangeably. While the combined hardening rule will be a good choice for cyclic deformations, all models are suitable for the folding process. By combining the BISO model with the CHAB model over-prediction is eliminated. The pure kinematic model is enough for the folding, but not enough for cyclic deformation. The springback shows that the AISI 1040 steel in the folding process is dependent on anisotropy. Its plastic deformability in the axial and radial directions is different. Therefore, the Hill48 criterion becomes a good choice because it can represent material anisotropy owing to its anisotropy-dependent coefficients Residual stress on the ring does not cause any plastic deformation. Thus, the folding process can be completed without any defect on the ring.