A Hyper-Pseudoelastic Model of Cyclic Stress-Softening Effect for Rubber Composites
Abstract
:1. Introduction
2. Basic Laws of Stress-Softening Effect
- (1)
- The area enclosed by the loading–unloading curve gradually decreases with the increase of the cyclic loading–unloading order.
- (2)
- In the presence of residual deformation, a negative load must be applied to completely restore the rubber composites to a non-deformed state; that is, the unloading nominal stress is less than 0 when the principal stretch equals 1. In addition, the residual deformation increases with the increase of the loading–unloading order, and the residual deformation of the rubber composites mainly occurs during the first loading–unloading process.
- (3)
- The nominal stress difference corresponding to the previous loading–unloading curve is greater than that corresponding to the subsequent loading–unloading curve under the same stretch, which actually confirms the basic law (1).
- (4)
- The previous loading curves are always above the subsequent loading curves, and the unloading curves also have the same feature. And there is an intersection point between the previous unloading curve and the subsequent loading curve.
- (5)
- When the maximum stretch amplitude remains unchanged, the stress-softening effect corresponding to the first loading–unloading process is the most obvious. After several loading–unloading, the stress response tends to be stable. A similar phenomenon was also observed by Dorfmann et al. [18], Simo [33], and Sasso et al. [38].
- (6)
- Compared with 20 phr filled rubber composite, 60 phr filled rubber composite has a more obvious stress-softening effect, which indicates that the stress-softening effect is more significant with the increase of filler content in rubber composites.
3. Hyper-Pseudoelastic Model of Cyclic Stress-Softening Effect
3.1. No Residual Deformation Effect
3.2. Residual Deformation Effect
4. Results and Discussion
4.1. No Residual Deformation Effect
4.2. Residual Deformation Effect
5. Conclusions
- (1)
- A detailed analysis of the cyclic loading–unloading experimental results of rubber composites has been carried out. The basic laws of stress-softening effect are summarized to guide the derivation of the hyper-pseudoelastic model. It can be found that the stress-softening effect and residual deformation of rubber composites corresponding to different loading–unloading orders are different, especially the nominal stress–stretch curves of initial loading–unloading and subsequent loading–unloading are significantly different.
- (2)
- The hyper-pseudoelastic model in the cyclic loading–unloading process is proposed, in which a strain energy evolution function similar to (1-D) in continuum damage mechanics is introduced to characterize the cyclic stress-softening effect. The specific expressions of the strain energy evolution function with or without residual deformation effect are given, and the constraints that the strain energy evolution function needs to satisfy are also obtained based on the basic laws of the stress-softening effect. The hyper-pseudoelastic model establishes the theoretical relationship between strain energy and cyclic loading–unloading order directly, which provides great convenience in deriving the stress response corresponding to arbitrary loading–unloading order.
- (3)
- The influences of the material parameters on the cyclic loading–unloading curve are discussed. The research results show that g mainly controls the degree of stress softening of rubber composites, while A mainly plays a role in fine-tuning the stress–stretch curve. Additionally, the dissipation energy is larger when g is smaller, or A is larger corresponding to the same loading–unloading order.
- (4)
- Based on the nominal stress–stretch experimental results of cyclic loading–unloading processes, the calibration method of material parameters and specific expression of strain energy evolution function with residual deformation effect are obtained. Further, the hyper-pseudoelastic model is verified by comparing the theoretical results with experimental results. The proposed model can predict the cyclic stress-softening effect of rubber composites with different filler contents effectively.
Author Contributions
Funding
Institutional Review Board Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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20 phr | Hyperelastic material parameters | |||||
0.05 | 4.221 | −0.815 | −0.156 | −0.286 | −4.63 | |
Pseudoelastic material parameters | ||||||
0.491 | 0.596 | 0.963 | 0.753 | 0.438 | 1.952 | |
60 phr | Hyperelastic material parameters | |||||
−1.528 | −1.011 | 0.223 | 4.205 | −1.134 × 10−3 | −4.399 | |
Pseudoelastic material parameters | ||||||
0.354 | 0.496 | 1.25 | 0.965 | 0.3 | 0.16 |
20 phr | ||||||
0.466 | 0.343 | −0.011 | 0.028 | −0.085 | −0.136 | |
0.468 | −0.526 | −7.003 | 2.46 | −0.375 | ||
60 phr | ||||||
0.34 | 1.33 | −0.08 | 0.104 | −0.173 | 0.375 | |
0 | 0.479 | −3.688 | 5.479 | 8.99 |
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Dong, Y.; Fu, Y.; He, C.; Fang, D. A Hyper-Pseudoelastic Model of Cyclic Stress-Softening Effect for Rubber Composites. Polymers 2023, 15, 3033. https://doi.org/10.3390/polym15143033
Dong Y, Fu Y, He C, Fang D. A Hyper-Pseudoelastic Model of Cyclic Stress-Softening Effect for Rubber Composites. Polymers. 2023; 15(14):3033. https://doi.org/10.3390/polym15143033
Chicago/Turabian StyleDong, Yifeng, Yutong Fu, Chunwang He, and Daining Fang. 2023. "A Hyper-Pseudoelastic Model of Cyclic Stress-Softening Effect for Rubber Composites" Polymers 15, no. 14: 3033. https://doi.org/10.3390/polym15143033