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A Microstructural Damage Model toward Simulating the Mullins Effect in Double-Network Hydrogels

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Abstract

The damage models based on the eight-chain model and the affine full-chain network model are not adequate to describe the damage behaviors in double-network (DN) hydrogels. To overcome this limitation, we propose a combined chain stretch model with new damage flow rules. It is demonstrated that the new proposed micro-chain stretch is a reduced form of the complete representation for the transversely isotropic tensor function. As a result, the damage models based on the eight-chain model and the affine model are incorporated as special cases. The effects of chain affineness and network entangling are simultaneously involved in the new model, while only one of these two effects can be characterized in either the eight-chain model or the affine model. It is further shown that the new model can effectively capture the Mullins features of the DN hydrogels and achieve better agreement with the experimental data than the affine model and the eight-chain model.

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References

  1. Liu Z, Toh W, Ng TY. Advances in mechanics of soft materials: a review of large deformation behavior of hydrogels. Int J Appl Mech. 2015;7:1530001.

    Article  Google Scholar 

  2. Ullah F, Othman MBH, Javed F, Ahmad Z, Akil HM. Classification, processing and application of hydrogels: a review. Mat Sci Eng: C. 2015;57:414–33.

    Article  Google Scholar 

  3. Lei J, Li Z, Xu S, Liu Z. Recent advances of hydrogel network models for studies on mechanical behaviors. Acta Mech Sin. 2021:1-20.

  4. Eslahi N, Abdorahim M, Simchi A. Smart polymeric hydrogels for cartilage tissue engineering: a review on the chemistry and biological functions. Biomacromol. 2016;17:3441–63.

    Article  Google Scholar 

  5. Ionov L. Biomimetic hydrogel-based actuating systems. Adv Funct Mater. 2013;23:4555–70.

    Article  Google Scholar 

  6. Dong L, Agarwal AK, Beebe DJ, Jang H. Adaptive liquid microlenses activated by stimuli-responsive hydrogels. Nature. 2006;442:551–4.

    Article  Google Scholar 

  7. Gong JP, Katsuyama Y, Kurokawa T, Osada Y. Double-network hydrogels with extremely high mechanical strength. Adv Mater. 2003;15:1155–8.

    Article  Google Scholar 

  8. Mai TT, Matsuda T, Nakajima T, Gong JP. Distinctive characteristics of internal fracture in tough double network hydrogels revealed by various modes of stretching. Macromolecules. 2018;51:5245–57.

    Article  Google Scholar 

  9. Creton C. 50th anniversary perspective: networks and gels: soft but dynamic and tough. Macromolecules. 2017;50:8297–316.

    Article  Google Scholar 

  10. Nakajima T. Generalization of the sacrificial bond principle for gel and elastomer toughening. Polym J. 2017;49:477–85.

    Article  Google Scholar 

  11. Diani J, Fayolle B, Gilormini P. A review on the Mullins effect. Eur Polymer J. 2009;45(3):601–12.

    Article  Google Scholar 

  12. Haque MA, Kurokawa T, Gong JP. Super tough double network hydrogels and their application as biomaterials. Polymer. 2012;53:1805–22.

    Article  Google Scholar 

  13. Liu Y, Zhang H, Zheng Y. A micromechanically based constitutive model for the inelastic and swelling behaviors in double network hydrogels. J Appl Mech. 2016;83(2):021008.

    Article  Google Scholar 

  14. Lu H, Wang X, Shi X, Yu K, Fu YQ. A phenomenological model for dynamic response of double-network hydrogel composite undergoing transient transition. Compos Part B Eng. 2018;151:148–53.

    Article  Google Scholar 

  15. Qi Y, Caillard J, Long R. Fracture toughness of soft materials with rate-independent hysteresis. J Mech Phys Solid. 2018;118:341–64.

    Article  MathSciNet  Google Scholar 

  16. Vernerey FJ, Brighenti R, Long R, Shen T. Statistical damage mechanics of polymer networks. Macromolecules. 2018;51:6609–22.

    Article  Google Scholar 

  17. Külcü İD. Characterization of stress softening and self-healing in a double network hydrogel. Results in Physics. 2019;12:1826–33.

    Article  Google Scholar 

  18. Lu T, Wang Z, Tang J, Zhang W, Wang T. A pseudo-elasticity theory to model the strain-softening behavior of tough hydrogels. J Mech Phys Solid. 2020;137:103832.

    Article  MathSciNet  Google Scholar 

  19. Xiao R, Mai TT, Urayama K, Gong JP, Qu S. Micromechanical modeling of the multi-axial deformation behavior in double network hydrogels. Int J Plast. 2021;137:102901.

    Article  Google Scholar 

  20. Kuhn W, Grün F. Beziehungen zwischen elastischen Konstanten und Dehnungsdoppelbrechung hochelastischer Stoffe. Kolloid-Zeitschrift. 1942;101:248–71.

    Article  Google Scholar 

  21. Treloar LRG. The Physics of Rubber Elasticity. 3rd ed. Oxford: Clarendon Press; 1975.

    Google Scholar 

  22. Cohen A, Padé A. approximant to the inverse Langevin function. Rheol Acta. 1991;30:270–3.

    Article  Google Scholar 

  23. James HM, Guth E. Theory of elastic properties of rubber. J Chem Phys. 1943;11:455–81.

    Article  Google Scholar 

  24. Flory PJ, Rehner J. Statistical mechanics of cross-linked polymer networks: I. Rubberlike Elast J Chem Phys. 1943;11:512–20.

    Article  Google Scholar 

  25. Arruda EM, Boyce MC. A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J Mech Phys Solid. 1993;41:389–412.

    Article  Google Scholar 

  26. Treloar LRG, Riding G. A non-Gaussian theory of rubber in biaxial strain. I. Mechanical properties. Proc. R. Soc. London, Ser. A. 1979; 369:261–280.

  27. Boyce MC, Arruda EM. Constitutive models of rubber elasticity: a review. Rubber Chem Technol. 2000;73:504–23.

    Article  Google Scholar 

  28. Steinmann P, Hossain M, Possart G. Hyperelastic models for rubber-like materials: consistent tangent operators and suitability for Treloar’s data. Arch Appl Mech. 2012;82:1183–217.

    Article  Google Scholar 

  29. Hossain M, Steinmann P. More hyperelastic models for rubber-like materials: consistent tangent operators and comparative study. J Mech Beha Mater. 2013;22:27–50.

    Article  Google Scholar 

  30. Miehe C, Göktepe S, Lulei F. A micro-macro approach to rubber-like materials–part I: the non-affine micro-sphere model of rubber elasticity. J Mech Phys Solid. 2004;52:2617–60.

    Article  Google Scholar 

  31. Zheng QS, Spencer AJM. Tensors which characterize anisotropies. Int J Eng Sci. 1993;31:679–93.

    Article  MathSciNet  Google Scholar 

  32. Xiao H. On minimal representations for constitutive equations of anisotropic elastic materials. J Elast. 1996;45:13–32.

    Article  MathSciNet  Google Scholar 

  33. Marckmann G, Verron E, Gornet L, Chagnon G, Charrier P, Fort P. A theory of network alteration for the Mullins effect. J Mech Phys Solid. 2002;50:2011–28.

    Article  Google Scholar 

  34. Hossain M, Possart G, Steinmann P. A finite strain framework for the simulation of polymer curing. Part I: Elasticity Comput Mech. 2009;44(5):621–30.

    MathSciNet  MATH  Google Scholar 

  35. Dal H, Kaliske M. A micro-continuum-mechanical material model for failure of rubber-like materials: application to ageing-induced fracturing. J Mech Phys Solid. 2009;57(8):1340–56.

    Article  Google Scholar 

  36. Bažant P, Oh BH. Efficient numerical integration on the surface of a sphere. ZAMM. 1986;66:37–49.

    Article  MathSciNet  Google Scholar 

  37. Itskov M. On the accuracy of numerical integration over the unit sphere applied to full network models. Comput Mech. 2016;57:859–65.

    Article  MathSciNet  Google Scholar 

  38. Verron E. Questioning numerical integration methods for microsphere (and microplane) constitutive equations. Mech Mater. 2015;89:216–28.

    Article  Google Scholar 

  39. Mao Y, Talamini B, Anand L. Rupture of polymers by chain scission. Extreme Mech Lett. 2017;13:17–24.

    Article  Google Scholar 

  40. Li B, Bouklas N. A variational phase-field model for brittle fracture in polydisperse elastomer networks. Int J Solids Struct. 2020;182:193–204.

    Article  Google Scholar 

  41. Lei J, Li Z, Xu S, Liu Z. A mesoscopic network mechanics method to reproduce the large deformation and fracture process of cross-linked elastomers. J Mech Phys Solid. 2021;156:04599.

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grant No. 12022204, the Zhejiang Provincial Natural Science Foundation of China under Grant No. LD22A020001, and the Fundamental Research Funds for the Central Universities, China (Grant No. 2021FZZX001-16).

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Authors and Affiliations

  1. State Key Laboratory of Fluid Power and Mechatronic System, Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Department of Engineering Mechanics, Zhejiang University, Hangzhou, 310027, China

    Lin Zhan & Rui Xiao

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  1. Lin Zhan
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  2. Rui Xiao
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Correspondence to Rui Xiao.

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Zhan, L., Xiao, R. A Microstructural Damage Model toward Simulating the Mullins Effect in Double-Network Hydrogels. Acta Mech. Solida Sin. 35, 682–693 (2022). https://doi.org/10.1007/s10338-022-00316-5

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