Abstract
Highly entangled hydrogels exhibit excellent mechanical properties, including high toughness, high stretchability, and low hysteresis. By considering the evolution of randomly distributed entanglements within the polymer network upon mechanical stretches, we develop a constitutive theory to describe the large stretch behaviors of these hydrogels. In the theory, we utilize a representative volume element (RVE) in the shape of a cube, within which there exists an averaged chain segment along each edge and a mobile entanglement at each corner. By employing an explicit method, we decouple the elasticity of the hydrogels from the sliding motion of their entanglements, and derive the stress-stretch relations for these hydrogels. The present theoretical analysis is in agreement with experiment, and highlights the significant influence of the entanglement distribution within the hydrogels on their elasticity. We also implement the present developed constitutive theory into a commercial finite element software, and the subsequent simulations demonstrate that the exact distribution of entanglements strongly affects the mechanical behaviors of the structures of these hydrogels. Overall, the present theory provides valuable insights into the deformation mechanism of highly entangled hydrogels, and can aid in the design of these hydrogels with enhanced performance.
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GUO, X., DONG, X. Y., ZOU, G. J., GAO, H. J., and ZHAI, W. Strong and tough fibrous hydrogels reinforced by multiscale hierarchical structures with multimechanisms. Science Advances, 9(2), eadf7075 (2023)
HUA, M. T., WU, S. W., MA, Y. F., ZHAO, Y. S., CHEN, Z. L., FRENKEL, I., STRZALKA, J., ZHOU, H., ZHU, X. Y., and HE, X. M. Strong tough hydrogels via the synergy of freeze-casting and salting out. nature, 590(7847), 594–599 (2021)
LIU, C., MORIMOTO, N., JIANG, L., KAWAHARA, S., NORITOMI, T., YOKOYAMA, H., MAYUMI, K., and ITO, K. Tough hydrogels with rapid self-reinforcement. Science, 372(6546), 1078–1081 (2021)
NIAN, G. D., KIM, J., BAO, X. Y., and SUO, Z. G. Making highly elastic and tough hydrogels from doughs. Advanced Materials, 34(50), 2206577 (2022)
WANG, Z., ZHENG, X. J., OUCHI, T., KOUZNETSOVA, T. B., BEECH, H. K., AV-RON, S., MATSUDA, T., BOWSER, B. H., WANG, S., JOHNSON, J. A., KALOW, J. A., OLSEN, B. D., GONG, J. P., RUBINSTEIN, M., and CRAIG, S. L. Toughening hydrogels through force-triggered chemical reactions that lengthen polymer strands. Science, 374(6564), 193–196 (2021)
ZHAO, X. H., CHEN, X. Y., YUK, H., LIN, S. T., LIU, X. Y., and PARADA, G. Soft materials by design: unconventional polymer networks give extreme properties. Chemical Reviews, 121(8), 4309–4372 (2021)
HAN, S. J., WU, Q. R., ZHU, J. D., ZHANG, J. Y., CHEN, A. B., SU, S., LIU, J. T., HUANG, J. R., YANG, X. X., and GUAN, L. H. Tough hydrogel with high water content and ordered fibrous structures as an artificial human ligament. Materials Horizons, 10(3), 1012–1019 (2023)
GONG, J. P., KATSUYAMA, Y., KUROKAWA, T., and OSADA, Y. Double-network hydrogels with extremely high mechanical strength. Advanced Materials, 15(14), 1155–1158 (2003)
JIA, Y. T., ZHOU, Z. D., JIANG, H. L., and LIU, Z. S. Characterization of fracture toughness and damage zone of double network hydrogels. Journal of the Mechanics and Physics of Solids, 169, 105090 (2022)
KIM, J., ZHANG, G. G., SHI, M. X. Z., and SUO, Z. G. Fracture, fatigue, and friction of polymers in which entanglements greatly outnumber cross-links. Science, 374(6564), 212–216 (2021)
LI, D. K., ZHAN, W., ZUO, W., LI, L. P., ZHANG, J., CAI, G. Y., and TIAN, Y. Elastic, tough and switchable swelling hydrogels with high entanglements and low crosslinks for water remediation. Chemical Engineering Journal, 450, 138417 (2022)
LIU, P. Y., ZHANG, Y., GUAN, Y., and ZHANG, Y. J. Peptide-crosslinked, highly entangled hydrogels with excellent mechanical properties but ultra-low solid content. Advanced Materials, 35(13), 2210021 (2023)
SHI, M. X. Z., KIM, J., NIAN, G. D., and SUO, Z. G. Highly entangled hydrogels with degradable crosslinks. Extreme Mechanics Letters, 59, 101953 (2023)
WANG, Y. C., NIAN, G. D., KIM, J., and SUO, Z. G. Polyacrylamide hydrogels VI: synthesis-property relation. Journal of the Mechanics and Physics of Solids, 170, 105099 (2023)
TANI, J., TAKAGI, T., and QIU, J. Intelligent material systems: application of functional materials. Applied Mechanics Reviews, 51(8), 505–521 (1998)
BOSNJAK, N. and SILBERSTEIN, M. N. Pathways to tough yet soft materials. Science, 374(6564), 150–151 (2021)
BUKOWSKI, C., ZHANG, T., RIGGLEMAN, R. A., and CROSBY, A. J. Load-bearing entanglements in polymer glasses. Science Advances, 7(38), eabg9763 (2021)
ZHU, J. K. and LUO, J. Effects of entanglements and finite extensibility of polymer chains on the mechanical behavior of hydrogels. Acta Mechanica, 229, 1703–1719 (2018)
EDWARDS, S. F. and VILGIS, T. The effect of entanglements in rubber elasticity. Polymer, 27(4), 483–492 (1986)
NIAN, X. C., YANG, Q. S., MA, L. H., and ZHANG, X. Y. Constitutive modeling for hydrogel with chain entanglements and application to adaptive hydrogel composite structures. Mechanics of Advanced Materials and Structures, 30(24), 5122–5136 (2023)
BAYAT, M. R., DOLATABADI, R., and BAGHANI, M. Transient swelling response of ph-sensitive hydrogels: a monophasic constitutive model and numerical implementation. International Journal of Pharmaceutics, 577, 119030 (2020)
BÖGER, L., NATEGHI, A., and MIEHE, C. A minimization principle for deformation-diffusion processes in polymeric hydrogels: constitutive modeling and FE implementation. International Journal of Solids and Structures, 121, 257–274 (2017)
MENG, Q. H. and SHI, X. H. A mechanistically motivated constitutive model of biopolymer hydrogels with structural evolution. Journal of the Mechanics and Physics of Solids, 173, 105205 (2023)
HUANG, R., ZHENG, S. J., LIU, Z. S., and NG, T. Y. Recent advances of the constitutive models of smart materials — hydrogels and shape memory polymers. International Journal of Applied Mechanics, 12(2), 2050014 (2020)
PAN, Z. Z. and BRASSART, L. Constitutive modelling of hydrolytic degradation in hydrogels. Journal of the Mechanics and Physics of Solids, 167, 105016 (2022)
WANG, Q. M. and GAO, Z. M. A constitutive model of nanocomposite hydrogels with nanoparticle crosslinkers. Journal of the Mechanics and Physics of Solids, 94, 127–147 (2016)
DAL, H., AÇIKGÖZ, K., and BADIENIA, Y. On the performance of isotropic hyperelastic constitutive models for rubber-like materials: a state of the art review. Applied Mechanics Reviews, 73(2), 020802 (2021)
LU, D. and CHEN, B. A constitutive theory for large stretch behaviors of slide-ring gels by considering molecular frictions. Soft Matter, 19(8), 1531–1539 (2023)
FALENDER, J. R., YEH, G. S. Y., and MARK, J. E. The effect of chain length distribution on elastomeric properties 1: comparisons between random and highly nonrandom networks. Journal of the American Chemical Society, 101(24), 7353–7356 (1979)
LI, B. and BOUKLAS, N. A variational phase-field model for brittle fracture in polydisperse elastomer networks. International Journal of Solids and Structures, 182–183, 193–204 (2020)
DARGAZANY, R. and ITSKOV, M. A network evolution model for the anisotropic Mullins effect in carbon black filled rubbers. International Journal of Solids and Structures, 46(16), 2967–2977 (2009)
ZHANG, H. H. and HU, Y. H. A statistical-chain-based theory for dynamic living polymeric gels with concurrent diffusion, chain remodeling reactions and deformation. Journal of the Mechanics and Physics of Solids, 172, 105155 (2023)
GHAREEB, A. and ELBANNA, A. An adaptive quasicontinuum approach for modeling fracture in networked materials: application to modeling of polymer networks. Journal of the Mechanics and Physics of Solids, 137, 103819 (2020)
LAVOIE, S. R., LONG, R., and TANG, T. Modeling the mechanics of polymer chains with deformable and active bonds. The Journal of Physical Chemistry B, 124(1), 253–265 (2020)
WANG, Q. M., GOSSWEILER, G. R., CRAIG, S. L., and ZHAO, X. H. Mechanics of mechanochemically responsive elastomers. Journal of the Mechanics and Physics of Solids, 82, 320–344 (2015)
GUO, Q. and ZAIRI, F. A micromechanics-based model for deformation-induced damage and failure in elastomeric media. International Journal of Plasticity, 140, 102976 (2021)
LU, T. Q., WANG, Z. T., TANG, J. D., ZHANG, W. L., and WANG, T. J. A pseudo-elasticity theory to model the strain-softening behavior of tough hydrogels. Journal of the Mechanics and Physics of Solids, 137, 103832 (2020)
ITSKOV, M. and KNYAZEVA, A. A rubber elasticity and softening model based on chain length statistics. International Journal of Solids and Structures, 80, 512–519 (2016)
YASUDA, Y., MASUMOTO, T., MAYUMI, K., TODA, M., YOKOYAMA, H., MORITA, H., and ITO, K. Molecular dynamics simulation and theoretical model of elasticity in slide-ring gels. ACS Macro Letters, 9(9), 1280–1285 (2020)
TRELOAR, L. R. G. The elasticity and related properties of rubbers. Reports on Progress in Physics, 36(7), 755 (1973)
CAI, S. Q. and SUO, Z. G. Mechanics and chemical thermodynamics of phase transition in temperature-sensitive hydrogels. Journal of the Mechanics and Physics of Solids, 59(11), 2259–2278 (2011)
MARKO, J. F. and SIGGIA, E. D. Statistical mechanics of supercoiled DNA. Physical Review E, 52(3), 2912–2938 (1995)
EPSTEIN, M. and SEGEV, R. Differentiable manifolds and the principle of virtual work in continuum mechanics. Journal of Mathematical Physics, 21 (5), 1243–1245 (2008)
LÁNCZOS, C. The Variational Principles of Mechanics, Courier Corporation, U. S. A. (2012)
NORIOKA, C., INAMOTO, Y., HAJIME, C., KAWAMURA, A., and MIYATA, T. A universal method to easily design tough and stretchable hydrogels. NPG Asia Materials, 13(1), 34 (2021)
JAMES, H. M. and GUTH, E. Theory of the elastic properties of rubber. The Journal of Chemical Physics, 11(10), 455–481 (1943)
ARRUDA, E. M. and BOYCE, M. C. A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. Journal of the Mechanics and Physics of Solids, 41(2), 389–412 (1993)
FAN, Q. Y., CHEN, B., and CAO, Y. Constitutive model reveals the defect-dependent viscoelasticity of protein hydrogels. Journal of the Mechanics and Physics of Solids, 125, 653–665 (2019)
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Project supported by the Key Research Project of Zhejiang Laboratory (No. K2022NB0AC03), the National Natural Science Foundation of China (No. 11872334), and the National Natural Science Foundation of Zhejiang Province of China (No. LZ23A020004)
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Liu, J., Lu, D. & Chen, B. Tuning mechanical behaviors of highly entangled hydrogels with the random distribution of mobile entanglements. Appl. Math. Mech.-Engl. Ed. 45, 277–294 (2024). https://doi.org/10.1007/s10483-024-3076-8
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DOI: https://doi.org/10.1007/s10483-024-3076-8