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A nonlinear visco-poroelasticity model for transversely isotropic gels

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Abstract

A polymeric gel contains a crosslinked polymer network and solvent. Gels can swell or shrink in response to external stimuli. Two kinetic processes are usually involved during the deformation of gels: the viscoelastic and poroelastic responses. Viscoelasticity of gels is generated from the local rearrangement of the polymers, while poroelasticity of gels is generated from the solvent migration. The coupled time-dependent behaviors of gels can be formulated by coupling a rheological spring-dashpot model with a diffusion-deformation model of gels. In this work, we build a general framework of coupled visco-poroelasticity for transversely isotropic gels and study how the mechanical anisotropy could induce anisotropic time-dependent behaviors of gels even though their kinetic properties are isotropic. The constitutive model is implemented into a finite element code in commercial software. Several numerical simulations are performed to investigate the time-dependent deformation and frequency-dependent energy dissipation under different loading directions. The results show that even though the viscoelasticity and the poroelasticity are isotropic, the time-dependent behaviors along different directions are different due to the mechanical anisotropy. The fibers aligned in the transversely isotropic gel enhance the elastic feature of the material, and thus influence the dissipation time scales and the amount of energy loss for both viscoelasticity and poroelasticity.

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References

  1. Hoare TR, Kohane DS (2008) Hydrogels in drug delivery: progress and challenges. Polymer (Guildf) 49:1993–2007. https://doi.org/10.1016/J.POLYMER.2008.01.027

    Article  Google Scholar 

  2. Qiu Y, Park K (2001) Environment-sensitive hydrogels for drug delivery. Adv Drug Deliv Rev 53:321–339. https://doi.org/10.1016/S0169-409X(01)00203-4

    Article  Google Scholar 

  3. Li J, Mooney DJ (2016) Designing hydrogels for controlled drug delivery. Nat Rev Mater 1:16071. https://doi.org/10.1038/natrevmats.2016.71

    Article  Google Scholar 

  4. Jen AC, Wake MC, Mikos AG (1996) Hydrogels for cell immobilization. Biotechnol Bioeng 50:357–364. https://doi.org/10.1002/(SICI)1097-0290(19960520)50:4%3c357:AID-BIT2%3e3.0.CO;2-K

    Article  Google Scholar 

  5. Lee KY, Mooney DJ (2001) Hydrogels for tissue engineering. Chem Rev 101:1869–1880. https://doi.org/10.1021/cr000108x

    Article  Google Scholar 

  6. Hoffman AS (2002) Hydrogels for biomedical applications. Adv Drug Deliv Rev 54:3–12. https://doi.org/10.1016/S0169-409X(01)00239-3

    Article  Google Scholar 

  7. Gerlach G, Guenther M, Sorber J et al (2005) Chemical and pH sensors based on the swelling behavior of hydrogels. Sensors Actuators B Chem 111–112:555–561. https://doi.org/10.1016/J.SNB.2005.03.040

    Article  Google Scholar 

  8. Bassil M, Davenas J, EL Tahchi M (2008) Electrochemical properties and actuation mechanisms of polyacrylamide hydrogel for artificial muscle application. Sens Actuators B Chem 134:496–501. https://doi.org/10.1016/J.SNB.2008.05.025

    Article  Google Scholar 

  9. Richter A, Paschew G, Klatt S et al (2008) Review on hydrogel-based pH sensors and microsensors. Sensors 8:561–581. https://doi.org/10.3390/s8010561

    Article  Google Scholar 

  10. Buenger D, Topuz F, Groll J (2012) Hydrogels in sensing applications. Prog Polym Sci 37:1678–1719. https://doi.org/10.1016/J.PROGPOLYMSCI.2012.09.001

    Article  Google Scholar 

  11. Kim C-C, Lee H-H, Oh KH, Sun J-Y (2016) Highly stretchable, transparent ionic touch panel. Science (80-) 353:682–687. https://doi.org/10.1126/science.aaf8810

    Article  Google Scholar 

  12. Grigoryan B, Paulsen SJ, Corbett DC et al (2019) Multivascular networks and functional intravascular topologies within biocompatible hydrogels. Science (80-) 364:458–464. https://doi.org/10.1126/science.aav9750

    Article  Google Scholar 

  13. Lee A, Hudson AR, Shiwarski DJ et al (2019) 3D bioprinting of collagen to rebuild components of the human heart. Science 365:482–487. https://doi.org/10.1126/science.aav9051

    Article  Google Scholar 

  14. Akizuki S, Mow VC, Müller F et al (1986) Tensile properties of human knee joint cartilage: I. Influence of ionic conditions, weight bearing, and fibrillation on the tensile modulus. J Orthop Res 4:379–392. https://doi.org/10.1002/jor.1100040401

    Article  Google Scholar 

  15. Pinsky PM, Datye DV (1991) A microstructurally-based finite element model of the incised human cornea. J Biomech 24:907–922. https://doi.org/10.1016/0021-9290(91)90169-N

    Article  Google Scholar 

  16. Spilker RL, Donzelli PS, Mow VC (1992) A Transversely Isotropic Biphasic Finite-Element Model of the Meniscus. J Biomech 25:1027–1045. https://doi.org/10.1016/0021-9290(92)90038-3

    Article  Google Scholar 

  17. Feng Y, Okamoto RJ, Namani R et al (2013) Measurements of mechanical anisotropy in brain tissue and implications for transversely isotropic material models of white matter. J Mech Behav Biomed Mater 23:117–132. https://doi.org/10.1016/j.jmbbm.2013.04.007

    Article  Google Scholar 

  18. Haque MA, Kamita G, Kurokawa T et al (2010) Unidirectional alignment of lamellar bilayer in hydrogel: one-dimensional swelling, anisotropic modulus, and stress/strain tunable structural color. Adv Mater. https://doi.org/10.1002/adma.201002509

    Article  Google Scholar 

  19. Marelli B, Ghezzi CE, James-Bhasin M, Nazhat SN (2015) Fabrication of injectable, cellular, anisotropic collagen tissue equivalents with modular fibrillar densities. Biomaterials 37:183–193. https://doi.org/10.1016/j.biomaterials.2014.10.019

    Article  Google Scholar 

  20. Miyamoto N, Shintate M, Ikeda S et al (2013) Liquid crystalline inorganic nanosheets for facile synthesis of polymer hydrogels with anisotropies in structure, optical property, swelling/deswelling, and ion transport/fixation. Chem Commun 49:1082–1084. https://doi.org/10.1039/C2CC36654A

    Article  Google Scholar 

  21. Inadomi T, Ikeda S, Okumura Y et al (2014) Photo-induced anomalous deformation of poly(N-isopropylacrylamide) gel hybridized with an inorganic nanosheet liquid crystal aligned by electric field. Macromol Rapid Commun 35:1741–1746. https://doi.org/10.1002/marc.201400333

    Article  Google Scholar 

  22. Mredha TI, Guo YZ, Nonoyama T, Nakajima T (2018) A facile method to fabricate anisotropic hydrogels with perfectly aligned hierarchical fibrous structures. Adv Mater 1704937:1–8. https://doi.org/10.1002/adma.201704937

    Article  Google Scholar 

  23. Humphrey JD, Yin FCP (1987) On constitutive relations and finite deformations of passive cardiac tissue: I. A pseudostrain-energy function. J Biomech Eng 109:298–304. https://doi.org/10.1115/1.3138684

    Article  Google Scholar 

  24. Holzapfel GA (2006) Determination of material models for arterial walls from uniaxial extension tests and histological structure. J Theor Biol 238:290–302. https://doi.org/10.1016/j.jtbi.2005.05.006

    Article  MathSciNet  MATH  Google Scholar 

  25. Weiss JA, Maker BN, Govindjee S (1996) Finite element implementation of incompressible, transversely isotropic hyperelasticity. Comput Methods Appl Mech Eng 135:107–128. https://doi.org/10.1016/0045-7825(96)01035-3

    Article  MATH  Google Scholar 

  26. Gou K, Pence TJ (2016) Hyperelastic modeling of swelling in fibrous soft tissue with application to tracheal angioedema. J Math Biol 72:499–526. https://doi.org/10.1007/s00285-015-0893-0

    Article  MathSciNet  MATH  Google Scholar 

  27. Agoras M, Lopez-Pamies O, Ponte Castañeda P (2009) A general hyperelastic model for incompressible fiber-reinforced elastomers. J Mech Phys Solids 57:268–286. https://doi.org/10.1016/j.jmps.2008.10.014

    Article  MATH  Google Scholar 

  28. Zhurov AI, Limbert G, Aeschlimann DP, Middleton J (2007) A constitutive model for the periodontal ligament as a compressible transversely isotropic visco-hyperelastic tissue. Comput Methods Biomech Biomed Eng 10:223–235. https://doi.org/10.1080/13639080701314894

    Article  Google Scholar 

  29. Anssari-Benam A, Bucchi A, Screen HRC, Evans SL (2017) A transverse isotropic viscoelastic constitutive model for aortic valve tissue. R Soc Open Sci 4:160585. https://doi.org/10.1098/rsos.160585

    Article  MathSciNet  Google Scholar 

  30. Gao X, Shi Z, Kuśmierczyk P et al (2016) Time-dependent rheological behaviour of bacterial cellulose hydrogel. Mater Sci Eng, C 58:153–159. https://doi.org/10.1016/j.msec.2015.08.019

    Article  Google Scholar 

  31. Limbert G, Middleton J (2004) A transversely isotropic viscohyperelastic material application to the modeling of biological soft connective tissues. Int J Solids Struct 41:4237–4260. https://doi.org/10.1016/j.ijsolstr.2004.02.057

    Article  MATH  Google Scholar 

  32. Biot MA (1955) Theory of elasticity and consolidation for a porous anisotropic solid. J Appl Phys 26:182–185. https://doi.org/10.1063/1.1721956

    Article  MathSciNet  MATH  Google Scholar 

  33. Mow VC, Kuei SC, Lai WM, Armstrong CG (1980) Biphasic creep and stress relaxation of articular cartilage in compression: theory and experiments. J Biomech Eng 102:73–84. https://doi.org/10.1115/1.3138202

    Article  Google Scholar 

  34. Hong W, Zhao X, Zhou J, Suo Z (2008) A theory of coupled diffusion and large deformation in polymeric gels. J Mech Phys Solids 56:1779–1793. https://doi.org/10.1016/J.JMPS.2007.11.010

    Article  MATH  Google Scholar 

  35. Doi M (2009) Gel dynamics. J Phys Soc Jpn 78:52001. https://doi.org/10.1143/JPSJ.78.052001

    Article  Google Scholar 

  36. Zhang J, Zhao X, Suo Z, Jiang H (2009) A finite element method for transient analysis of concurrent large deformation and mass transport in gels. J Appl Phys 105:93522. https://doi.org/10.1063/1.3106628

    Article  Google Scholar 

  37. Duan Z, Zhang J, An Y, Jiang H (2013) Simulation of the transient behavior of gels based on an analogy between diffusion and heat transfer. J Appl Mech. https://doi.org/10.1115/1.4007789

    Article  Google Scholar 

  38. Bouklas N, Landis CM, Huang R (2015) A nonlinear, transient finite element method for coupled solvent diffusion and large deformation of hydrogels. J Mech Phys Solids 79:21–43. https://doi.org/10.1016/J.JMPS.2015.03.004

    Article  MathSciNet  MATH  Google Scholar 

  39. Cai S, Hu Y, Zhao X, Suo Z (2010) Poroelasticity of a covalently crosslinked alginate hydrogel under compression. J Appl Phys 108:113514. https://doi.org/10.1063/1.3517146

    Article  Google Scholar 

  40. Chester SA, Di Leo CV, Anand L (2015) A finite element implementation of a coupled diffusion-deformation theory for elastomeric gels. Int J Solids Struct 52:1–18. https://doi.org/10.1016/J.IJSOLSTR.2014.08.015

    Article  Google Scholar 

  41. Lucantonio A, Nardinocchi P, Teresi L (2013) Transient analysis of swelling-induced large deformations in polymer gels. J Mech Phys Solids 61:205–218. https://doi.org/10.1016/J.JMPS.2012.07.010

    Article  MathSciNet  Google Scholar 

  42. Nardinocchi P, Pezzulla M, Teresi L (2015) Anisotropic swelling of thin gel sheets. Soft Matter 11:1492–1499. https://doi.org/10.1039/c4sm02485k

    Article  Google Scholar 

  43. Liu Y, Zhang H, Zhang J, Zheng Y (2015) Constitutive modeling for polymer hydrogels: a new perspective and applications to anisotropic hydrogels in free swelling. Eur J Mech A/Solids 54:171–186. https://doi.org/10.1016/J.EUROMECHSOL.2015.07.001

    Article  MathSciNet  MATH  Google Scholar 

  44. BN S, Shuolun W, CS A (2020) Modeling deformation-diffusion in polymeric gels. Poromechanics VI:141–148

    Google Scholar 

  45. Bosnjak N, Wang S, Han D et al (2019) Modeling of fiber-reinforced polymeric gels. Mech Res Commun 96:7–18. https://doi.org/10.1016/J.MECHRESCOM.2019.02.002

    Article  Google Scholar 

  46. Hu Y, Zhao X, Vlassak JJ, Suo Z (2010) Using indentation to characterize the poroelasticity of gels. Appl Phys Lett. https://doi.org/10.1063/1.3370354

    Article  Google Scholar 

  47. Lai Y, Hu Y (2018) Probing the swelling-dependent mechanical and transport properties of polyacrylamide hydrogels through AFM-based dynamic nanoindentation. Soft Matter 14:2619–2627. https://doi.org/10.1039/c7sm02351k

    Article  Google Scholar 

  48. Lai Y, Hu Y (2017) Unified solution for poroelastic oscillation indentation on gels for spherical, conical and cylindrical indenters. Soft Matter 13(4):852–861

    Article  Google Scholar 

  49. Lai Y, He D, Hu Y (2019) Indentation adhesion of hydrogels over a wide range of length and time scales. Extreme Mech Lett 31:100540

    Article  Google Scholar 

  50. Hu Y, Chan EP, Vlassak JJ, Suo Z (2011) Poroelastic relaxation indentation of thin layers of gels. J Appl Phys 110(8):086103

    Article  Google Scholar 

  51. Hu Y, You J-O, Auguste DT, Suo Z, Vlassak JJ (2012) Indentation: a simple, nondestructive method for characterizing the mechanical and transport properties of pH-sensitive hydrogels. J Mater Res 27(1):152–160

    Article  Google Scholar 

  52. Hu Y, Chen X, Whitesides GM, Vlassak JJ, Suo Z (2011) Indentation of polydimethylsiloxane submerged in organic solvents. J Mater Res 26(6):785–795

    Article  Google Scholar 

  53. Hu Y, Suo Z (2012) Viscoelasticity and poroelasticity in elastomeric gels. Acta Mech Solida Sin 25:441–458. https://doi.org/10.1016/S0894-9166(12)60039-1

    Article  Google Scholar 

  54. Wang X, Hong W (2012) A visco-poroelastic theory for polymeric gels. Proc R Soc A Math Phys Eng Sci 468:3824–3841. https://doi.org/10.1098/rspa.2012.0385

    Article  MathSciNet  MATH  Google Scholar 

  55. Caccavo D, Lamberti G (2017) PoroViscoElastic model to describe hydrogels’ behavior. Mater Sci Eng C 76:102–113. https://doi.org/10.1016/J.MSEC.2017.02.155

    Article  Google Scholar 

  56. Chester SA (2012) A constitutive model for coupled fluid permeation and large viscoelastic deformation in polymeric gels. Soft Matter 8:8223–8233

    Article  Google Scholar 

  57. He D, Hu Y (2020) Nonlinear visco-poroelasticity of gels with different rheological parts. J Appl Mech. https://doi.org/10.1115/1.4046966

    Article  Google Scholar 

  58. Pioletti DP, Rakotomanana LR, Benvenuti J-F, Leyvraz P-F (1998) Viscoelastic constitutive law in large deformations: application to human knee ligaments and tendons. J Biomech 31:753–757. https://doi.org/10.1016/S0021-9290(98)00077-3

    Article  Google Scholar 

  59. Feynman RP, Leighton RB, Sands M (1965) The Feynman lectures on physics; vol. i. Am J Phys 33:750–752. https://doi.org/10.1119/1.1972241

    Article  Google Scholar 

  60. Flory PJ, Rehner J (1943) Statistical mechanics of cross-linked polymer networks I. Rubberlike elasticity. J Chem Phys 11:512–520. https://doi.org/10.1063/1.1723791

    Article  Google Scholar 

  61. Holzapfel GA (2000) Nonlinear solid mechanics a continuum approach for engineering. Wiley, Chichester

    MATH  Google Scholar 

  62. Huggins ML (1941) Solutions of long chain compounds. J Chem Phys 9:440. https://doi.org/10.1063/1.1750930

    Article  Google Scholar 

  63. Flory PJ (1953) Principles of polymer chemistry. Cornell University Press, New York

    Google Scholar 

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Acknowledgements

The materials are based upon work supported by the National Science Foundation (NSF, Grant No. 1554326). Y.H. also acknowledges the funding support from Air Force Office of Scientific Research (AFOSR) under Award No. FA9550-19-1-0395 (Dr. B.-L. “Les” Lee, Program Manager).

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  1. George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, 801 Ferst Drive, Atlanta, GA, 30332, USA

    Dongjing He & Yuhang Hu

  2. School of Chemical and Biomolecular Engineering, Georgia Institute of Technology, 311 Ferst Drive, Atlanta, GA, 30332, USA

    Yuhang Hu

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  1. Dongjing He
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He, D., Hu, Y. A nonlinear visco-poroelasticity model for transversely isotropic gels. Meccanica 56, 1483–1504 (2021). https://doi.org/10.1007/s11012-020-01219-w

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