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Acta Mechanica

, Volume 230, Issue 7, pp 2625–2637 | Cite as

A semi-analytical solution for finite bending of a functionally graded hydrogel strip

  • Mohammad Reza Bayat
  • Arash Kargar-Estahbanaty
  • Mostafa BaghaniEmail author
Original Paper
  • 44 Downloads

Abstract

In this paper, a semi-analytical formulation is developed to examine the swelling-induced finite bending of a functionally graded hydrogel strip, when the strip is embedded in a solvent bath of an assigned chemical potential. The cross-link density of the hydrogel polymeric network varies through the strip thickness either linearly or exponentially. As a result of inhomogeneous swelling ratio through the hydrogel strip thickness, the strip bends in a circle. In contrast to earlier solution methods, the initial configuration is mapped to the deformed state without assuming any intermediary virtual state, using a total deformation gradient tensor. The swelling response of the hydrogel is studied utilizing the Flory–Huggins model for the free energy changes due to the mixing and deformation of the hydrogel network. In order to validate the presented method, FEM is employed to solve the finite bending of the functionally graded hydrogel strip. Using the presented method, the effects of the hydrogel network cross-link density distribution on the radial and tangential stress fields, strip bending curvature, and semi-angle are studied. In contrast to hydrogel-based multilayers, continuous stress and deformation field are found for the functionally graded hydrogel strip. Also, multiple tangential stress-free axes are observed for functionally graded hydrogel strips under bending configuration.

Notes

References

  1. 1.
    Hong, W., Liu, Z., Suo, Z.: Inhomogeneous swelling of a gel in equilibrium with a solvent and mechanical load. Int. J. Solids Struct. 46(17), 3282–3289 (2009).  https://doi.org/10.1016/j.ijsolstr.2009.04.022 CrossRefzbMATHGoogle Scholar
  2. 2.
    Drozdov, A.J.A.M.: Volume phase transition in thermo-responsive hydrogels: constitutive modeling and structure–property relations. Acta Mech. 226(4), 1283–1303 (2015).  https://doi.org/10.1007/s00707-014-1251-9 MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bayat, M.R., Baghani, M.: Finite element modeling and design of pH/temperature sensitive hydrogel based biphasic twisting actuators. Sci. Iran. (2018).  https://doi.org/10.24200/sci.2018.20603 Google Scholar
  4. 4.
    Lai, F., Li, H., Zou, H.J.A.M.: A modeling analysis for effect of elastic modulus on kinetics of ionic-strength-sensitive hydrogel. Acta Mech. 226(6), 1957–1969 (2015).  https://doi.org/10.1007/s00707-014-1290-2 CrossRefGoogle Scholar
  5. 5.
    Kargar-Estahbanaty, A., Baghani, M., Shahsavari, H., Faraji, G.: A combined analytical-numerical investigation on photosensitive hydrogel micro-valves. Int. J. Appl. Mech. 9(07), 1750103 (2017).  https://doi.org/10.1142/S1758825117501034 CrossRefGoogle Scholar
  6. 6.
    Peppas, N.A., Hilt, J.Z., Khademhosseini, A., Langer, R.: Hydrogels in biology and medicine: from molecular principles to bionanotechnology. Adv. Mater. 18(11), 1345–1360 (2006).  https://doi.org/10.1002/adma.200501612 CrossRefGoogle Scholar
  7. 7.
    Wong, A.P., Perez-Castillejos, R., Love, J.C., Whitesides, G.M.: Partitioning microfluidic channels with hydrogel to construct tunable 3-D cellular microenvironments. Biomaterials 29(12), 1853–1861 (2008).  https://doi.org/10.1016/j.biomaterials.2007.12.044 CrossRefGoogle Scholar
  8. 8.
    Beebe, D.J., Moore, J.S., Bauer, J.M., Yu, Q., Liu, R.H., Devadoss, C., Jo, B.-H.: Functional hydrogel structures for autonomous flow control inside microfluidic channels. Nature 404(6778), 588–590 (2000).  https://doi.org/10.1038/35007047 CrossRefGoogle Scholar
  9. 9.
    Kleverlaan, M., van Noort, R.H., Jones, I.: Deployment of swelling elastomer packers in Shell E&P. In: SPE/IADC Drilling Conference 2005. Society of Petroleum Engineers (2005)Google Scholar
  10. 10.
    Benjamin, C.C., Lakes, R.S., Crone, W.C.: Measurement of the stiffening parameter for stimuli-responsive hydrogels. Acta Mech. 229(9), 3715–3725 (2018).  https://doi.org/10.1007/s00707-018-2201-8 CrossRefGoogle Scholar
  11. 11.
    Drozdov, A.D., deClaville Christiansen, J.: Time-dependent response of hydrogels under multiaxial deformation accompanied by swelling. Acta Mech. 229(12), 5067–5092 (2018).  https://doi.org/10.1007/s00707-018-2288-y CrossRefGoogle Scholar
  12. 12.
    Wu, Z., Zhong, Z.: A nonlinear theory accounting for stress-induced orientational transitions in nematic gels. Acta Mech. 224(6), 1243–1250 (2013).  https://doi.org/10.1007/s00707-013-0871-9 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Zhu, J., Luo, J.: Effects of entanglements and finite extensibility of polymer chains on the mechanical behavior of hydrogels. Acta Mech. 229(4), 1703–1719 (2018).  https://doi.org/10.1007/s00707-017-2060-8 MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hu, Z., Zhang, X., Li, Y.: Synthesis and application of modulated polymer gels. Science 269(5223), 525–527 (1995).  https://doi.org/10.1126/science.269.5223.525 CrossRefGoogle Scholar
  15. 15.
    Abdolahi, J., Baghani, M., Arbabi, N., Mazaheri, H.: Analytical and numerical analysis of swelling-induced large bending of thermally-activated hydrogel bilayers. Int. J. Solids Struct. 99, 1–11 (2016).  https://doi.org/10.1016/j.ijsolstr.2016.08.017 CrossRefGoogle Scholar
  16. 16.
    Abdolahi, J., Baghani, M., Arbabi, N., Mazaheri, H.: Finite bending of a temperature-sensitive hydrogel tri-layer: an analytical and finite element analysis. Compos. Struct. 164, 219–228 (2017).  https://doi.org/10.1016/j.compstruct.2016.12.063 CrossRefGoogle Scholar
  17. 17.
    Guvendiren, M., Yang, S., Burdick, J.A.: Swelling-induced surface patterns in hydrogels with gradient crosslinking density. Adv. Funct. Mater. 19(19), 3038–3045 (2009).  https://doi.org/10.1002/adfm.200900622 CrossRefGoogle Scholar
  18. 18.
    Guvendiren, M., Burdick, J.A., Yang, S.: Kinetic study of swelling-induced surface pattern formation and ordering in hydrogel films with depth-wise crosslinking gradient. Soft Matter 6(9), 2044–2049 (2010).  https://doi.org/10.1039/B927374C CrossRefGoogle Scholar
  19. 19.
    Guvendiren, M., Burdick, J.A., Yang, S.: Solvent induced transition from wrinkles to creases in thin film gels with depth-wise crosslinking gradients. Soft Matter 6(22), 5795–5801 (2010).  https://doi.org/10.1039/C0SM00317D CrossRefGoogle Scholar
  20. 20.
    Wu, Z., Bouklas, N., Liu, Y., Huang, R.: Onset of swell-induced surface instability of hydrogel layers with depth-wise graded material properties. Mech. Mater. 105, 138–147 (2017).  https://doi.org/10.1016/j.mechmat.2016.11.005 CrossRefGoogle Scholar
  21. 21.
    Wu, Z., Bouklas, N., Huang, R.: Swell-induced surface instability of hydrogel layers with material properties varying in thickness direction. Int. J. Solids Struct. 50(3), 578–587 (2013).  https://doi.org/10.1016/j.ijsolstr.2012.10.022 CrossRefGoogle Scholar
  22. 22.
    Timoshenko, S.: Analysis of bi-metal thermostats. J. Opt. Soc. Am. 11(3), 233–255 (1925).  https://doi.org/10.1364/JOSA.11.000233 CrossRefGoogle Scholar
  23. 23.
    Roccabianca, S., Gei, M., Bigoni, D.: Plane strain bifurcations of elastic layered structures subject to finite bending: theory versus experiments. IMA J. Appl. Math. (2010).  https://doi.org/10.1093/imamat/hxq020 MathSciNetzbMATHGoogle Scholar
  24. 24.
    Lucantonio, A., Nardinocchi, P., Pezzulla, M.: Swelling-induced and controlled curving in layered gel beams. Proc. R. Soc. A Math. Phys. Eng. Sci. 470(2171), 1–16 (2014).  https://doi.org/10.1098/rspa.2014.0467 CrossRefGoogle Scholar
  25. 25.
    Chester, S.A., Anand, L.: A thermo-mechanically coupled theory for fluid permeation in elastomeric materials: application to thermally responsive gels. J. Mech. Phys. Solids 59(10), 1978–2006 (2011).  https://doi.org/10.1016/j.jmps.2011.07.005 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Morimoto, T., Ashida, F.: Temperature-responsive bending of a bilayer gel. Int. J. Solids Struct. 56–57, 20–28 (2015).  https://doi.org/10.1016/j.ijsolstr.2014.12.009 CrossRefGoogle Scholar
  27. 27.
    Arbabi, N., Baghani, M., Abdolahi, J., Mazaheri, H., Mashhadi, M.M.: Finite bending of bilayer pH-responsive hydrogels: a novel analytic method and finite element analysis. Compos. Part B Eng. 110, 116–123 (2017).  https://doi.org/10.1016/j.compositesb.2016.11.006 CrossRefGoogle Scholar
  28. 28.
    Kargar-Estahbanaty, A., Baghani, M., Arbabi, N.: Developing an analytical solution for photo-sensitive hydrogel bilayers. J. Intell. Mater. Syst. Struct. (2018).  https://doi.org/10.1177/1045389X18754353 Google Scholar
  29. 29.
    Hong, W., Zhao, X., Zhou, J., Suo, Z.: A theory of coupled diffusion and large deformation in polymeric gels. J. Mech. Phys. Solids 56(5), 1779–1793 (2008).  https://doi.org/10.1016/j.jmps.2007.11.010 CrossRefzbMATHGoogle Scholar
  30. 30.
    Flory, P.J.: Thermodynamics of high polymer solutions. J. Chem. Phys. 10(1), 51–61 (1942).  https://doi.org/10.1063/1.1723621 CrossRefGoogle Scholar
  31. 31.
    Flory, P.J., Rehner Jr., J.: Statistical mechanics of cross-linked polymer networks II. Swelling. J. Chem. Phys. 11(11), 521–526 (1943).  https://doi.org/10.1063/1.1723792 CrossRefGoogle Scholar
  32. 32.
    Huggins, M.L.: Solutions of long chain compounds. J. Chem. Phys. 9(5), 440–440 (1941).  https://doi.org/10.1063/1.1750930 CrossRefGoogle Scholar
  33. 33.
    Treloar, L.R.G.: The Physics of Rubber Elasticity. Oxford University Press, Oxford (1975)Google Scholar
  34. 34.
    Kierzenka, J., Shampine, L.F.: A BVP solver that controls residual and error. J. Numer. Anal. Ind. Appl. Math. 3(1–2), 27–41 (2008). ISSN: 1790-8140MathSciNetzbMATHGoogle Scholar
  35. 35.
    Byrd, R.H., Gilbert, J.C., Nocedal, J.: A trust region method based on interior point techniques for nonlinear programming. Math. Program. 89(1), 149–185 (2000).  https://doi.org/10.1007/PL00011391 MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Almasi, A., Baghani, M., Moallemi, A., Baniassadi, M., Faraji, G.: Investigation on thermal stresses in FGM hyperelastic thick-walled cylinders. J. Therm. Stresses 41(2), 204–221 (2018).  https://doi.org/10.1080/01495739.2017.1395719 CrossRefGoogle Scholar
  37. 37.
    Freund, L.B., Suresh, S.: Thin Film Materials: Stress, Defect Formation and Surface Evolution. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mechanical Engineering, College of EngineeringUniversity of TehranTehranIran

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