Abstract
Soft materials are known for their plethora of biomedical applications, intricate structure–property correlation and nonlinear mechanical response. Multiple length–time scale phenomena and hierarchical structure results in their nonlinearity. Phenomenological and continuum mechanical models have been developed to predict their mechanics, which have mostly been very material-specific with inability to predict the mechanics of different types of soft materials simultaneously. This shortcoming has been addressed in the present work, wherein a generic nonlinear viscoelastic model has been proposed to predict the mechanical response of hydrogels, sponges, and xerogels. A fractal derivative viscoelastic model is proposed considering a fractal Maxwell model in parallel with a nonlinear spring. In particular, this model is chosen to qualitatively mimic the material nonlinearity inherent in soft materials. The fractal dashpot in combination with the nonlinear spring accounts for the power law time-dependent rheology of generic soft materials. These two different aspects in the form of nonlinear stiffness and non-Newtonian rheology account for mechanics of most soft materials. The present model is shown to fit well the existing literature results for mechanical response of a multitude of soft material classes with different test conditions and loading rates, which is one of the salient features of the model, apart from its simplistic mathematical framework. Further, a parametric study is reported on the mechanics of nanocellulose loaded poly(vinyl alcohol) xerogel. The model predictions are observed to be in conjunction with the experimental observations.
Similar content being viewed by others
References
Abuzeid, O.M.: A linear viscoelastic creep-contact model of a flat fractal surface: Kelvin-Voigt medium. Ind. Lubr. Tribol. 56(6), 334–340 (2004)
Abuzeid, O.M., Eberhard, P.: Linear viscoelastic creep model for the contact of nominal flat surfaces based on fractal geometry: standard linear solid (SLS) material. J. Tribol. 129(3), 461–466 (2007)
Aegerter, M.A., Leventis, N., Koebel, M.M.: Aerogels Handbook. Springer, Berlin (2011)
Ahearne, M., Yang, Y., El Haj, A.J., Then, K.Y., Liu, K.-K.: Characterizing the viscoelastic properties of thin hydrogel-based constructs for tissue engineering applications. J. R. Soc. Interface 2(5), 455–463 (2005)
Aime, S., Cipelletti, L., Ramos, L.: Power law viscoelasticity of a fractal colloidal gel. J. Rheol. 62, 1429 (2018)
Annarasa, V., Popov, A.A., De Focatiis, D.S.A.: A phenomenological constitutive model for the viscoelastic deformation of elastomers. Mech. Time-Depend. Mater. (2020). https://doi.org/10.1007/s11043-020-09452-2
Babu, A.N.S., Rajan, A., Pramanik, R., Arunachalakasi, A.: A thermodynamically-consistent phenomenological viscoplastic model for hydrogels. Mater. Res. Express 6(8), 085418 (2019). https://doi.org/10.1088/2053-1591/ab2a49
Bacca, M., McMeeking, R.M.: A viscoelastic constitutive law for hydrogels. Meccanica 52(14), 3345–3355 (2017)
Budday, S., Sommer, G., Birkl, C., Langkammer, C., Haybaeck, J., Kohnert, J., Bauer, M., Paulsen, F., Steinmann, P., Kuhl, E., et al.: Mechanical characterization of human brain tissue. Acta Biomater. 48, 319–340 (2017)
Cai, W., Chen, W., Xu, W.: Characterizing the creep of viscoelastic materials by fractal derivative models. Int. J. Non-Linear Mech. 87, 58–63 (2016)
Chaimoon, K., Chindaprasirt, P.: An anisotropic hyperelastic model with an application to soft tissues. Eur. J. Mech. A, Solids 78, 103845 (2019)
Dai, L., Tian, C., Xiao, R.: Modeling the thermo-mechanical behaviour and constrained recovery performance of cold-programmed amorphous shape-memory polymers. Int. J. Plast. 127, 102654 (2020)
Drury, J.L., Mooney, D.J.: Hydrogels for tissue engineering: scaffold design variables and applications. Biomaterials 24(24), 4337–4351 (2003)
Fakhouri S., Hutchens, S.B., Crosby, A.J.: Puncture mechanics of soft solids. Soft Matter 11(23), 4723–4730 (2015)
Ficarella, E., Lamberti, L., Papi, M., De Spirito, M., Pappalettere, C.: Viscohyperelastic calibration in mechanical characterization of soft matter. In: Mechanics of Biological Systems and Materials, vol. 6, pp. 33–37. Springer, Berlin (2017)
Garcia-Gonzalez, D., Jérusalem, A., Garzon-Hernandez, S., Zaera, R., Arias, A.: A continuum mechanics constitutive framework for transverse isotropic soft tissues. J. Mech. Phys. Solids 112, 209–224 (2018)
Gong, X., Wang, Y., Tian, Z., Zheng, X., Chen, L.: Controlled production of spruce cellulose gels using an environmentally “green” system. Cellulose 21(3), 1667–1678 (2014)
Gudimetla, M.R., Doghri, I.: A finite strain thermodynamically-based constitutive framework coupling viscoelasticity and viscoplasticity with application to glassy polymers. Int. J. Plast. 98, 197–216 (2017)
Han, L., Xu, J., Lu, X., Gan, D., Wang, Z., Wang, K., Zhang, H., Yuan, H., Weng, J.: Biohybrid methacrylated gelatin/polyacrylamide hydrogels for cartilage repair. J. Phys. Chem. B 5(4), 731–741 (2017)
He, G., Liu, Y., Deng, X., Fan, L.: Constitutive modeling of viscoelastic–viscoplastic behavior of short fiber reinforced polymers coupled with anisotropic damage and moisture effects. Acta Mech. Sin. 35(3), 495–506 (2019)
Hei, X., Chen, W., Pang, G., Xiao, R., Zhang, C.: A new visco-elasto-plastic model via time-space fractional derivative. Mech. Time-Depend. Mater. 22, 129–141 (2018)
Heymans, N., Bauwens, J.C.: Fractal rheological models and fractional differential equations for viscoelastic behaviour. Rheol. Acta 33, 210–219 (1994)
Hu, Y., Suo, Z.: Viscoelasticity and poroelasticity in elastomeric gels. Acta Mech. Solida Sin. 25(5), 441–458 (2012)
Johnson, B., Bauer, J.M., Niedermaier, D.J., Crone, W.C., Beebe, D.J.: Experimental techniques for mechanical characterization of hydrogels at the microscale. Exp. Mech. 44(1), 21 (2004)
Karimi, A., Navidbakhsh, M., Beigzadeh, B.: A visco-hyperelastic constitutive approach for modeling polyvinyl alcohol sponge. Tissue Cell 46(1), 97–102 (2014)
Katti, A., Shimpi, N., Roy, S., Lu, H., Fabrizio, E.F., Dass, A., Capadona, L.A., Leventis, N.: Chemical, physical, and mechanical characterization of isocyanate cross-linked amine-modified silica aerogels. Chem. Mater. 18(2), 285–296 (2006)
Kelly, J.F., McGough, R.J.: Fractal ladder models and power law wave equations. J. Acoust. Soc. Am. 126(4), 2072–2081 (2009)
Korchagin, V., Dolbow, J., Stepp, D.: A theory of amorphous viscoelastic solids undergoing finite deformations with application to hydrogels. Int. J. Solids Struct. 44(11–12), 3973–3997 (2007)
Li, W., Wang, D., Yang, W., Song, Y.: Compressive mechanical properties and microstructure of PVA–HA hydrogels for cartilage repair. RSC Adv. 6(24), 20166–20172 (2016)
Lin, J., Zheng, S., Xiao, R., Yin, J., Wu, Z., Zheng, Q., Qian, J.: Constitutive behaviours of tough physical hydrogels with dynamic metal-coordinated bonds. J. Mech. Phys. Solids 139, 103935 (2020)
Liu, K., Ovaert, T.C.: Poro-viscoelastic constitutive modeling of unconfined creep of hydrogels using finite element analysis with integrated optimization method. J. Mech. Behav. Biomed. Mater. 4(3), 440–450 (2011)
Lu, H., Wang, X., Shi, X., Yu, K., Fu, Y.Q.: A phenomenological model for dynamic response of double-network hydrogel composite undergoing transient transition. Composites, Part B, Eng. 151, 148–153 (2018)
Mainardi, F., Masina, E., Spada, G.: A generalization of the Becker model in linear viscoelasticity: Creep, relaxation and internal friction. Mech. Time-Depend. Mater. 23(3), 283–294 (2019)
Murali Krishnan, J., Deshpande, A.P., Sunil Kumar, P.B.: Rheology of Complex Fluids. Springer, Berlin (2010)
Ould Eleya, M.M., Ko, S., Gunasekaran, S.: Scaling and fractal analysis of viscoelastic properties of heat-induced protein gels. Food Hydrocoll. 18, 315–323 (2004)
Panda, D., Konar, S., Bajpai, S.K., Arockiarajan, A.: Synthesis and viscoelastic characterization of microstructurally aligned silk fibroin sponges. J. Mech. Behav. Biomed. Mater. 71, 362–371 (2017)
Panda, D., Konar, S., Bajpai, S.K., Arockiarajan, A.: Thermodynamically-consistent constitutive modeling of aligned silk fibroin sponges: theory and application to uniaxial compression. Int. J. Solids Struct. 138, 144–154 (2018)
Pazos, V., Mongrain, R., Tardif, J.C.: Polyvinyl alcohol cryogel: optimizing the parameters of cryogenic treatment using hyperelastic models. J. Mech. Behav. Biomed. Mater. 2(5), 542–549 (2009)
Pramanik, R., Arockiarajan, A.: Influence of mechanical compressive loads on microstructurally aligned PVA xerogels. Mater. Lett. 236, 222–224 (2019)
Pramanik, R., Ganivada, B., Ram, F., Shanmuganathan, K., Arockiarajan, A.: Influence of nanocellulose on mechanics and morphology of polyvinyl alcohol xerogels. J. Mech. Behav. Biomed. Mater. 90, 275–283 (2019)
Pramanik, R., Narayanan, A., Rajan, A., Konar, S., Arockiarajan, A.: Transversely isotropic freeze-dried PVA hydrogels: theoretical modelling and experimental characterization. Int. J. Eng. Sci. 144, 103144 (2019)
Rajagopal, K.R.: Non-linear elastic bodies exhibiting limiting small strain. Math. Mech. Solids 16(1), 122–139 (2011)
Rajagopal, K.R., Srinivasa, A.R.: A Gibbs-potential-based formulation for obtaining the response functions for a class of viscoelastic materials. Proc. R. Soc. A, Math. Phys. Eng. Sci. 467(2125), 39–58 (2011)
Rajan, A., Pramanik, R., Narayanan, A., Arockiarajan, A.: Mechanics of viscoelastic buckling in slender hydrogels. Mater. Res. Express 6(5), 055320 (2019)
Rich, S.I., Wood, R.J., Majidi, C.: Untethered soft robotics. Nat. Electron. 1(2), 102–112 (2018)
Sandolo, C., Coviello, T., Matricardi, P., Alhaique, F.: Characterization of polysaccharide hydrogels for modified drug delivery. Eur. Biophys. J. 36(7), 693–700 (2007)
Sanginario, V., Ginebra, M.P., Tanner, K.E., Planell, J.A., Ambrosio, L.: Biodegradable and semi-biodegradable composite hydrogels as bone substitutes: morphology and mechanical characterization. J. Mater. Sci., Mater. Med. 17(5), 447–454 (2006)
Serra-Aguila, A., Puigoriol-Forcada, J.M., Reyes, G., Menacho, J.: Viscoelastic models revisited: characteristics and interconversion formulas for generalized Kelvin–Voigt and Maxwell models. Acta Mech. Sin. 35(6), 1191–1209 (2019)
Siegel, R.A., Gu, Y., Lei, M., Baldi, A., Nuxoll, E.E., Ziaie, B.: Hard and soft micro-and nanofabrication: an integrated approach to hydrogel-based biosensing and drug delivery. J. Control. Release 141(3), 303–313 (2010)
Sun, Y., Chen, C.: Fractional order creep model for coral sand. Mech. Time-Depend. Mater. 23(4), 465–476 (2019)
Suo, Z.: Mechanics of stretchable electronics and soft machines. Mater. Res. Soc. Bull. 37(3), 218–225 (2012)
Tibbitt, M.W., Anseth, K.S.: Hydrogels as extracellular matrix mimics for 3d cell culture. Biotechnol. Bioeng. 103(4), 655–663 (2009)
Toh, S.W., Loh, X.J.: Advances in hydrogel delivery systems for tissue regeneration. Mater. Sci. Eng. C 45, 690–697 (2014)
Tokarev, I., Minko, S.: Stimuli-responsive porous hydrogels at interfaces for molecular filtration, separation, controlled release, and gating in capsules and membranes. Adv. Mater. 22(31), 3446–3462 (2010)
Vemaganti, K., Madireddy, S., Kedari, S.: On the inference of viscoelastic constants from stress relaxation experiments. Mech. Time-Depend. Mater. 24(1), 1–24 (2020)
Volokh, K.: Mechanics of Soft Materials. Springer, Berlin (2016)
Wang, Y., Maurel, G., Couty, M., Detcheverry, F., Merabia, S.: Implicit medium model for fractal aggregate polymer nanocomposites: linear viscoelastic properties. Macromolecules 52(5), 2021–2032 (2019)
Wineman, A.S., Rajagopal, K.R.: Mechanical Response of Polymers: An Introduction. Cambridge University Press, Cambridge (2000)
Xiao, R., Tian, C.: A constitutive model for strain hardening behaviours of pre-deformed amorphous polymers: incorporating a dissipative dynamics of molecular orientation. J. Mech. Phys. Solids 125, 472–487 (2019)
Xiao, R., Sun, H., Chen, W.: A finite deformation fractional viscoplastic model for the glass transition behaviour of amorphous polymers. Int. J. Non-Linear Mech. 93, 7–14 (2017)
Zhang, Y.S., Khademhosseini, A.: Advances in engineering hydrogels. Science 356(6337), eaaf3627 (2017)
Zhang, L., Zhao, J., Zhu, J., He, C., Wang, H.: Anisotropic tough poly (vinyl alcohol) hydrogels. Soft Matter 8(40), 10439–10447 (2012)
Zhirikova, Z.M., Aloyev, V.Z.: Application of model of the viscoelastic body and the fractal analysis for the description of process of flowability of polymeric nanocomposites. Mater. Sci. Forum 935, 150–154 (2018)
Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for profit sectors.
Author information
Authors and Affiliations
Contributions
R Pramanik: Conceptualization, Writing - original draft
F Soni: Resources
K Shanmuganathan: Supervision
A Arockiarajan: Project administration, Writing - review & editing
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they do not have any conflict of interest.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Pramanik, R., Soni, F., Shanmuganathan, K. et al. Mechanics of soft polymeric materials using a fractal viscoelastic model. Mech Time-Depend Mater 26, 257–270 (2022). https://doi.org/10.1007/s11043-021-09486-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11043-021-09486-0