Abstract
A model of the indentation of a circular elastic membrane (neo-Hookean material) on an incompressible liquid layer and an effective numerical solution method are presented. For the interface between the indenter and the membrane, stick or slip contact conditions are considered. The solution procedure identifies the liquid pressure and, for slip contact, the radius in the reference configuration displaced to the edge of the indenter. Assuming a range of material and test performance parameters regarding deep tonometry of soft subcutaneous tissue, the predictions for the distributions of stretches, line tensions and profiles of the indented membrane are analyzed. Among the key results of this work are the dependencies of the total indentation force and the liquid pressure on the Young’s modulus of the membrane’s material that appeared to be approximately linear for a range of Young’s modulus values. The profiles of the membrane for different indentation depths are close to parabolic and can be used to relate the longitudinal line tension at the edge of the indenter to the total indentation force.
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Aernouts, J., Aerts, J.R.M., Dirckx, J.J.J.: Mechanical properties of human tympanic membrane in the quasi-static regime from in situ point indentation measurements. Hear. Res. 290, 45–54 (2012)
Argatov, I.I., Sabina, F.J.: Spherical indentation of a transversaly isotropic elastic half-space reinforced with a thin layer. Int. J. Eng. Sci. 50, 132–143 (2012)
Bando, K., Oiso, Y.: Indentation analysis of microcapsule with initial stetch. J. Biomech. Sci. Eng. 6, 268277 (2013)
Bernal, R., Tassius, C., Melo, F., Geminard, J.-C.: Mechanical characterization of elastic membranes: cell mechanics applications. Appl. Phys. Lett. 90(6), 063903–3 (2007)
Choi, A.P.C., Zheng, Y.P.: Estimation of Young’s modulus and Poisson’s ratio of soft tissue from indentation using two different-sized indentors: finite element analysis of the finite deformation effect. Med. Biol. Eng. Comput. 43, 258–264 (2005)
Homsy, A.: Solid on liquid deposition, a review of technological solutions. Microelectron. Eng. 141, 267–279 (2015)
Hu, Y., Zhao, X., Vlassak, J.J., Suo, Z.: Using indentation to characterize the poroelasticity of gels. Appl. Phys. Lett. 96, 121904-1-3 (2010)
Iivarinen, J.T., Korhonen, R.K., Julkunen, P., Jurvelin, J.S.: Experimental and computational analysis of soft tissue elastic modulus in forearm using a manual indentation device. Med. Eng. Phys. 33, 1245–1253 (2011)
Kalra, A., Lowe, A., Al-Jumaily, A.M.: Mechanical behaviour of skin: a review. J. Mater. Sci. Eng. 5(4), 1000254 (2016)
Kruglikov, I.L., Scherer, P.E.: General theory of skin reinforcement. PLoS ONE 12(8), e0182865–15 (2017)
Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells, 2nd edn. Cambridge University Press, New York (1998)
Liu, N., Olszewski, W.L.: Use of tonometry to assess lower extremity lymphedema. Lymphology 25, 155–158 (1992)
Long, R., Shull, K.R., Hui, C.-Y.: Large deformation adhesive contact mechanics of circular membranes with a flat rigid substrate. J. Mech. Phys. Solids 58, 1225–1242 (2010)
Long, R., Hui, C.Y.: Axisymmetric membrane in adhesive contact with rigid substrates: analytical solutions under large deformation. Int. J. Solids Struct. 49, 672–683 (2012)
Nowak, J., Kaczmarek, M.: Modelling deep tonometry of lymphedematous tissue. Phys. Mesomech. 21(1), 6–14 (2018)
Pailler-Mattei, C., Bec, S., Zahouani, H.: In vivo measurements of elastic mechanical properties of human skin by indentation tests. Med. Eng. Phys. 30, 599–606 (2008)
Selvadurai, A.P.S., Shi, M.: Fluid pressure loading of a hyperelastic membrane. Int. J. Nonlinear Mech. 47(2), 228–239 (2012)
Sen, S., Subramanian, S., Sischer, D.: Indentation and adhesive probing of a cell membrane with AFM: theoretical model and experiments. Biophys. J. 89, 3203–3213 (2005)
Sohail, T., Nadler, B.: On the contact of an inflated spherical membrane-fluid structure with a rigid conical indenter. Acta Mech. 218, 225–235 (2011)
Taffetani, M., Vella, D.: Regimes of wrinkling in pressurized elastic shells. Phil. Trans. R. Soc. A 375, 20160330 (2017)
Taber, L.A.: Large deformation mechanics of the enucleated eyeball. J. Biomech. Eng. 106, 229–234 (1984)
Toyras, J., Lyyra-Laitinen, T., Niinimaki, M., Lindgren, M., Nieminen, M.T., Kiviranta, I., Jurvelin, J.S.: Estimation of the Young’s modulus of articular cartilage using an arthroscopic indentation instrument and ultrasonic measurement of tissue thickness. J. Biomech. 34, 251–256 (2001)
Tran, H.V., Charleus, F., Rachik, M., Ehrlacher, A., Ho Ba Tho, M.C.: In vivo characterization of the mechanical properties of human skin derived from MRI and indentation techniques. Comput. Method Biomech. Biol. Eng. 10(6), 401–407 (2007)
Vella, D., Ajdri, A., Vaziri, A., Boudaoud, A.: The indentation of pressurized elastic shells: from polymeric capsules to yeast cells. J. R. Soc. Interface 9, 448–455 (2012)
Yoo, L., Reed, J., Shin, A., Kung, J., Gimzewski, J.K., Poukens, V., Goldberg, R.A., Mancini, R., Taban, M., Moy, R., Demer, J.L.: Characterization of ocular tissue using microindentation and Hertzian viscoelastic models. Invest. Ophthalmol. Vis. Sci. 52(6), 3475–3482 (2011)
Zaleska, M.T., Olszewski, W.L., Durlik, M., Kaczmarek, M.K., Freidenrich, B.: Tonometry of deep tissues for setting effective compression pressures in lymphedema of limbs. Lymphat. Res. Biol. 16, 193–200 (2018)
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This work was partially supported by the National Science Centre in Poland under Grant UMO-2013/11/B/ST8/03589.
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Fąs, T., Kazimierska-Drobny, K. & Kaczmarek, M. Indentation of a circular membrane on an incompressible liquid. Acta Mech 229, 4779–4790 (2018). https://doi.org/10.1007/s00707-018-2248-6
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DOI: https://doi.org/10.1007/s00707-018-2248-6