Abstract
The stress-strain state of a pressurized spherical shell and circular cylinder are studied by means of the exact 3D theory of elasticity and the 2D approximate shell theories of moderate thicknesses. Both the sphere and the cylinder are made of transversally isotropic material. The first problem models the changes in the fluid pressure inside the human eye due to injected additional volume of liquid. The pressurized cylinder is one of simplest model of the human vessels. The algebraic relationships for deflections and stresses are derived. Both described shell theories take into account the variation of the shell thickness, which can be important for soft materials, e.g. the human eye’s tissue. The asymptotical analysis of the exact 3D solutions has been performed and the accuracy of the approximate solutions, obtained with the approximate theories is analyzed. The effect of the thickness changes are also discussed.
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References
Altenbach, H.: Theories for laminated and sandwich plates. An overview. Mech. Compos. Mater. 34, 333–349 (1998)
Ambartsumyan, S.: Theory of Anisotropic Plates. Technomic Publishing, Stamford (1970)
Atanackovic, T.M., Guran, A.: Theory of Elasticity for Scientist and Engineers. Birkhäuser, Berlin (2000)
Bauer, S.M., Kachanov, A.B., Semenov, B.N., Sliesoraityte, I.: The effect of cornea thickness on measurements of intraocular pressure by applanation methods. In: Proceeding of the Ocular Biomechanics Conference, pp. 69–73 (2007)
Bauer, S.M., Zimin, B.A., Tovstik, P.E.: Simplest Models of Plates and Shells in Ophthalmology (in Russ.). St. Petersburg University Press, St. Petersburg (2000)
Bauer, S.M., Voronkova, E.B.: On the deformation of the lamina cribrosa under intraocular pressure. Russ. J. Biomech. 5(1), 73–82 (2001)
Bauer, S.M., Mironov, A.N.: On the mathematical simulation of the stress-strain state of the eye shell undergoing the scleral buckling procedure. Acta. Bioeng. Biomech. 4, 726–727 (2002)
Bauer, S.M., Lyubimov, G.A., Tovstik, P.E.: On the mathematical simulation of the measuring of the intraocular pressure by Maklakov method. Technische Mechanik 24, 231–235 (2004)
Bauer, S.M., Lyubimov, G.A., Tovstik, P.E.: Mathematical modelling of Maklakoffs method for mesuring the intraocular pressure. Fluid Dyn. 40, 20–33 (2005)
Bauer, S.M.: Mechanical models of the development of glaucoma. In: Guran, A., Smirnov, A.L., Steigmann, D.J., Vaillancourt, R. (eds.) Advances in Mechanics of Solids in Memory of Prof. E.M. Haseganu, pp. 153–178. World Scientific Publishing, Singapore (2006)
Bauer, S.M., Karamshina, L.A., Kachanov, A.B.: Mechanical models of the measurements of intraocular pressure by Goldmann and Maklakov applanation tonometers after refractive surgery (in Russ.). Russ. J. Biomech. 16(3(57)), 25–32 (2012)
Iomdina, E.N.: Mechanical properties of the human eye tissues. Modern Problems of Biomechanics, pp. 183–200. Moscow University Press, Moscow (2006)
Kotliar, K., Maier, M., Bauer, S., Feucht, N., Lohmann, C., Lanzl, I.: Effect of intravitreal injections and volume changes on intraocular pressure: clinical results and biomechanical model. Acta. Ophthamol Scand. 85, 777–781 (2007)
Lang, G.K.: Ophthalmology. Thieme, Stuttgart (2000)
Lyubimov, G.A.: Opportunities of the elastometry method for investigating for elastic properties of the eyeball shell. Fluid Dyn. 45, 169–176 (2010)
Palii, O.M., Spiro, V.E.: Anisotropic Shells in Shipbuildings. Theory and Analysis (in Russ.). Sudostroenie, Leningrad (1977)
Reddy, J.N., Wang, C.M.: An overview of the relationships between of the classical and shear deformation plate theories. Compos. Sci. Technol. 60, 2327–2335 (2000)
Rodionova, V.A., Titaev, V.F., Chernykh, K.F.: Applied Theory of Anisotropic Plares and Shells (in Russ.). St. Petersburg University Press, St. Petersburg (1996)
Stein, A.A.: Pressure-volume dependence for eyeball under external load. Fluid Dyn. 45, 177–186 (2010)
Tovstik, P.E., Tovstik, T.P.: On the 2d models of plates and shells including the transversal shear. Z. Angew. Math. Mech. 87, 160–171 (2007)
Voronkova, E.B., Bauer, S.M., Eriksson, A.: Nonclassical theories of shells in application to soft biological tissue. In: Altenbach, H., Eremeyev, V.A. (eds.) Shell-like Structures, Advanced Structured Materials, vol. 15, pp. 647–654. Springer, Berlin (2011)
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Bauer, S.M., Voronkova, E.B. (2015). Nonclassical Shell Theories in Ocular Biomechanics. In: Altenbach, H., Mikhasev, G. (eds) Shell and Membrane Theories in Mechanics and Biology. Advanced Structured Materials, vol 45. Springer, Cham. https://doi.org/10.1007/978-3-319-02535-3_4
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DOI: https://doi.org/10.1007/978-3-319-02535-3_4
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