Definition
By geometry of a surface one usually means characterization of its metric and curvature properties in surface curvilinear coordinates. Due to a large variety of surface shapes, it is convenient to use the common tensor notation. In shell theory, the most useful concepts are the surface covariant differentiation, description of surface curves and surface divergence theorems of vector and tensor fields.
Introduction
Geometry of a surface embedded into the three-dimensional Euclidean point space was presented in many classical monographs, for example, by Eisenhart (1947) and do Carmo (1976). Within the needs of theoretical description required in shell structures, appropriate introductions were worked out as parts of the books by Green and Zerna (1954), Naghdi (1963), Chernykh (1964), Flügge (1972), Pietraszkiewicz (1977), Başar and Krätzig (2001), Ciarlet (2005), and Lebedev et al. (2010).
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References
Başar Y, Krätzig WB (2001) Theory of shell structures, 2nd edn. VDI Verlag, Düsseldorf
Chernykh KF (1964) Linear theory of shells, part 2 (in Russian). University Press, Leningrad. English translation: NASA-TT-F-II 562, 1968
Ciarlet PG (2005) An introduction to differential geometry with application to elasticity. Springer, Berlin
Do Carmo MP (1976) Differential geometry of curves and surfaces. Prentice Hall, Upper Saddle River
Eisenhart LP (1947) An introduction to differential geometry with use of the tensor calculus. University Press, Princeton
Flügge W (1972) Tensor analysis and continuum mechanics. Springer, Berlin et al
Green AE, Zerna W (1954) Theoretical elasticity. Clarendon Press, Oxford
Lebedev LP, Cloud MJ, Eremeyev VA (2010) Tensor analysis with applications in mechanics. World Scientific, Singapore
Naghdi PM (1963) Foundations of elastic shell theory. In: Sneddon IN, Hill R (eds) Progress in solid mechanics IV. North-Holland, Amsterdam, pp 1–90
Pietraszkiewicz W (1977) Introduction to the non-linear theory of shells. Mitteilungen aus dem Institut für Mechanik 10. Ruhr-Universität, Bochum
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Pietraszkiewicz, W. (2018). Surface Geometry, Elements. In: Altenbach, H., Öchsner, A. (eds) Encyclopedia of Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53605-6_186-1
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DOI: https://doi.org/10.1007/978-3-662-53605-6_186-1
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