In the field of surface growth, there are growth processes that result in the surface of an object changing shape over time. As the object grows, its surface may change from flat to curved, or change curvature. Two points on the surface may also change in distance as a result of deformations of the object or accretion of new matter onto the object. The shape of the surface and its changes can be described in terms of non-Euclidean geometry and in particular, Riemannian geometry with a space- and time-dependent curvature.
Examples of non-Euclidean surface growth are found in the mechanics of growing gravitational bodies,E. I. Rashba, Construction sequence dependent stresses in massive bodies due to their weight, Proc. Inst. Struct. Mech. Acad. Sci. Ukrainian SSR 18, 23 (1953).V. E. Naumov, Mechanics of growing deformable solids: A review, J. Eng. Mech. 120, 207 (1994).J. G. Bentler and J. F. Labuz, Performance of a Cantilever retaining wall, J. Geotech. Geoenviron. Eng. 132, 1062 (2006).S. A. Lychev, Geometric aspects of the theory of incompatible deformations in growing solids, in Mechanics for Materials and Technologies, ed. by H. Altenbach, R. Goldstein, and E.Murashkin, Advanced Structured Materials, 46 (Springer, New York, 2017). propagating fronts of phase transitions, epitaxial growth of nanostructures and additive 3D printing, growth of plants,R. R. Archer, Growth Stresses and Strains in Trees, Springer Series in Wood Science (Springer-Verlag, Berlin, 1987) and cell motility
References
Further reading A. V. Manzhirov and S. A. Lychev, Mathematical modeling of additive manufacturing technologies, in: Proceedings of the World Congress on Engineering 2014, Lecture Notes in Engineering and Computer Science (IAENG, London, UK, 2014), 2, pp. 1404–1409.
A. D. Drozdov, Viscoelastic Structures: Mechanics of Growth and Aging (Academic Press, New York, 1998).