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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.[1][2]
Examples of non-Euclidean surface growth are found in the mechanics of growing gravitational bodies,[3][4][5][6][7][8] propagating fronts of phase transitions,[9] epitaxial growth of nanostructures and additive 3D printing,[10] growth of plants,[11] and cell motility[12]
References
edit- ^ Truskinovsky, Lev; Zurlo, Giuseppe (2019-05-03). "Nonlinear elasticity of incompatible surface growth". Physical Review E. 99 (5). American Physical Society (APS): 053001. arXiv:1901.06182. Bibcode:2019PhRvE..99e3001T. doi:10.1103/physreve.99.053001. ISSN 2470-0045. PMID 31212512.
- ^ Zurlo, Giuseppe; Truskinovsky, Lev (2017-07-26). "Printing Non-Euclidean Solids". Phys. Rev. Lett. 119 (4). American Physical Society (APS): 048001. arXiv:1703.03082. Bibcode:2017PhRvL.119d8001Z. doi:10.1103/PhysRevLett.119.048001. ISSN 2470-0045. PMID 29341729.
- ^ 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).
- ^ Brown, C. B.; Goodman, L. E. (1963-12-17). "Gravitational stresses in accreted bodies". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 276 (1367). The Royal Society: 571–576. Bibcode:1963RSPSA.276..571B. doi:10.1098/rspa.1963.0227. ISSN 2053-9169.
- ^ 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).
- ^ Bacigalupo, Andrea; Gambarotta, Luigi (2012). "Effects of Layered Accretion on the Mechanics of Masonry Structures". Mechanics Based Design of Structures and Machines. 40 (2). Informa UK Limited: 163–184. doi:10.1080/15397734.2011.628622. ISSN 1539-7734.
- ^ 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).
- ^ Wildeman, Sander; Sterl, Sebastian; Sun, Chao; Lohse, Detlef (2017-02-23). "Fast Dynamics of Water Droplets Freezing from the Outside In". Physical Review Letters. 118 (8). American Physical Society (APS): 084101. arXiv:1701.06818. Bibcode:2017PhRvL.118h4101W. doi:10.1103/physrevlett.118.084101. ISSN 0031-9007. PMID 28282160.
- ^ Ge, Qi; Sakhaei, Amir Hosein; Lee, Howon; Dunn, Conner K.; Fang, Nicholas X.; Dunn, Martin L. (2016-08-08). "Multimaterial 4D Printing with Tailorable Shape Memory Polymers". Scientific Reports. 6 (1). Springer Science and Business Media LLC: 31110. Bibcode:2016NatSR...631110G. doi:10.1038/srep31110. ISSN 2045-2322. PMC 4976324. PMID 27499417.
- ^ R. R. Archer, Growth Stresses and Strains in Trees, Springer Series in Wood Science (Springer-Verlag, Berlin, 1987)
- ^ Dafalias, Yannis F.; Pitouras, Zacharias (2007-12-06). "Stress field in actin gel growing on spherical substrate". Biomechanics and Modeling in Mechanobiology. 8 (1). Springer Science and Business Media LLC: 9–24. doi:10.1007/s10237-007-0113-y. ISSN 1617-7959. PMID 18058144.
Further reading
edit- 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).
- Lind, Johan U.; Busbee, Travis A.; Valentine, Alexander D.; Pasqualini, Francesco S.; Yuan, Hongyan; et al. (2016-10-24). "Instrumented cardiac microphysiological devices via multimaterial three-dimensional printing". Nature Materials. 16 (3). Springer Science and Business Media LLC: 303–308. doi:10.1038/nmat4782. ISSN 1476-1122. PMC 5321777. PMID 27775708.
- Lychev, S.; Koifman, K. (2019). Geometry of Incompatible Deformations: Differential Geometry in Continuum Mechanics. De Gruyter. doi:10.1515/9783110563214. ISBN 978-3-11-056201-9.
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