A number of processes of surface growth in areas ranging from mechanics of growing gravitational bodies[1][2][3][4][5][6] through propagating fronts of phase transitions,[7] epitaxial growth of nanostructures and 3D printing,[8] growth of plants,[9] and cell mobility[10] require non-Euclidean description because of incompatibility of boundary conditions and different mechanisms of developing stresses at interfaces. Indeed, these mechanisms result in the curving of initially flat elements of the body and changing separation between different elements of it (especially in the soft matter). Gradual accumulation of deformations under the influx of accumulating mass results in the memory-conscious grows of the body and makes strains the subject of long-range forces. As a result of all above factors, generic non-Euclidean growth is described in terms of Riemannian geometry with a space- and time-dependent curvature.[11][12]

References

  1. ↑ 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).
  2. ↑ 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. The Royal Society. 276 (1367): 571–576. doi:10.1098/rspa.1963.0227. ISSN 2053-9169.
  3. ↑ V. E. Naumov, Mechanics of growing deformable solids: A review, J. Eng. Mech. 120, 207 (1994).
  4. ↑ J. G. Bentler and J. F. Labuz, Performance of a Cantilever retaining wall, J. Geotech. Geoenviron. Eng. 132, 1062 (2006).
  5. ↑ Bacigalupo, Andrea; Gambarotta, Luigi (2012). "Effects of Layered Accretion on the Mechanics of Masonry Structures". Mechanics Based Design of Structures and Machines. Informa UK Limited. 40 (2): 163–184. doi:10.1080/15397734.2011.628622. ISSN 1539-7734.
  6. ↑ 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).
  7. ↑ Wildeman, Sander; Sterl, Sebastian; Sun, Chao; Lohse, Detlef (2017-02-23). "Fast Dynamics of Water Droplets Freezing from the Outside In". Physical Review Letters. American Physical Society (APS). 118 (8): 084101. arXiv:1701.06818. doi:10.1103/physrevlett.118.084101. ISSN 0031-9007.
  8. ↑ 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. Springer Science and Business Media LLC. 6 (1): 31110. doi:10.1038/srep31110. ISSN 2045-2322. PMC 4976324.
  9. ↑ R. R. Archer, Growth Stresses and Strains in Trees, Springer Series in Wood Science (Springer-Verlag, Berlin, 1987)
  10. ↑ Dafalias, Yannis F.; Pitouras, Zacharias (2007-12-06). "Stress field in actin gel growing on spherical substrate". Biomechanics and Modeling in Mechanobiology. Springer Science and Business Media LLC. 8 (1): 9–24. doi:10.1007/s10237-007-0113-y. ISSN 1617-7959.
  11. ↑ Truskinovsky, Lev; Zurlo, Giuseppe (2019-05-03). "Nonlinear elasticity of incompatible surface growth". Physical Review E. American Physical Society (APS). 99 (5): 053001. arXiv:1901.06182. doi:10.1103/physreve.99.053001. ISSN 2470-0045.
  12. ↑ Zurlo, Giuseppe; Truskinovsky, Lev (2017-07-26). "Printing Non-Euclidean Solids". Phys. Rev. Lett. American Physical Society (APS). 119 (4): 048001. arXiv:1703.03082. doi:10.1103/PhysRevLett.119.048001. ISSN 2470-0045.

F. Sozio, M.F. Shojaei, S. Sadik, and A. Yavari, Nonlinear mechanics of thermoelastic accretion, \emph{Zeitschrift f\"ur Angewandte Mathematik und Physik (ZAMP)} \textbf{71}(3), 2020, 87.

F. Sozio and A. Yavari, Nonlinear mechanics of accretion, \emph{Journal of Nonlinear Science} \textbf{29}(4), 2019, 1813-1863.

F. Sozio and A. Yavari, Nonlinear mechanics of surface growth for cylindrical and spherical elastic bodies, \emph{Journal of the Mechanics and Physics of Solids} \textbf{98}, 2017, pp. 12-48.

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).
  • 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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