Non-Euclidean surface growth

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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.{{cite journal | last1=Truskinovsky | first1=Lev | last2=Zurlo | first2=Giuseppe | title=Nonlinear elasticity of incompatible surface growth | journal=Physical Review E | publisher=American Physical Society (APS) | volume=99 | issue=5 | date=2019-05-03 | issn=2470-0045 | doi=10.1103/physreve.99.053001 | page=053001| pmid=31212512 |arxiv=1901.06182| bibcode=2019PhRvE..99e3001T }}{{cite journal | last1=Zurlo | first1=Giuseppe | last2= Truskinovsky | first2= Lev | title=Printing Non-Euclidean Solids | journal=Phys. Rev. Lett. | publisher=American Physical Society (APS) | volume= 119 | issue=4 | date=2017-07-26 | issn=2470-0045 | doi=10.1103/PhysRevLett.119.048001 | page= 048001 | pmid=29341729 | url = https://link.aps.org/doi/10.1103/PhysRevLett.119.048001| arxiv=1703.03082 | bibcode=2017PhRvL.119d8001Z }}

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).{{cite journal |first1=C. B. |last1=Brown |first2= L. E. |last2=Goodman| title=Gravitational stresses in accreted bodies | journal=Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences | publisher=The Royal Society | volume=276 | issue=1367 | date=1963-12-17 | issn=2053-9169 | doi=10.1098/rspa.1963.0227 | pages=571–576|bibcode=1963RSPSA.276..571B }}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).{{cite journal | last1=Bacigalupo | first1=Andrea | last2=Gambarotta | first2=Luigi | title=Effects of Layered Accretion on the Mechanics of Masonry Structures | journal=Mechanics Based Design of Structures and Machines | publisher=Informa UK Limited | volume=40 | issue=2 | year=2012 | issn=1539-7734 | doi=10.1080/15397734.2011.628622 | pages=163–184}}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,{{cite journal | last1=Wildeman | first1=Sander | last2=Sterl | first2=Sebastian | last3=Sun | first3=Chao | last4=Lohse | first4=Detlef | title=Fast Dynamics of Water Droplets Freezing from the Outside In | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=118 | issue=8 | date=2017-02-23 | issn=0031-9007 | doi=10.1103/physrevlett.118.084101 | page=084101| pmid=28282160 |arxiv=1701.06818| bibcode=2017PhRvL.118h4101W }} epitaxial growth of nanostructures and additive 3D printing,{{cite journal | last1=Ge | first1=Qi | last2=Sakhaei | first2=Amir Hosein | last3=Lee | first3=Howon | last4=Dunn | first4=Conner K. | last5=Fang | first5=Nicholas X. | last6=Dunn | first6=Martin L. | title=Multimaterial 4D Printing with Tailorable Shape Memory Polymers | journal=Scientific Reports | publisher=Springer Science and Business Media LLC | volume=6 | issue=1 | date=2016-08-08 | issn=2045-2322 | doi=10.1038/srep31110 | page=31110|doi-access=free| pmid=27499417 | pmc=4976324 | bibcode=2016NatSR...631110G }} growth of plants,R. R. Archer, Growth Stresses and Strains in Trees, Springer Series in Wood Science (Springer-Verlag, Berlin, 1987) and cell motility{{cite journal | last1=Dafalias | first1=Yannis F. | last2=Pitouras | first2=Zacharias | title=Stress field in actin gel growing on spherical substrate | journal=Biomechanics and Modeling in Mechanobiology | publisher=Springer Science and Business Media LLC | volume=8 | issue=1 | date=2007-12-06 | issn=1617-7959 | doi=10.1007/s10237-007-0113-y | pages=9–24| pmid=18058144 }}