##plugins.themes.bootstrap3.article.main##
Abstract
In this paper, general infinitesimal deformations of higher orders of simply connected surfaces are considered in three-dimensional Euclidean space E3, which are important in the study of their continuous deformations. The problem of finding the displacement vectors of these strains is reduced to the study and solution of a system of n equations (or basic equations) of general infinitesimal deformations of finite order n, which are obtained with respect to an arbitrarily chosen coordinate system on the surface. It is shown that for closed surfaces of positive Gaussian curvature, the mathematical model of this problem in the conjugate-isothermal coordinate system is a system of n heterogeneous equations of a complex form, which in the case of an ovaloid is reduced to an system of n integral equations. Using tensor methods, the apparatus of the theory of generalized analytic functions, and methods of functional analysis, it is proved that a regular ovaloid in E3 "as a whole" admits a general infinitesimal deformation of finite order n, which is uniquely determined by predefined 3n functions. Their geometric meaning is found: setting them is equivalent to setting the values of the variations of the normal unit vector and the area element up to the order n inclusively. The strain vector fields are determined up to constant vectors. It was established that the ovaloid will be tough with respect to the general infinitesimal deformations of finite order n if and only if all values of the variations of the normal unit vector and the area element up to order n inclusive are identically equal to zero. A sphere of radius R is considered as a confirming example of the surface. The displacement vectors are found in explicit form.
##plugins.themes.bootstrap3.article.details##
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution (CC-BY) 4.0 License that allows others to share the work with an acknowledgment of the work’s authorship and initial publication in this journal.
Provided they are the owners of the copyright to their work, authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal’s published version of the work (e.g., post it to an institutional repository, in a journal or publish it in a book), with an acknowledgment of its initial publication in this journal.
Authors are permitted and encouraged to post their work online (e.g., in institutional repositories, disciplinary repositories, or on their website) prior to and during the submission process.
References
2. П. Колобов. О бесконечно малых деформациях поверхности с сохранением площади. Учен.записки Кабардино-Балкарского ун-та, сер.физика, матем, №30, с.65-68., 1966
3. Б. Караев. Бесконечно малые изгибания высших порядков в тензорном изложении. Известия АН Туркменской ССР, Т.6, вып.2., с.116-122, 1969
4. П. Марков. Бесконечно малые изгибания высших порядков многомерных поверхностей. Укр.геом.сб., вып.8, с.87-94, 1982
5. Л. Безкоровайная. О бесконечно малых ареальных деформациях овальных поверхностей. Известия вузов, Математика, №5 (252), с.69-71, 1983
6. Н. Дерманец. Определение А-деформаций высших порядков по заданным значениям вариаций нормали. УкрНИИНТИ.-№56, Ук-84, Деп.-40 с. От 13.01.84, 1984
7. С. Климентов. О продолжении бесконечно малых изгибаний высших порядков односвязной поверхности положительной кривизны. Мат.заметки, Т.36, вып.3, с.393-403, 1984
8. И. Векуа. Обобщенные аналитические функции. М.: Наука.-509 с., 1988
9. Л. Безкоровайна. Структура множини розв’язків системи рівнянь для загальної нескінченно малої деформації. Тези доповідей міжнародної конференції, «Геометрія в Одесі -2004», Одеса, с.7-8, 2004
10. В. Фоменко. О жесткости овалоидов относительно G-деформаций при условии стационарности заданной функции главных радиусов кривизны. Соврем.проблемы математики и механики.- М.: Из-во МГУ, Т.6. Математика. Вып.3, с.177-186, 2011
11. Д. Жуков. Бесконечно малые MG-деформации овалоидов. Владикавказский матем.журнал, т.15, вып.2, с.36-45, 2013
12. L. Bezkorovaina. Surfaces generated by the real and imaginary parts of analytic functions: a-deformations occurring independently or simultaneously. Ukrainian Mathematical Journal, Vol. 70, no. 4, Apr. 2018, p. 447-63, 2018.
13. L. Bezkorovaina, T. Vashpanova. A - deformations of a surface with stationary lengths of lgt-lines. Ukrainian Mathematical Journal,Vol. 62, no. 7, July 2010, pp. 878 884, 2010.
14. I. Hinterleitner, V. Kiosak. Ric-vector fields on conformally at spaces. Proceedings of American Institute of Physics,1191, p. 98-103, https://doi.org/10.1063/1.3275604, 2009.
15. V. Kiosak, V. Matveev. There exist no 4-dimensional geodesically equivalent metrics with the same stress-energy tensor. Journal of Geometry and Physics, 78, 1-11, https://doi.org/10.1016/j.geomphys.2014.01.002, 2014.
16. T. Podousova, N. Vashpanova. A continuation a-deformations of surfaces of positive curvature with boundary. Proceedings of the International Geometry Center, Vol.7, no. 3, p. 38-48, http://dx.doi.org/10.15673/2072-9812.3/2014.40572, 2014.
17. Y.Vashpanov, T. Podousova, Yong Suk Kim, Jung-Young Son. Determination of geometric parameters of cracks in concrete by image processing. Advances in Civil Engineering,Volume 2019 ,Article ID 2398124 , 17 p. https://doi.org/10.1155/2019/2398124, 2019
18. N. Vashpanova, T. Podousova, J. Fedchenko. Canonical deformations of pseudo-riemannian spaces. AIP Conference Proceedings 2164, 040005 ; https://doi.org/10.1063/1.5130797, 2019.