Restrictions on stress components in the vertices of regular triangular and quadrangular pyramids embedded in elastic body | Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika – Tomsk State University Journal of Mathematics and Mechanics. 2019. № 62. DOI: 10.17223/19988621/62/10

Restrictions on stress components in the vertices of regular triangular and quadrangular pyramids embedded in elastic body

In the framework of continuous model of deformable bodies, each point of the continuum is associated with an elementary volume. The concepts of continuum mechanics regarding material properties and state parameters (stresses, strains) are applicable to this volume. In the paper, this statement extends to singular points which are the vertices of triangular and quadrangular pyramids embedded in an elastic body. The restrictions on the stress components at the considered points are studied. It is shown that the number of restrictions determines a non-classical formulation of the problem of mechanics of a deformable body. The dependences for material constants of the bonded elements, which lead to an unlimited increase in the stresses in the vertices of triangular and quadrangular pyramids immersed in an elastic medium, are found to be the same. Moreover, these dependences coincide with those known for a circular cone and a spatial edge. The investigation results will find application in the mechanics of composite materials when studying the samples by indentation or interaction with prismatic cantilevers.

Download file

Counter downloads: 127

Keywords

внутренняя особая точка, неклассическая задача, концентрация напряжений, элементарный объем, internal singular point, non-classical problem, stress concentration, elementary volume

Authors

Name	Organization	E-mail
Pestrenin Valeriy M.	Perm State National Research University	PestreninVM@mail.ru
Pestrenina Irena V.	Perm State National Research University	IPestrenina@gmail.com
Landik Lidiya V.	Perm State National Research University	LidiaLandik@gmail.com

Всего: 3

References

Williams M.L. Stress singularities resulting from various boundary conditions in angular corners in extension // J. App. Mech. 1952. V. 19. P. 526-528.

Андреев А.В. Суперпозиция степенно-логарифмических и степенных сингулярных решений в двумерных задачах теории упругости // Вестник Пермского национального исследовательского политехнического университета. Механика. 2014. № 1. С. 5-30.

Рязанцева Е. А. Метод граничных состояний в задачах теории упругости с сингулярностями физического и геометрического характера: дис. канд. физ.-мат. наук: 01.02.04 / Е.А.Рязанцева. Липецк, 2015. 215 с.

Kσvalenko, M.D., Menshova, I.V., Kerzhaev, A.P. et al. Mixed boundary value problems in the theory of elasticity in an infinite strip // Acta Mech. 2018. 229. 4339. DOI: 10.1007/s00707-018-2244-x.

He Z., Kotousov A. On Evaluation of Stress Intensity Factor from In-Plane and Transverse Surface Displacements // Experim. Mech. 2016. V. 56. No. 8. P. 1385-1393. DOI: 10.1007/s11340-016-0176-8.

Xu W., Tong Z., Leung, A.Y.T., Xu X., Zhou Z. Evaluation of the stress singularity of an interface V-notch in a bimaterial plate under bending // Eng. Frac. Mech. 2016. V. 168. P. 1125. DOI: 10.1016/j.engfracmech.2016.09.009.

Koguchi H., Antonio da Costa J. Analysis of the stress singularity field at a vertex in 3D-bonded structures having a slanted side surface // Int. J. Solids and Struc. 2010. V. 47. P. 3131-3140. DOI: 10.1016/j.ijsolstr.2010.07.015

NKemzi B. On solution of Lame equations in axisymmetric domains with conical points // Math. Methods Appl. Sciences. 2005. V. 28. No. l. P. 29-41.

Корепанова Т.О., Матвеенко В.П., Шардаков И.Н. Аналитические построения собственных решений для изотропных конических тел и их приложения для оценки сингулярности напряжений // Докл. АН. 2014. Т. 457. № 3. С. 286-291. DOI: 10.7868/ s0869565214210105.

Koguchi H., Muramoto T. The order of stress singularity near the vertex in three-dimensional joints // Int. J. Solids and Structures. 2000. V. 37(35). P. 4737-4762. DOI: 10.1016/S0020-7683(99)00159-6.

Mittelstedt C., Beder W. Efficient computation of order and mode of three-dimensional stress singularities in linear elasticity by the boundary finite element method // Int. J. Solids and Structures. 2006. V. 43. No. 10. P. 2868-2903. DOI: 10.1016/j.ijsolstr.2005.05.059.

Lee Y., Jeon I., Im S. The stress intensities of three-dimensional corner singularities in a laminated composite // Int. J. Solids and Struc. 2006. V. 43(9). P. 2710-2722. DOI: 10.1016/j.ijsolstr.2005.06.050.

Zhixue W. A method for eliminating the effect of 3-D bi-material interface corner geometries on stress singularity // Engineering Fracture Mechanics. 2005. V. 73(7). P. 953-962. DOI: 10.1016/j.engfracmech.2005.10.010.

Koguchi H., da Costa J. A. Analysis of the stress singularity field at a vertex in 3D-bonded structures having a slanted side surface // Int. J. Solids and Struc. 2010. V. 47. P. 3131-3140. DOI: 10.1016/j.ijsolstr.2010.07.015.

Kovaienko M.D., Menshova I.V., Kerzhaev A.P. On the exact solutions of the biharmonic problem of the theory of elasticity in a half-strip // A.P.Z. Angew. Math. Phys. 2018. No. 69. P. 121-138. DOI: 10.1007/s00033-018-1013-y.

Apel Т., Mehrmann V., Wathns D. Structured eigenvalue methods for the computation of corner singularities in 3D anisotropic elastic structures // Comput. Methods Appl. Mech. Engng. 2002. No. 191. P. 4459-4473.

Wu Z. A method for eliminating the effect of 3-D bi-material interface corner geometries on stress singularity // Eng. Fract. Mech. 2006. V. 73. No 7. P. 953-962. DOI: 10.1016/ j.engfracmech.2005.10.010

Miyazaki T., Inoue T., Noda N.A. Practical method for analyzing singular index and intensity of singular stress field for three dimensional bonded plate (Conference Paper) // IOP Conference Series: Materials Science and Engineering. 2018. V. 372(1). P. 0120022018.

Ping X., Chen M., Zhu W., Xiao Y., Wu W. Computations of Singular Stresses Along ThreeDimensional Corner Fronts by a Super Singular Element Method // Int. J. Comp. Methods. 2017. V. 14(6). P. 1750065

Пестренин В.М., Пестренина И.В., Ландик Л.В. Ограничения на параметры напряженного состояния в вершине кругового конуса // Вестник Томского государственного университета. Математика и механика. 2018. № 52. С. 89-101. DOI: 10.17223/ 19988621/52/9.

Pestrenin V. M., Pestrenina I. V., Landik L.V. Nonstandart problems for structural elements with spatial composite ribs// Mechanics of Composite Materials. 2015. V. 51. No. 4. P. 489504. DOI: 10.1007∕s11029-015-9520-9.

Pestrenin V.M., Pestrenina I.V. Constraints on stress components at the internal singular point of an elastic compound structure // Mechanics of Composite Materials. 2017. V. 53. No. 1. P. 107-116. DOI: 10.1007/s11029-017-9644-1.

Pestrenin V. M., Pestrenina I. V., Landik L.V. Otress state at the vertex of a composite wedge, one side of which slides without friction along a rigid surface // Latin American Journal of Solids and Structures. 2017. V. 14. No. 11. P. 2067-2088. DOI: 10.1590/1679-78253826.

Pestrenin V.M., Pestrenina I.V., Landik L.V. Restrictions on the stress components in the edge points of the homogeneous elastic body // Engineering Solid Mechanics. 2019. V. 7. No. 3. P. 229-246. DOI: 10.5267/j.esm.2019.5.001.