FAR EASTERN BRANCH OF THE RUSSIAN ACADEMY OF SCIENCES
INSTITUTE OF APPLIED MATHEMATICS KHABAROVSK DIVISION
Far Eastern Mathematical Journal
I. Singularity removal in the elasticity theory solutions based on a non-euclidean model of a continuous medium: the case of zero and first harmonics |
Guzev M.A., Chernysh E.V. |
2025, issue 1, P. 21-38 DOI: https://doi.org/10.47910/FEMJ202502 |
Abstract |
| A representation for singularities in the zero and first harmonics of the classical elastic stress field was obtained using the Airy stress function for a plane-strained state of a continuous medium. For a non-Euclidean model of a continuous medium, the structure of the internal stress field of a plane-strained state was shown to consist of a classical elastic stress field and a non-classical stress field determined through the incompatibility function of deformations. The singular contribution of the zero and first harmonics of the non-classical stress field is calculated. The requirement of regular behavior of the internal stress field allowed to compensate for the singularity in the elasticity theory solution by choosing a singularity of the non-classical stress field. |
Keywords: Airy stress function, non-Euclidean model of a continuous medium, deformation incompatibility. |
Download the article (PDF-file) |
References |
| [1] Cherepanov G.P., Mekhanika khrupkogo razrusheniya, Nauka, M., 1974. [2] Barber J.R., “Wedge Problems”, In Elasticity. Part of the book series: Solid Mechanics and Its Applications, v. 172, Springer, Dordrecht, 2010, 149–170. [3] Cai W., Arsenlis A., Weinberger C., Bulatov V., “A non-singular continuum theory of dislocations”, Journal of the Mechanics and Physics of Solids, v. 54(3), 2006, 561–587. [4] Lazar M., “Non-singular dislocation loops in gradient elasticity”, Physics Letters A, v. 376(21), 2012, 1757–1758. [5] Lazar M., “The fundamentals of non-singular dislocations in the theory of gradient elasticity: Dislocation loops and straight dislocations”, International Journal of Solids and Structure, v. 50(2), 2013, 352–262. [6] Lazar M., Po G., “The non-singular Green tensor of Mindlin’s anisotropic gradient elasticity with separable weak non-locality”, Physics Letters A, v. 379(24-25), 2018, 1538–1543. [7] Po G., Lazar M., Admal N. C., Ghoniem N., “A non-singular theory of dislocations in anisotropic crystals”, International Journal of Plasticity, v. 379(24-25), 2018, 1–22. [8] Kioseoglou J., Konstantopoulos I., Ribarik et G. al., “Nonsingular dislocation and crack fields: implications to small volumes”, Microsystem Technologies, v. 15, 2009, 117–121. [9] Aifantis E.C., “A note on gradient elasticity and nonsingular crack fields”, J. Mech. Behav. Mater., v. 20, 2011, 103–105. [10] Konstantopoulos I., Aifantis E.C., “Gradient elasticity applied to a crack”, J. Mech. Behav. Mater., v. 22, 2013, 193–201. [11] Parisis K., Konstantopoulos I., Aifantis E. C., “Nonsingular Solutions of GradEla Models for Dislocations: An Extension to Fractional GradEla”, J. Mech. Behav. Mater., v. 3(03n04), 2018, A. 1840013. [12] Vasil'ev V.V., “Singulyarnye resheniya v zadachakh mekhaniki i matematicheskoi fiziki”, Izvestiya Rossiiskoi akademii nauk. Mekhanika tverdogo tela, t. 4, 2018, 48–65. [13] Vasil'ev V.V., Lur'e S.A., “O singulyarnosti resheniya v ploskoi zadache teorii uprugosti dlya konsol'noi polosy”, Izvestiya Rossiiskoi akademii nauk. Mekhanika tverdogo tela., t. 4, 2013, 40–49. [14] Vasil'ev V.V., Lur'e S.A., “Nelokal'nye resheniya singulyarnykh zadach matematicheskoi fiziki i mekhaniki”, Prikladnaya matematika i mekhanika, t. 82, 2018, 459–471. [15] Vasil'ev V.V., Lur'e S.A., “Differentsial'nye uravneniya i problema singulyarnosti reshenii v prikladnoi mekhanike i matematike”, Prikladnaya mekhanika i tekhnicheskaya fizika, t. 64, 2023, 114–127. [16] Kondo K., “On the geometrical and physical foundations of the theory of yielding”, Proc. Japan. Nat. Congr. Apll. Mech., v. 2, 1952, 41–47. [17] Bilby B.A., Bullough R., Smith E., “Continuous distributions of dislocations: a new application of the methods of non-Reimannian geometry”, Proceedings of the Royal Society A, v. 231, 1955, 263–273. [18] Stojanovic R., “On the stress functions in elastodynamics”, Acta Mechanica, v. 24(3), 1976, 209–217. [19] Kroner E., “Incompatibility, defects, and stress functions in the mechanics of generalized continua”, International Journal of Solids and Structures, v. 21(7), 1985, 747–756. [20] Edelen D.G.B., “A correct, globally defined solution of the screw dislocation problem in the gauge theory of defects”, International Journal of Engineering Science, v. 34(1), 1996, 81–86. [21] Guzev M.A., Non-Euclidean models of elastoplastic materials with structure defects, Lambert Academic Publishing, 2010. [22] Guzev M.A., Liu W., Qi C., “Non-Euclidean model for description of residual stresses in planar deformations”, Applied Mathematical Modelling, v. 90, 2021, 615–623. [23] Guzev M.A., Riabokon E.P., “Construction of Nonsingular Stress Fields for Non-Euclidean Model in Planar Deformations”, Journal of Siberian Federal University. Mathematics & Physic, v. 14(6), 2021, 815–821. [24] Godunov S.K., Elementy mekhaniki sploshnoi sredy, Nauka, M., 1978. [25] Guzev M.A., “Struktura kinematicheskogo i silovogo polya v Rimanovoi modeli sploshnoi sredy”, PMTF, t. 52(5), 2011, 39–48. |