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IssuesArchive of Issues2018-4pp.440-453

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S.A. Lurie and D.B. Volkov-Bogorodskii, "Green Tensor and Solution of the Boussinesq Problem in the Generalized Theory of Elasticity," Mech. Solids. 53 (4), 440-453 (2018)
Year 2018 Volume 53 Number 4 Pages 440-453
DOI 10.3103/S0025654418040106
Title Green Tensor and Solution of the Boussinesq Problem in the Generalized Theory of Elasticity
Author(s) S.A. Lurie (Ishlinsky Institute for Problems in Mechanics RAS, pr. Vernadskogo 101, str. 1, Moscow, 119526 Russia; Institute of Applied Mechanics of Russian Academy of Sciences, Leningradsky pr. 7, Moscow, 125040, Russia, salurie@mail.ru)
D.B. Volkov-Bogorodskii (Institute of Applied Mechanics of Russian Academy of Sciences, Leningradsky pr. 7, Moscow, 125040, Russia)
Abstract The fundamental spatial problems of the theory of elasticity such as the problem of constructing Green tensor and the Boussinesq problem of the action of a concentrated force on a half-space are considered. According to the classical theory of elasticity, these problems are singular. It is shown that an analytical solution of such problems can be constructed by the Papkovich-Neuber representation without invoking symmetry conditions. This makes it possible to present the solution of the problems under consideration in a single form and allows us to write an explicit solution of half-space loaded by a concentrated vector-force having non-zero projections onto the normal to the plane bounding the half-space and onto the plane itself.

This paper deals with the generalized regular solutions of the considered fundamental problems of the elasticity. The solutions are limited at a singular point and damp at infinity.
Keywords generalized theory of elasticity, Papkovich-Neuber representation, regular solution, Green tensor, Boussinesq problem
References
1.  V.V. Vasiliev, "Stress Tensor Symmetry and Singular Solutions in the Theory of Elasticity," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 2, 62-72 (2010) [Mech. Sol. (Engl. Transl.) 45 (2), 205-213 (2010)].
2.  V.V. Vasiliev and S.A. Lurie, "On the Solution Singularity in the Plane Elasticity Problem for a Cantilever Strip," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 4, 40-49 (2013) [Mech. Sol. (Engl. Transl.) 48 (4), 388-396 (2013)].
  
3.  V.V. Vasiliev and S.A. Lurie, "Continuous Model with Microstructure," Comp. Nanostruct. 7 (1), 2-10 (2015).
4.  V.V. Vasiliev and S.A. Lurie, "Generalized Theory of Elasticity," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 4, 16-27 (2015) [Mech. Sol. (Engl. Transl.) 50 (4), pp. 379-388 (2015)].
5.  V.V. Vasiliev and S.A. Lurie, "Generalized Solution of the Problem on a Circular Membrane Loaded by a Lumped Force," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 3, 115-119 (2061) [Mech. Sol. (Engl. Transl.) 51 (3), 334-338 (2011)].
6.  V.V. Vasiliev and S.A. Lurie, "New Solution of the Plane Problem for an Equilibrium Crack," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 5, 61-67 (2016) [Mech. Sol. (Engl. Transl.) 51 (5), 557-561 (2016)].
7.  V.V. Vasiliev and S.A. Lurie, "New Solution of Axisymmetric Contact Problem of Elasticity," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 5, 12-21 (2017) [Mech. Sol. (Engl. Transl.) 52 (5), 479-487 (2017)].
8.  V.V. Vasiliev and S.A. Lurie, "Non-local Solutions of Singular Problems of Mathematical Physics and Mechanics," Prikl. Mat. Mech. 82 (4), 459-471 (2018).
9.  M.Yu. Gutkin, "Nanoscopics of Dislocations and Disclinations in Gradient Elasticity," Rev. Adv. Mater. Sci. No. 1, 27-60 (2000).
10.  M. Lazar and G.A. Maugin, "A Note on Line Forces in Gradient Elasticity," Mech. Res. Commun. 33, 674-680 (2006).
11.  M. Lazar, "The Fundamentals of Non-singular Dislocations in the Theory of Gradient Elasticity: Dislocation Loops and Straight Dislocations," Int. J. Sol. Struct. 50, 352-362 (2013).
12.  P.F. Papkovich, Theory of Elasticity (Oborongiz, Moscow-Leningrad, 1939) [in Russian].
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Received 02 March 2018
Link to Fulltext https://link.springer.com/article/10.3103/S0025654418040106
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