|
ИСТИНА |
Войти в систему Регистрация |
ФНКЦ РР |
||
Application of the Wiener-Hopf approach to incorrectly posed BVP of plane elasticityтезисы доклада Тезисы
- Автор: Galybin A.N.
- Сборник: http://www.newton.ac.uk/event/whtw01/timetable
- Тезисы
- Год издания: 2019
- Аннотация: Usually the Wiener-Hopf method is associated with integrals transforms applied to a system of differential equations or to integral equations of the relevant convolution type, which leads to the Riemann BVP for semi-infinite domains. The latter is reduced to factorisation problem for its coefficient on an infinite contour parallel to the real (or imaginary) axis in the complex plane. For standard formulations the index of the Riemann BVP (determined by the increment of the argument of the coefficient on the contour) is usually zero, which provides the uniqueness of the solution of the Riemann BVP and hence an integral (or differential) equation of the considered problem possesses unique solution as well. However for some incorrectly posed BVP of plane elasticity it has been shown [1-2] that the index can be any finite number. No solution exists if the index is negative or, if it is a positive integer, the total solution is a linear combination of independent solutions defined by the index and it includes a finite number of arbitrary constants. In this study we consider a non-classical formulation of plane elasticity for half-planes in which no magnitudes of stresses and/or displacements are given on the boundary but only their orientations are known. In particular we present three formulations that involves two boundary conditions as follows 1. Principal directions of the stress tensor and their normal derivative; 2. Orientations of the displacement vector and their normal derivative; 3. Principal directions of stresses and orientations of displacements. In all these cases we do not use integral transforms but reduce these problems to a system of singular integral equations, SIE, based on the properties of the Kolosov-Muskhelishvili complex potentials [3]. These SIE can be presented for any contours but in the case of halt-plane they can be solved analytically by factorisation typical for the Wiener-Hopf techniques but complemented by the investigation of the solvability of the index of the Riemann problem. This is necessary to derive the general solution following the well-known approach due to Gakhov [4]. It is worth noting that the index, in general, is independent of the contour and thus the number of the solutions found for the case of half-plane remains for other contours as well. In contrast to Gakhov [4] we do not reduce the systems of two real SIE to one complex SIE of the general type although it can readily be done for Problems 1 and 2. For Problem 3 no direct reduction is possible as this would lead to the presence of the derivatives of the unknown functions in the complex dominant SIE, which makes it impossible direct application of Gakhov’s approach. In all cases the total solution of the system is defined by two partial indexes that are indexes of the corresponding Riemann problems. The solutions exist if the cumulative index is non-negative; otherwise no bounded solutions can be found in terms of complex potentials. REFERENCES 1.Galybin, A.N., Mukhamediev, Sh.A. 1999. Plane elastic boundary value problem posed on orientation of principal stresses. Journal of the Mechanics and Physics of Solids, 47, 2381-2409. 2. Galybin, A.N. 2011. Boundary value problems of plane elasticity involving orientations of displacements and tractions. Journal of Elasticity, 102, 15-30. 3. Muskhelishvili, N.I. 1963. Some basic problems of the mathematical theory of elasticity. P. Noordhoff, Groningen, the Netherlands. 4. Gakhov, F.D. 1977. Boundary value problems. Moscow, Nauka.
- Добавил в систему: Галыбин Александр Николаевич