We have constructed the fundamental system of solutions of the axially symmetric problem of the theory of elasticity for an unbounded body with a sheet of volume forces, normal to a chosen plane, and moment dipoles, which is a mathematical model of the internal boundary layer of a certain type. With the help of such layers, one succeeds in formulating some inverse problems of elasticity and, hence, the related problems of the control of stress-strain state on the corresponding surfaces. We have also formulated and solved the generalized Kelvin problem and, according to it, for the plane of distributed normal load, and established the law of distribution of the moment dipoles (sheet parameters), which provides the vertical displacements assigned for points of the plane, in particular, zero, by the corresponding tension.
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References
V. V. Bozhydarnyk and H. T. Sulym, Elements of the Theory of Elasticity [in Ukrainian], Svit, Lviv (1994).
V. S. Vladimirov, Generalized Functions in Mathematical Physics [in Russian], Nauka, Moscow (1979).
V. A. Halazyuk and H. T. Sulym, “On one class of the fundamental solutions of static problem of the theory of elasticity in cylindrical coordinates,” Mat. Metody Fiz.-Mekh. Polya, 43, No. 1, 84–93 (2000).
I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products [in Russian], Nauka, Moscow (1971).
A. G. Naumovets, “The use of surface phase transitions for the control of the properties of surfaces,” in: Promising Materials and Technologies, Vol. 2, Akademperiodika, Kyiv (2003), pp. 319–351.
M. O. Rvachov, “On the discontinuity of displacements in a continuous elastic space,” Dopov. Akad. Nauk Ukr. SSR, Ser. A, No. 10, 39–41 (1983).
I. Sneddon, Fourier Transforms, McGraw-Hill, New York (1951).
H. Sulym, Foundations of the Mathematical Theory of Thermoelastic Equilibrium of Deformable Solids with Thin Inclusions [in Ukrainian], NTSh Res.-Publ. Center, Lviv (2007).
C. Truesdell, A First Course in Rational Continuum Mechanics, Academic Press, New York (1971).
E. Kossecka, “Surface distributions of double forces,” Arch. Mech. Stosow., 23, No. 3, 313–328 (1971).
W. Thomson, “Note on the integration of the equations of equilibrium of an elastic solid,” Cambr. Dublin Math. J., 97–99 (1848); see also: Sir William Thomson, Mathematical and Physical Papers, Vol. 1, Cambridge Univ. Press, Cambridge (1882).
W. Thomson, “On a mechanical representation of electric, magnetic, and galvanic forces,” Cambr. Dublin Math. J., II, 76–80 (1847); see also: Sir William Thomson, Mathematical and Physical Papers, Vol. 1, Cambridge Univ. Press, Cambridge (1882).
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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 53, No. 4, pp. 132–142, October–December, 2010.
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Halazyuk, V.A., Sulym, H.T. Fundamental system of solutions of the axially symmetric problem of the theory of elasticity for a body with a plane sheet of volume moment dipoles and forces. J Math Sci 181, 457–469 (2012). https://doi.org/10.1007/s10958-012-0698-2
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DOI: https://doi.org/10.1007/s10958-012-0698-2