Inverse Problems of the Inhomogeneous Theory of Elasticity for Thick-Walled Shells

Inverse Problems of the Inhomogeneous Theory of Elasticity for Thick-Walled Shells

V. Andreev

Department of Strength of Materials, Moscow State University of Civil Engineering, Russia

| |
| | Citation



The inhomogeneous theory of elasticity considers bodies, the mechanical characteristics of which (the modulus of elasticity and Poisson’s ratio) are functions of the coordinates. If indirect problems of the inhomogeneous theory of elasticity are identified, and the stress-strain state of the body has well-known functions of mechanical characteristics, the essence of inverse problems is to determine the functions of the inhomogeneity for a given stress state of the body. One of the first solutions to such an inverse problem was published in the work of Lekhnitskii (“Radial distribution of stresses in the wedge and half-plane with variable modulus of elasticity”. PMM, XXVI(1), pp. 146–151, 1962). In this article, we consider one-dimensional inverse problems for thick-walled cylindrical and spherical shells that are subjected to internal and external pressures in a non-varying temperature field. The aim of this work is to identify the dependence of the elastic modulus on the radial co-ordinate for which the equivalent stress according to a particular theory of strength will be constant at all points of the body (such structures are called equal stress), or the equivalent stress in all points will be equal to the strength of the material (such structures are called equal strength). For example, the author has proven that the limit loads on resulting equal-strength inhomogeneous shells can be significantly increased.


equal-strength structure, equal-stress structure, inhomogeneity, inverse problem, modulus of elasticity, theory of elasticity, thick-walled shell


[1] Andreev, V.I., Nekotoryje zadachi i metody mehaniki neodnorodnyh tel [Some problems  and methods of mechanics of nonhomogeneous bodies], Izdat, ASV: Moscow, 286 pp, 2002.

[2] Andreev, V.I. & Potekhin, I.A., Modelirovanie I sozdanie ravnoprochnogo tsilindra na osnove iteratsionnogo podkhoda [Modeling and creation equal-strength cylinder based on an iterative approach]. International Journal for Computational Civil and Structural Engineering, 4(1), pp. 79–84, 2008.

[3] Andreev, V.I. & Potekhin, I.A., Optimizatsija po prochnosti tolstostennykh obolochek [Optimization in strength of thick-walled shells], MGSU: Moscow, 86 pp, 2011.

[4] Paturojev, V.V., Polimerbetony [Polymer concretes], Strojizdat: Moscow, 286 pp, 1987.

[5] Genijev, G.A. & Kissuk, V.N., K voprosu obobshchenija teorii prochnosti betona [Generalization of the concrete strength theory]. Beton I zhelezobeton, 2, pp. 16–19, 1965.

[6] Karpenko N.I., Obshchie modeli mehaniki zhelezobetona [General models of mechanics of reinforced concrete], Strojizdat: Moscow, 416 pp, 1996.

[7] Kamke, E., Differentialgleichungen, akademie verlag: Leipzig, 576 pp, 1959.

[8] Andreev, V.I., Optimization of thick-walled shells based on solutions of inverse problems of the elastic theory for inhomogeneous bodies. Computer Aided Optimum Design in Engineering XII (OPTI XII). WIT Press, pp. 189–201, 2012.

[9] Andreev, V.I. & Minayeva, A.S., Postroenie na osnove pervoy teorii prochnosti modeli ravnonapriazhonnogo tsilindra podverzhennogo silovym i temperaturnym nagruzkam [Building on the fi rst theory of strength the model of equal stress cylinder subjected to power and thermal loads]. Privolzhskiy zhurnal, 4, pp. 34–39, 2011.

[10] Andreev, V.I. & Bulushev, S.V., The solution of the return problem for the equa l-stress thick-walled spherical shell subject to power and temperature loadings according to the fi rst and third theories of strength, Proceedings of XXI Russian-Slovak-Polish Seminar “Theoretical Foundation of Civil Engineering”, Warszawa, pp. 93–98, 2012.