Novosibirsk, Russia, May, 30 – June, 4, 2011

"Modern Problems of Applied Mathematics and Mechanics: Theory, Experiment and Applications", devoted to the 90th anniversary of professor Nikolai N. Yanenko

## Dilman V.L. Eroshkina T.V.## Mathematical modelling of а critical state of less strong layers of a thin- walled cylindrical shell.## Reporter: Dilman V.L.Mathematical modelling of а critical state of less strong layers of a thin- walled cylindrical shell. The stress-strained state of thin-walled cylindrical shells from the reinforced material, subject to a monotonous static loading by an intrinsic pressure, and also an axial thrust, during their plastic straining is observed. Shells can be homogeneous or contain layers from less strong material and had lengthways, across or under an angle to forming. An instance of such shells are the pipes of the big diameter containing factory longitudinal or spiral and assembly transverse welded joints. It is supposed, that inside of a layer, in the basic material of the joint or on boundary line between them places defect. Under the critical state means the moment of loss of stability of process of a plastic straining. Construction of mathematical model of this state is based on bringing of diverse situations (various factors of mechanical heterogeneity, different angles less strong layer to an axis of a shell, various thickness of a layer and a shell, the location of defects and their geometrical characteristics) to problems of flat deformation for mechanically non-uniform joints. The latter are usually not predetermined regional problems for systems of the equations of hyperbolic type. Various approaches used by authors to predetermine them lead to various mathematical models of the critical state of non-uniform joints. Models are investigated analytical and, to some degree, by numerical methods. Comparison of models allows to size up extent of their adequacy at a qualitative level. On this basis are found in an analytic form critical axial and hoop stresses in a wall of a pipe, dependence of critical pressure on parameters of a shell and conditions of loading, limiting thickness of a wall of a shell depending on the same parameters and working pressure are specified. Because of the complexity (in some cases) of analytical expressions programs that represent the desired value as depending on known parameters are written in package MATLAB
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