On some features of soft shells of revolution static problems solution at large deformations


DOI: 10.34759/trd-2020-114-04

Аuthors

Korovaytseva E. A.

Institute of Mechanics Lomonosov Moscow State University, 1, Michurinsky prospect, Moscow, 119192, Russia

e-mail: katrell@mail.ru

Abstract

The presented work demonstrates specifics of thin-walled structures mechanics at strong nonlinearity problems solution computational algorithms realization by an example of soft shell of revolution static inflation at the large deformations problem. The problem solution algorithm under consideration realizes the method of differentiation by parameter. The feature of the algorithm consists in using specific method of nonlinear Cauchy problem solving with right-hand side given on discrete mesh of variable step.

Studying the properties of the system of equations under consideration revealed the necessity for solution regularization preceding of the suggested algorithm realization. Thus, the initial stage of solution construction uses the well-posed system of equations of technical theory of soft shells, as its solution is close to the solution of the problem being considered, but without having a poor stipulation.

Analysis of various parameters of computational algorithm or properties of soft shells theory equations impact on calculation results cannot be performed in analytical form. Thus, for these issues studying numerical experiments concerning solution of hemisphere bloating problem with different conditions of the edge fixation are used. Physical properties of shell material are supposed to be described by Treloar relations. For the obtained results illustration only loading diagrams as pressure-deflection curves are used. The effect of the following parameters of the computational algorithm on the computational results was studied. These were the preliminary internal pressure magnitude, solution continuation parameter selection; the magnitude of initial and maximum allowable solution continuation parameter step; the number of parameter steps on which solution regularization was used; the ways of stress state calculation at developing the resolving system of equations.

It is worth mentioning that in the presented work, apparently for the first time, the author managed to obtain the value of deflection of the soft hemispherical shell, being inflated, w0=35R (where R is the radius of the non-deformed shell) only in 60 solution continuation parameter steps. The maximum value of deflection herewith obtained prior to the loss in calculation stability reaches 108R. In this case the solution error is equal to 5%, and the number of solution continuation parameter steps is 280.

A significant, difficult-to-predict dependence of computational results on the algorithm parameters mentioned above was revealed, up to obtaining wrong results. Thus, setting a miniscule value of the preliminary pressure may lead to the convergence problem of the solution correction right at the first step. An excessively large value of the step by the parameter leads to significant distortion of the results, while a too small one leads to the computation stability loss. Conclusions on the effect of basic stress state computing method and a number of steps by the parameter, at which the system of equations regularization is performed, may appear different for various boundary conditions at sphere equator and various methods of the solution continuation parameter selection.

Thus, judging by the performed studies, when solving problems of the stress-strain state, being followed by large displacements and deformations analysis, applicability checking of any `previously developed recommendations concerning calculations in each concrete case is necessary. Realization of the selected solution algorithm or parameters assignment should not be restricted by only one method. Studying the problem solution behavior in a wide range of possible changes of the algorithm parameters is necessary; and in the absence of analytical solution, the repetitiveness of the results at various values of the assigned parameters can be the indicator of the obtained results certainty. Probably, such numerical experiments may and should be performed by the technique similar to the methods of experimental mechanics, and properties determining of this or that algorithm in application to the solution of this or that type of problems is similar to determining mechanical properties of materials at various types of the stress-strain state.

Keywords:

soft shell, high-elastic material, large deformations, nonlinear boundary value problem, method of differentiation by parameter

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