Mathematical modeling of the process of nonlinear deformation of thin-walled structures

Abstract

Objective. The objective is to develop a unified method for solving a general nonlinear boundary value problem associated with discontinuous phenomena, which allows identifying all the characteristic features of the behavior of thin-walled systems under load. The issues of nonlinear deformation, loss of stability of the initial equilibrium shape and post-critical behavior are considered using the example of a thin spherical shell. Method. The problem is solved by numerical and analytical methods, representing a set of methods of catastrophe theory and the finite difference method of increased accuracy. The main attention is paid to the mathematical aspects of the phenomena under consideration. Result. The parameters of the stressstrain state of subcritical, critical and postcritical deformation are determined using a spherical shell as an example. The relationships between the limit and bifurcation values of the load parameters are obtained, allowing us to determine the group of the limit state of the achieved level of the stress-strain state of the structure. Conclusion. The solution of the general problem allows us to obtain complete and necessary information to determine the degree of danger of the states of structures and ensure their reliability.