Numerical study of the stability of a shell of minimal surface on a circular plan with regard to geometric nonlinearity under thermal and power loading

Abstract

Great progress has been made in the field of thin elastic shells calculation, both in the field of mathematical theory, which, based on the Kirchoff-Love hypothesis in surface theory, is engaged in the construction of various forms of important design equations and the development of accurate methods for their solution, and in the field of structural and applied mechanics, which is based on the specified initial parameters and accepted additional working hypotheses, which are determined by experimental data, and which deals with simplification of calculation schemes and methods of their solution, which will be convenient for engineering calculations.

The geometric nonlinearity of the equations is achieved by taking into account the quadratic term in the expressions of the component of the membrane deformation tensor and changes in the deformed shape of the middle surface of the shell. The solution of the system of nonlinear equations is constructed by the iterative method of continuation of the solution in the parameter, taking into account the Newton-Kantorovich method.

When operating thin shells, they can be subjected to harsh conditions - under the influence of various temperature and force loads. Large, inhomogeneous temperature fields affect the mechanical properties of the material and can cause large deformations, and can be the most determining factor affecting the strength and load-bearing capacity of the shell as a whole.

Geometrically nonlinear problems are mainly used to formulate structural stability problems. In most cases, the stability problem can be solved by reducing it to a linear formulation under natural oscillations.

Geometrically nonlinear problems are problems of elasticity theory in which nonlinearity in the dependence of strains and displacements is taken into account, while stresses and strains are related linearly. Taking into account nonlinear components of deformations is necessary for the calculation of flexible thin-walled structures.

In the numerical experiment, the eigenvalues of the stability factor are equal to 1.013, which means that there is no margin of safety and stability in the shell, and we can further use these results for multicriteria parametric optimization, and the results of the study are confirmed by the authors' methodology for objects where optimization of the shell geometry is taken into account.