Mixed equations of the soft shells theory

DOI: 10.34759/trd-2019-108-1

Аuthors

Korovaytseva E. A.

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

e-mail: katrell@mail.ru

Abstract

The article proposes a new variant of resolve relationships of the soft shells theory. This variant differs from the existing ones by the numerical realization convenience. Both equations of large deformations theory and technical theory of soft shells are considered. The principle of virtual displacements is applied to derive equations of the shell equilibrium. Expressions for components finite deformations are used in the V.V. Novozhilov’s form. Functions of generalized forces were introduced to the final notation of variation equation. It allows setting down the equations of the soft shells theory in the form used while boundary problems formalization. As the result, the system of equations describing the soft shells behavior at large deformations includes the equations of equilibrium, additional projection relationships, dependencies, describing the true forces reduction to the initial state metric, and geometry relationships. For the system of equations closure, physical relationships, linking real forces with deformations, are used. Thus, the finally composed system consists of 23 equations with 23 unknowns.

While considering the technical theory of soft shells, deformations are assumed small, and forces, acting in the shell being deformed, are represented as a sum of two terms, corresponding to the main and additional stress states. The initial state geometry is considered as known. Forces and displacements corresponding to the additional state are being determined from the system of equations, obtained as the result of variation equation components linearization relative to the main state. As in the case of large deformations, functions of generalized forces are introduced for the additional state. As the result, a system of three equations of equilibrium is formed for the main state. The system of equations for additional state consists of 18 equations, and includes equilibrium equations, additional projection equations, geometrical equations and physical relationships. The system is supplemented by the boundary conditions.

The obtained relations are not pretending on fundamental changes or add-ons, but unlike the known relations they do not require reducing to a single equation relative to one variable. They can be reduced to Cauchy normal form, convenient for application of numerical methods for direct integration, and standard methods of ill-conditioned boundary problems solution.

Keywords:

soft shells, large deformation, technical theory of soft shells, normal form of differential equations

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