Schalentheorien

Schalentheorien, also known as shell theories, are a class of mathematical models used to describe the behavior of thin, curved structures. These theories are fundamental in fields such as structural engineering, mechanical engineering, and aerospace engineering, where they are applied to analyze and design components like car bodies, aircraft fuselages, domes, and pressure vessels.

The core idea behind shell theories is to simplify the three-dimensional elasticity problem of a solid object into a two-dimensional problem defined on the mid-surface of the shell. This simplification is valid when the thickness of the shell is significantly smaller than its other dimensions. By making assumptions about the deformation and stress distribution through the thickness, shell theories reduce the computational complexity while still providing accurate predictions for many practical applications.

There are various types of shell theories, each with different levels of approximation and applicability. Common examples include Kirchhoff-Love theory, which assumes that lines initially normal to the mid-surface remain normal and unchanged in length after deformation, and Reissner-Mindlin theory, which allows for shear deformation through the thickness. More advanced theories, such as those based on three-dimensional elasticity or including higher-order deformation effects, exist for situations where simpler theories may not be sufficient.

The development and application of shell theories involve techniques from differential geometry and calculus of variations. Solutions are often obtained through analytical methods for simple geometries and boundary conditions, or through numerical methods like the finite element method for more complex problems. The accuracy of shell theories depends on the validity of their underlying assumptions for the specific problem being analyzed.