Singular problems in shell theory. Computing and asymptotics
Asymptotic methods are widely used in shell theory; see, e.g., [\textit{A. L. Gol'denveizer}, Theory of elastic thin shells. New York: Pergamon (1961)] and [\textit{S. M. Bauer, S. B. Filippov, A. L. Smirnov} and \textit{P. E. Tovstik}, Asymptotic methods in mechanics with applications to thin shells and plates. Asymptotic methods in mechanics. Providence, RI: American Mathematical Society. CRM Proc. Lect. Notes. 3, 3--140 (1993; Zbl 0793.34037)]. The results obtained are valid, as a rule, at the physical level of rigor. Their rigorous mathematical justification is an interesting challenge attracting mathematicians -- is enough to recall the classic papers by M. I. Vishik and L. A. Lyusternik initiated by Gol'denveizer's results. The book under review is devoted to a mathematically rigorous study of singularities in linear elastic shell theory which appear for very small thickness. The authors use the so-called (in Western literature) Koiter-Sanders linear theory (Novoshilov-Balabukh theory in the Russian literature). Also covered are issues of numerical study of very thin shells. The book includes the following chapters: 1. Geometrical formalism of shell theory. 2. Singularities and boundary layers in thin elastic shell theory. 3. Anisotropic error estimates in the layers. 4. Numerical simulations with anisotropic adaptive mesh. 5. Singularities of parabolic inhibited shells. 6. Singularities of hyperbolic inhibited shells. 7. Singularities of elliptic well-inhibited shells. 8. Generalities on boundary conditions for equations and systems: introduction to sensitive problems. 9. Numerical simulations for sensitive shells. 10. Examples of non-inhibited shell problems (non-geometrically rigid problems). This well-written book is a reader-friendly and well organized research work in the field of mathematical theory of shells. It can be recommended to highly-qualified experts in this field.
scientific article; zbMATH DE number 5763814