This monograph deals with the construction of a mathematical theory of elastic plates and shells, and with the corresponding two-dimensional boundary value problems. The book comprises materials presented in the Seminar of VIAM from 1968 till 1997, on a number of Symposia of ISIMM, and within a special course of lectures for students at Tbilisi State University.
After a short first introductory section, methods of reduction based on various simplifying hypotheses are presented in sections 2-5. This includes, in particular, the construction of refined theories for anisotropic nonhomogeneous plates with variable thickness without physical and geometrical restrictions. Further, in section 6, the author introduces Vekua’s method of reduction. On this basis, in section 7 a new theory of plates is constructed in which the differential operator is factorized in such a way that an approximate solution can be obtained by a parallel procedure. Section 8 is devoted to applications, section 9 presents refined theories for piezoelectric and electrically conductive elastic plates, section 10 includes the case of nonlinear elasticity, and section 11 gives some extensions to shells.
The remaining sections are concerned with the approximate solutions of two-dimensional boundary value problems that arose in the sections before. In particular, section 12 considers two-dimensional boundary value problems for strongly-elliptic differential systems, and sections 13-16 discuss one-dimensional boundary value problems for second-order nonlinear ordinary differential equations with Newtonian boundary conditions. The final section 17 concludes with some applications and numerical results.
The book is written on a high mathematical level, and in this regard it is interesting for researchers in the refined plate and shell theories.