The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. PrefaceThis work is devoted to an updated exploratory survey of results concerning elastic cusped shells, plates, and beams and cusped prismatic shell-fluid interaction problems. It also contains some up to now non-published results and new problems to be investigated. Mathematically, the corresponding problems lead to nonclassical, in general, boundary value and initial-boundary value problems for governing degenerate elliptic and hyperbolic systems in static and dynamical cases, respectively. Two principally different approaches of investigation are used:(1) to get results for 2D (two-dimensional) and 1D (one-dimensional) problems from results of the corresponding 3D (three-dimensional) problems and (2) to investigate directly governing degenerate and singular systems of 2D and 1D problems. In both the cases, it is important to study the relationship of 2D and 1D problems with 3D problems. On the one hand, it turned out that the second approach allows to investigate such 2D and 1D problems whose corresponding 3D problems are not possible to study within the framework of the 3D model of the theory of elasticity. On the other hand, the second approach is historically approved, since first the 1D and 2D models were created and only then the 3D model was constructed. Hence, the second approach gives a good chance for the further development (generalization) of the 3D model.The present work is addressed to engineers interested in the mathematical aspects of practical problems and mathematicians interested in engineering applications. Both can find new challenging problems expecting their resolution. It will also be very useful for students of advanced courses specializing in mechanics of continua, structural mechanics, mathematical modelling, partial differential equations and applications.

We construct variational hierarchical two-dimensional models for elastic, prismatic shells of variable thickness vanishing at boundary. With the help of variational methods, existence and uniqueness theorems for the corresponding two-dimensional boundary value problems are proved in appropriate weighted functional spaces. By means of the solutions of these two-dimensional boundary value problems, a sequence of approximate solutions in the corresponding three-dimensional region is constructed. We establish that this sequence converges in the Sobolev space H 1 to the solution of the original three-dimensional boundary value problem. (2000): 74K20, 74K25. Mathematics Subject Classifications

The present paper is devoted to construction of differential hierarchical models for elastic prismatic shells of variable thickness with microtemperatures. To this end, Vekua's dimension reduction method, based on the Fourier-Legendre expansions, is applied to basic equations of the linear theory of thermoelasticity of homogeneous isotropic bodies with microtemperatures. The special emphasis is placed on cusped prismatic shells.

The present paper is devoted to the two-dimensional version of statics of the linear theory of elastic materials with inner structure whose particles, in addition to the classical displacement and temperature fields, possess microtemperatures. Some results of the classical theories of elasticity and thermoelasticity are generalized. The Green's formulas in the case under consideration are obtained, basic boundary value problems are formulated, and uniqueness theorems are proved. The fundamental matrix of solutions for the governing system of the model and the corresponding single and double layer thermoelastopotentials are constructed. Properties of the potentials are studied. Applying the potential method, for the first and second boundary value problems, we construct singular integral equations of the second kind and prove the existence theorems of solutions for the bounded and unbounded domains. This paper describes the use of the L A T E X 2 " mmaauth.cls class file for setting papers for Mathematical Methods in the Applied Sciences.

In the present paper on the basis of the linear theory of thermoelasticity of homogeneous isotropic bodies with microtemperatures the zeroth order approximation of hierarchical models of elastic prismatic shells with microtemperatures in the case of constant thickness (but, in general, with bent face surfaces) is considered. The existence and uniqueness of solutions of basic boundary value problems when the projections of the bodies under consideration are bounded and unbounded domains with closed contours are established. The ways of solving boundary value problems in explicit forms and of their numerical solution are indicated.