Nonclassical Thermoelastic Problems in Nonlinear Dynamics of Shells

This monograph, addressing researchers as well as engineers, is devoted to nonclassical thermoelastic modelling of the nonlinear dynamics of shells. Differential equations of different dimensionality and different type have to be combined and nonlinearities of different geometrical, physical or elasto-plastic categories are addressed. Special emphasis is given to the Bubnov-Galerkin method. It can be applied to many problems in the theory of plates and shells, even those with very complex geometries, holes and various boundary conditions. The authors made every effort to keep the text intelligible for both practitioners and graduate students, although they offer a rigorous treatment of both purely mathematical and numerical approaches presented so that the reader can understand, analyse and track the nonlinear dynamics of spatial systems (shells) with thermomechanical behaviours.

This monograph describes some approaches to the nonlinear theory of plates and shells. By nonclassical approaches we mean the desciption of problems with mathematical models of different sizes (two-and three-dimensional dif ferential equations) and different types (differential equations of hyperbolic and parabolic type in the spatial coordinates). The nonlinearities investigated are also of various categories: geometrical, physical, elasto-plastic, and peri odic. Creating such types of mathematical models and their detailed justifica tion allows us to achieve the most accurate description of the real behaviour of shell-type structures. These models allow us to include interaction between the strain and temperature fields and coupling between the displacement field and the external influence of a transonic gas flow. The mathematical treatment of such models helps us greatly in obtaining reliable results by numerical computation. It appears that the most dangerous situation for thin shallow shells is the conjunction of a static load with dynamic interactions. Such combined loads very often cause buckling of shell structures, and in many cases a series of bucklings, which can cause fracture. The failure of a structure usually needs a small amount of time. Therefore the lifetime of a shell structure depends strongly on nonelastic deflections and it is important to mathematically model shell structures as precisely as possible. This monograph is one of several devoted to this subject. Now we shall briefly describe the contents of the book. Note that not all of the results presented here have been published in textbook format.

Covers a vast amount of material and it will help to promote the work done in the Eastern countries mostly not available in English so far Includes supplementary material: sn.pub/extras

Contenu
1 Introduction.- 2 Coupled Thermoelasticity and Transonic Gas Flow.- 2.1 Coupled Linear Thermoelasticity of Shallow Shells.- 2.2 Cylindrical Panel Within Transonic Gas Flow.- 3 Estimation of the Errors of the Bubnov-Galerkin Method.- 3.1 An Abstract Coupled Problem.- 3.2 Coupled Thermoelastic Problem Within the Kirchhoff-Love Model.- 3.3 Case of a Simply Supported Plate Within the Kirchhoff Model.- 3.4 Coupled Problem of Thermoelasticity Within a Timoshenko-Type Model.- 4 Numerical Investigations of the Errors of the Bubnov-Galerkin Method.- 4.1 Vibration of a Transversely Loaded Plate.- 4.2 Vibration of a Plate with an Imperfection in the Form of a Deflection.- 4.3 Vibration of a Plate with a Given Variable Deflection Change.- 5 Coupled Nonlinear Thermoelastic Problems.- 5.1 Fundamental Relations and Assumptions.- 5.2 Differential Equations.- 5.3 Boundary and Initial Conditions.- 5.4 On the Existence and Uniqueness of a Solution.- 6 Theory with Physical Nonlinearities and Coupling.- 6.1 Fundamental Assumptions and Relations.- 6.2 Variational Equations of Physically Nonlinear Coupled Problems.- 6.3 Equations in Terms of Displacements.- 7 Nonlinear Problems of Hybrid-Form Equations.- 7.1 Method of Solution for Nonlinear Coupled Problems.- 7.2 Relaxation Method.- 7.3 Numerical Investigations and Reliability of the Results Obtained.- 7.4 Vibration of Isolated Shell Subjected to Impulse.- 7.5 Dynamic Stability of Shells Under Thermal Shock.- 7.6 Influence of Coupling and Rotational Inertia on Stability.- 7.7 Numerical Tests.- 7.8 Influence of Damping e and Excitation Amplitude A.- 7.9 Spatial-Temporal Symmetric Chaos.- 7.10 Dissipative Nonsymmetric Oscillations.- 7.11 Solitary Waves.- 8 Dynamics of Thin Elasto-Plastic Shells.- 8.1 Fundamental Relations.- 8.2 Method of Solution.- 8.3 Oscillations and Stability of Elasto-Plastic Shells.- 9 Unsolved Problems in Nonlinear Dynamics of Shells.- References.