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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.



1. Introduction

In this chapter, we give a brief discussion of the literature on the nonlinear theory of plates and shells, paying attention particularly to Eastern references, where many interesting results have been obtained and which are (unfortunately) still not well distributed among the worldwide scientific community.
Jan Awrejcewicz, Vadim A. Krys’ko

2. Coupled Thermoelasticity and Transonic Gas Flow

This chapter includes considerations of the coupled linear thermoelasticity of shallow shells and of a cylindrical panel within a transonic gas flow. First, the fundamental assumptions related to the stress-strain relation of the Timoshenko kinematic model are formulated, and then the differential equations are derived. Both the Timoshenko and the Kirchhoff-Love models are taken into account. Then the boundary and initial conditions are formulated. Next, an abstract Cauchy problem for a coupled system of two differential equations in a Hilbert space is considered. This includes the thermoelastic problems of shallow shells modelled by the Kirchhoff-Love and Timoshenko theories defined earlier. In Sect. 2.1.5, theorems related to the existence and uniqueness of a general, “classical” solution to the coupled abstract program are given, and then the corresponding theorems for coupled thermoelastic problems of shallow shells are formulated.
Jan Awrejcewicz, Vadim A. Krys’ko

3. Estimation of the Errors of the Bubnov-Galerkin Method

In Sect. 3.1, an abstract coupled problem is considered and a few theorems related to the estimation of the accuracy of the Bubnov-Galerkin method are formulated and proved. The error estimates hold for a system of differential equations of a rather general form with homogeneous boundary conditions, which corresponds to coupled thermoelastic problems for plates and shallow shells with variable thickness. In addition, a particular case of this problem (with nonhomogeneous initial conditions), where a prior estimate of the errors of the Bubnov-Galerkin method is most effective, is illustrated and discussed. Finally, a prior estimate for the Bubnov-Galerkin method to a problem generalizing a class of dynamical problems of elasticity (without a heat transfer equation) for both three-dimensional and thin-walled elements of structures is given.
Jan Awrejcewicz, Vadim A. Krys’ko

4. Numerical Investigations of the Errors of the Bubnov-Galerkin Method

In Chap. 3, many prior error estimates for the Bubnov-Galerkin method applied to coupled thermoelastic problems of shallow shells and plates were derived. The estimates related to the general case of a shell with a variable thickness subjected to time-varying mechanical and thermal loads possess an important theoretical meaning (they guarantee, for a wide class of problems, strong convergence of successive approximations to the exact solution with a velocity larger than the estimated velocity). However, in applications, when specific real problems have to be considered the estimates possess a more generalized meaning. This question is addressed, for instance, in the case of the estimates obtained in Sect. 3.3 for vibrations of a simply supported plate with constant thickness, subjected to mechanical and thermal loads that are constant in time. In this chapter, numerical results are given and the efficiency of the estimate used is verified.
Jan Awrejcewicz, Vadim A. Krys’ko

5. Coupled Nonlinear Thermoelastic Problems

In this chapter, we formulate fundamental assumptions and relations similar to those presented in Chap. 2 for coupled linear thermoelasticity problems of shallow shells. A Timoshenko-type model including the inertial effect of rotation of shell elements is used. Both the generalized heat transfer equation and the equations governing vibration of a shell are formulated in Sect. 5.2, and then some special cases of these equations are analysed. In the next section, boundary and initial conditions are attached to the differential equations. In Sect. 5.4, the existence and uniqueness of a solution as well as the convergence of the Bubnov-Galerkin method, are rigorously discussed.
Jan Awrejcewicz, Vadim A. Krys’ko

6. Theory with Physical Nonlinearities and Coupling

In this chapter, a theory of shells with physical nonlinearities and coupling is outlined. In Sect. 6.1, the fundamental assumptions and relations are introduced.
Jan Awrejcewicz, Vadim A. Krys’ko

7. Nonlinear Problems of Hybrid-Form Equations

This chapter is devoted to the analysis of some nonlinear problems governed by the hybrid form of the differential equations obtained earlier.
Jan Awrejcewicz, Vadim A. Krys’ko

8. Dynamics of Thin Elasto-Plastic Shells

In Chap. 6, a mathematical model governing the oscillations of a flexible shell with physical nonlinearity and coupling between the thermal and deformation fields has been presented. However, the loading and unoading processes overlap in the (σ i (e i ) diagram, and the remaining elasto-plastic deformation has not been taken into account. In a theoretical treatment of complicated shell oscillations, in order to analyse the stress-strain state properly, the elastoplastic deformation should be considered, as well as the fatigue behaviour of the material. Only in this case can the mathematical model be close to the real behaviour of a structure. The aim of this chapter is to describe the development of complex mathematical models of structures, including elastoplastic deformation and cyclic loading.
Jan Awrejcewicz, Vadim A. Krys’ko

9. Unsolved Problems in Nonlinear Dynamics of Shells

The problems in the field of nonlinear dynamics of shells that remain to be solved can be summarized as follows:
Formulating the important boundary problems for the nonlinear dynamics of shells without assuming slenderness and central bending (i.e. for arbitrary rotations).
Formulating the important boundary problems for the coupled theory of thermoelasticity of shells without assuming slenderness and central bending, which means formulating the boundary problem for an initial set of hyperbolic and parabolic equations or of hyperbolic equations.
Obtaining numerical results for coupled problems of thermoelasticity for a wider class of boundary conditions than that investigated in this book.
Formulating a mathematical theory of the initial-boundary conditions within the Timoshenko theory of shells, taking into account different types of nonlinearity (geometric and physical nonlinearities, nonlinear elastic-plastic properties, and structural nonlinearities).
Formulating the important initial-boundary conditions for the coupled theory of thermoelasticity for polymer shells.
Investigation of the coupling of deflection and temperature fields for polymers in the framework of nonlinear geometry.
Research on obtaining an energy solution for nonlinear initial-boundary conditions in the vicinity of corners and of points or curves where the type of boundary conditions changes.
Detailed analysis of the regions of applicability of various nonlinear initial-boundary conditions in the theory of shallow shells rectangular in plan.
Investigation of the effect of local thermal shock in a three-dimensional theory, without the hypothesis of a linear distribution of the thermal field through the shell thickness.
Detailed investigation of the initial-boundary conditions in the theory of multilayer shells with coupled temperature and strain fields, in geometrically linear and nonlinear cases.
Investigation of the influence of local nonideal thermal contact on the stress-strain state in the theory of multilayer plates and shells for different kinematic models (Kirchhoff-Love, Timoshenko, etc.).
Derivation of the important initial-boundary problems in the nonlinear theory of shallow shells, including coupling of strain and temperature fields, from initial-boundary conditions of the three-dimensional coupled theory of thermoelasticity (for hyperbolic-parabolic and hyperbolic sets of equations).
Research on the initial-boundary conditions for nonlinear coupled thermoelastic problems of shells applying the quantitative theory of differential equations.
Finding a class of nonlinear initial-boundary problems of mathematical physics for which prior estimates of the solution can be obtained using the methods presented in this book.
Investigation of spatial-temporal chaos in the theory of plates and shells including coupling of strain and temperature fields with geometric, physical and structural nonlinearities.
Problems of aeroelasticity of shells interacting with transonic and supersonic flows of an ideal gas, including coupling of the strain, temperature and gas fields.
Research on nonsymmetric buckling of shells subjected to longitudinal periodic loads.
Jan Awrejcewicz, Vadim A. Krys’ko


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