Unsolved Problems in Nonlinear Dynamics of Shells

Abstract

The problems in the field of nonlinear dynamics of shells that remain to be solved can be summarized as follows: 1. Formulating the important boundary problems for the nonlinear dynamics of shells without assuming slenderness and central bending (i.e. for arbitrary rotations). 2. 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. 3. Obtaining numerical results for coupled problems of thermoelasticity for a wider class of boundary conditions than that investigated in this book. 4. 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). 5. Formulating the important initial-boundary conditions for the coupled theory of thermoelasticity for polymer shells. 6. Investigation of the coupling of deflection and temperature fields for polymers in the framework of nonlinear geometry. 7. 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. 8. Detailed analysis of the regions of applicability of various nonlinear initial-boundary conditions in the theory of shallow shells rectangular in plan. 9. 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. 10. 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. 11. 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.). 12. 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). 13. Research on the initial-boundary conditions for nonlinear coupled thermoelastic problems of shells applying the quantitative theory of differential equations. 14. 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. 15. 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. 16. Problems of aeroelasticity of shells interacting with transonic and supersonic flows of an ideal gas, including coupling of the strain, temperature and gas fields. 17. Research on nonsymmetric buckling of shells subjected to longitudinal periodic loads.