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    Elastic and Thermoelastic Problems in Nonlinear Dynamics of Structural Members: Applications of the Bubnov-Galerkin and Finite Difference Methods

    Jan Awrejcewicz,Vadim A. Krysko|2020.04.30

    Second/New Edition (in bold the new material):

    1 Introduction (to be updated).- 2 Coupled Thermoelasticity and Transonic Gas Flow.- 2.1 Coupled Linear Thermoelasticity of Shallow Shells.- 2.1.1 Fundamental Assumptions.- 2.1.2 Differential Equations.- 2.1.3 Boundary and Initial Conditions.- 2.1.4 An Abstract Coupled Problem.- 2.1.5 Existence and Uniqueness of Solutions of Thermoelasticity Problems.- 2.2 Cylindrical Panel Within Transonic Gas Flow.- 2.2.1 Statement and Solution of the Problem.- 2.2.2 Stable Vibrating Panel Within a Transonic Flow.- 2.2.3 Stability Loss of Panel Within Transonic 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 Mathematical Model of Cylindrical/Spherical Shell Vibrations.- 9.1. Fundamental Relations and Assumptions. - 9.2. The Bubnov-Galerkin Method.- 9.2.1. Closed Cylindrical Shell.- 9.2.2. Cylindrical Panel.-  9.3. Reliability of the Obtained Results.- 9.4. On the Set up Method in the Theory of Flexible Shallow Shells.- 9.5. Dynamic Stability Loss of the Shells Under the Step-Type Function.- 10 Chaotic Vibrations of Cylindrical and Spherical Shells.- 10.1. Novel Models of Scenarios of Transition from Periodic to Chaotic Orbits.- 10.2. Sharkovskiy’s Periodicity Exhibited by PDEs Governing Dynamics of Flexible Shells.- 10.3. On the Space-Temporal Chaos.- 11 Mathematical Models of Chaotic Vibrations of Closed Cylindrical Shells with Circular Cross Section.- 11.1. On the Convergence of the Bubnov-Galerkin (BG) Method in the Case of Chaotic Vibrations of Closed Cylindrical Shells.- 11.2. Chaotic Vibrations of Closed Cylindrical Shells Versus Their Geometric Parameters and the Area of the External Load Action.- 12 Chaotic Dynamics of Flexible Closed Cylindrical Nanoshells under Local Load.- 12.1. Statement of the Problem.- 12.2. Algorithm of the Bubnov-Galerkin Method.- 12.3. Numerical Experiment.- 13 Contact Interaction of Two Rectangular Plates Made From Different Materials Taking into Account Physical Nonlinearity.- 13.1. Statement of the Problem.- 13.2. Reduction of PDEs to ODEs.- 13.2.1. Method of Kantorovich-Vlasov (MKV).- 13.2.3. Method of Variational Iteration (MVI).- 13.2.4. Method of Arganovskiy-Baglay-Smirnov (MABS).- 13.2.5. Combined Method (MC).- 13.2.6. Matching of the Methods of Kantorovich-Vlasov and Arganovskiy-Baglay-Smirnov (MKV+MABS).- 13.2.7. Matching of the Methods of Vaindiner and the Arganovskiy-Baglay-Smirnov (MV+MABS).- 13.2.8. Matching of the Methods of Vaindiner and the Method of Variational Iterations (MV+MVI).- 13.2.9. Numerical Example.- 13.3. Mathematical Background.- 13.3.1. Theorems on Convergence of MVI.- 13.3.2. Convergence Theorem.- 13.4. Contact Interaction of Two Square Plates.- 13.4.1. Computational examples.- 13.5. Dynamics of a Contact Interaction.- 14 Chaotic Vibrations of Flexible Shallow Axially Symmetric Shells vs. Different Boundary Conditions.- 14.1. Problem Statement and the Method of Ssolution.- 14.2. Quantification of True Chaotic Vibrations.- 14.3. Modes of Vibrations (Simple Support).- 14.4. Modes of Vibrations (Rigid Clamping).- 14.5. Investigation on the Occurrence of Ribs (Simple Nonmovable Shell Support).- 14.6. Shell Vibration Modes (Movable Clamping).- 15 Chaotic Vibrations of Two Euler-Bernoulli Beams with a Small Clearance.- 15.1. Mathematical Model.- 15.2. Principal Component Analysis (PCA).- 15.3. Numerical Experiment.- 15.4. Application of the Principal Component Analysis.- 15.5. Concluding Remarks.- 16 Unsolved Problems in Nonlinear Dynamics of Shells.- References.- Index.

     

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